CATEGORICAL ACTIONS ON SUPERCATEGORY O
by
NICHOLAS J. DAVIDSON
A DISSERTATION
Presented to the Department of Mathematics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
September 2016
DISSERTATION APPROVAL PAGE
Student: Nicholas J. Davidson
Title: Categorical Actions on Supercategory O
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Mathematics
by:
Jonathan Brundan Chair
Alexander Kleshchev Core Member
Victor Ostrik Core Member
Marcin Bownik Core Member
Dejing Dou Institutional Representative
and
Scott Pratt Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded September 2016
ii
c© 2016 Nicholas J. Davidson
iii
DISSERTATION ABSTRACT
Nicholas J. Davidson
Doctor of Philosophy
Department of Mathematics
September 2016
Title: Categorical Actions on Supercategory O
This dissertation uses techniques from the theory of categorical actions of Kac-
Moody algebras to study the analog of the BGG category O for the queer Lie
superalgebra. Chen recently reduced many questions about this category to its so-
called types A, B, and C blocks. The type A blocks were completely described in joint
work with Brundan in terms of the general linear Lie superalgebra. This dissertation
proves that the type C blocks admit the structure of a tensor product categorification
of the n-fold tensor power of the natural sp∞(C)-module. Using this result, we relate
the combinatorics for these blocks to Webster’s orthodox bases for the quantum group
of type C∞, verifying the truth of a recent conjecture of Cheng-Kwon-Wang. This
dissertation contains coauthored material.
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CURRICULUM VITAE
NAME OF AUTHOR: Nicholas J. Davidson
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
Boise State University, Boise, ID
Northwest Nazarene University, Nampa, ID
DEGREES AWARDED:
Doctor of Philosophy, Mathematics, 2016, University of Oregon
Master of Science, Mathematics, 2011, Boise State University
Bachelor of Arts, Mathematics, 2009, Northwest Nazarene University
AREAS OF SPECIAL INTEREST:
Representation theory
Categorical actions of Kac-Moody algebras
Superalgebra
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, Department of Mathematics, University of Oregon,
Eugene OR, 2010–2016
Graduate Teaching Assistant, Boise State University, Boise, ID, 2009–2011
PUBLICATIONS:
“Type C blocks in super category O,” in preparation (Joint with J. Brundan).
“Type A blocks in super category O,” submitted,
arXiv:1606.05775 (Joint with J. Brundan).
“Categorical Actions and Crystals”, to appear in Contemp. Math.,
arXiv:1603.08938v2 (Joint with J. Brundan).
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Modules over localized group rings, for groups mapping onto free groups, Master’s
thesis, Boise State University, 2011.
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ACKNOWLEDGEMENTS
I have had the incredible fortune to learn mathematics from individuals who, in
addition to being world-class researchers, are also dedicated and effective instructors
and generous human beings. Thank you, Tim Bergren, for teaching me that math
is more than just pushing around symbols, and that given enough time and effort,
you can make a lawnmower into a go cart. Thanks to my undergraduate professors
Gary Ganske, Bob Decloss, and Ed Korntved. Without your influence I probably
would have been an engineer! Thanks to my master’s thesis advisor, Jens Harlander,
who taught me that there are no zombies in group theory, that Euler characteristics
are best understood in terms of buying used cars, and that the greatest advantage in
earning a PhD is never having to wear a necktie.
Thanks to Rob Muth and Joey Iverson for peppermint donut homework
assignments and other helpful discussion.
Thank you Shun-Jen Cheng, Jae-Hoon Kwon, Weiqiang Wang, and Shunsuke
Tsuchioka for the helpful discussion in Korea. Also, thank you Seijin Oh for the
chance to eat octopus!
Thanks to Victor Ostrik for introducing me to Lie algebras and starting me on
this path, and to Sasha Kleshchev for putting the word “quiver” in my vocabulary.
Thanks to Marcin Bownik for guiding me through Baby Rudin, and to Dejing
Dou, who volunteered to read my thesis during his summer off. Thanks to Arkady
Berenstein for teaching me that canonical bases make life nicer.
Most of all, thanks to my advisor Jon Brundan. You taught me that
representation theory is great, even if it really doesn’t have much to do with quantum
physics. You also taught me to push myself through exhaustion, as my wife and I
vii
chased you through a Korean train station after a 14 hour flight, and then through
Seoul looking for a Japanese restaurant. I have appreciated your limitless patience and
guidance as I learned to formulate my own arguments and write them down coherently.
Your incredible proof-reading skills has provided me with countless opportunities to
improve my thesis, and because of your helpful suggestions, it has turned out better
than I ever could have hoped. Thanks for the advice all the way through graduate
school, and the guidance in the job market.
Thanks to my parents Bob and Michell for all of the support over the years, my
grandparents Bob and Mary Kay for the fishing trips and grilled cheese sandwiches,
and to my grandparents Jug and Eunice for all the soup, casseroles, and pies that a
guy could ever want.
viii
For my wife Amber, who was patient when I locked myself in the office, fed me when
I had no time to cook, and periodically forced me to take a break and have fun. I
couldn’t have done it without your love, support, and occasional kicks in the pants.
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TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Categorical actions: a broad overview . . . . . . . . . . . . . . 1
1.2. Super category O . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3. Super background . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4. Statement of results . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5. Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
II. CATEGORICAL ACTIONS . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1. Schurian and highest weight categories . . . . . . . . . . . . . 17
2.2. Type A and C combinatorics . . . . . . . . . . . . . . . . . . . 21
2.3. Quiver Hecke categories . . . . . . . . . . . . . . . . . . . . . . 25
2.4. Categorical actions . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5. Tensor product categorifications . . . . . . . . . . . . . . . . . 29
2.6. An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.7. Type A blocks revisited . . . . . . . . . . . . . . . . . . . . . . 41
III. TYPE C BLOCKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1. The supercategory sO . . . . . . . . . . . . . . . . . . . . . . 45
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Chapter Page
3.2. Special projective superfunctors . . . . . . . . . . . . . . . . . 53
3.3. Bruhat order revisited . . . . . . . . . . . . . . . . . . . . . . 60
3.4. Weak categorical action . . . . . . . . . . . . . . . . . . . . . . 66
3.5. Strong categorical action . . . . . . . . . . . . . . . . . . . . . 68
3.6. Proof of main theorem (type C) . . . . . . . . . . . . . . . . . 81
IV. APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.1. The root system Ck . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2. The category Ok . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3. Categorical actions on Ok . . . . . . . . . . . . . . . . . . . . . 85
4.4. Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5. Classification of Prinjectives . . . . . . . . . . . . . . . . . . . 90
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
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CHAPTER I
INTRODUCTION
This chapter contains excerpts from the introduction of coauthored material in
[BD2]. J. Brundan and I worked closely in the writing of that introduction.
1.1. Categorical actions: a broad overview
In the early 1990s, Lusztig constructed canonical bases in certain integrable
representations of quantum groups. These bases possess amazing integrality and
positivity properties. The study of categorical actions of Kac-Moody algebras has its
roots in the idea that these bases must be the shadows cast by some higher structures.
Some of the first examples studied include:
– The representation theory of symmetric groups and their Hecke algebras [LLT,
A, G].
– Rational representations of the general linear group [BK1].
– The BGG category O for the general linear Lie (super)algebra [BFK, B1].
Inspired by common elements in these examples, Chuang and Rouquier [CR]
unified them under the axiomatic framework of sl2-categorifications. Subsequently,
Rouquier [R] extended the ideas to categorical actions of arbitrary Kac-Moody
algebras. Independently, motivated instead by the low-dimensional topology problem
of categorifying Reshetikhin-Turaev invariants, Khovanov and Lauda [KL1, KL2]
came up with equivalent formulations of the same notions.
Let us give a brief overview of the idea of categorical actions. Let s be any any
complex, symmetrizable Kac-Moody algebra with simple roots {αi | i ∈ I}, weight
1
lattice P , Chevalley generators {ei, fi | i ∈ I}, and coroots hi := [ei, fi]. In classical
representation theory, one is often interested in integrable linear representations of s.
Roughly speaking, this is the data of:
– A complex vector space V with weight decomposition V =
⊕
λ∈P Vλ.
– Locally nilpotent maps ei : Vλ → Vλ+αi , and fi : Vλ+αi → Vλ for each i ∈ I and
λ ∈ P .
The linear maps are required to satisfy certain relations. For example, the
commutator eifi− fiei must act on Vλ as the scalar λ(hi). Put into fancier language,
classical representation theory studies representations of s in the category Vec, whose
objects are vector spaces, and whose morphisms are linear maps.
In higher representation theory, one replaces the category Vec with the 2-
category Cat of categories, functors, and natural transformations. A weak categorical
representation of s is the data of:
– A (suitably finite, additive, linear...) category C, equipped with a decomposition
C = ⊕λ∈P Cλ.
– Biadjoint functors Fi : Cλ → Cλ−αi and Ei : Cλ−αi → Cλ for each i ∈ I and
λ ∈ P .
These functors are necessarily exact, so they induce linear operators ei := [Ei] and
fi := [Fi] on the split Grothendieck group [C] := C⊗Z K0(C). We require that these
operators make [C] into an integrable linear representation of s, with λ-weight space
[Cλ].
The key idea of higher representation theory is that, rather than specifying
relations between functors on the Grothendieck group, one should look for natural
2
transformations between the functors which induce the required Grothendieck group
relations. This line of thinking leads to the notion of a strong categorical action,
first introduced by Chuang-Rouquier [CR] in the case s = sl2, and then for
general s in [R, KL2]. By exploiting the “higher structure” afforded by the natural
transformations, Chuang and Rouquier proved Broue´’s Abelian defect conjecture for
the symmetric group. This was the first example demonstrating that the techniques of
higher representation theory can provide more information than classical techniques
alone.
We will not give the full definition of strong categorical actions here; it may be
found in Chapter 2. To illustrate the general idea, the axioms assert amongst other
things that for every weight λ ∈ P and each i ∈ I for which λ(hi) is a positive
integer, the functors EiFi and FiEi : Cλ → Cλ should admit a distinguished natural
isomorphism:
ρi,λ : EiFi ⇒ FiEi ⊕ Idλ(hi)Cλ . (1.1)
On the level of Grothendieck groups, this isomorphism of functors induces the relation
that [ei, fi] acts on the λ-weight space [Cλ] as multiplication by the scalar λ(hi).
We remark that the data of a strong categorical action is equivalent to a strict 2-
functor U(s) → Cat , where U(s) is the Kac-Moody 2-category of Khovanov-Lauda
and Rouquier. This 2-category is discussed at length in [BD1].
1.2. Super category O
In this dissertation, we use the rich structure arising from categorical actions to
study the analog of the BGG category O associated to the queer Lie superalgebra
3
qn(k). Chen [C] reduced most questions about this category to the study of the so-
called types A, B, C blocks. Already in the early 2000s, Brundan [B2] had investigated
the type B blocks (which correspond to integer weights) and formulated a version of
the Kazhdan-Lusztig conjecture for characters of irreducibles in those blocks in terms
of certain canonical bases for the quantum group of type B∞. Recently, Cheng, Kwon
and Wang [CKW] noted the type A blocks (defined below) and type C blocks (which
correspond to half-integer weights) mirror combinatorics of the quantum groups of
type A∞ and C∞, respectively. This led to analogs of Brundan’s Kazhdan-Lusztig
conjecture for the type A and type C blocks.
In joint work with Brundan [BD2], we have proved the truth of the Cheng-Kwon-
Wang conjecture for type A blocks ([CKW, Conjecture 5.13]). In fact, we establish
an equivalence of categories between the type A blocks for qn(k) and the integral
blocks of category O for a general linear Lie superalgebra. This reduces the Cheng-
Kwon-Wang conjecture for type A blocks to the Kazhdan-Lusztig conjecture of [B1],
which was proved already in [CLW, BLW].
While the type A conjecture has been verified, the type B conjecture from [B2]
and the original type C conjecture [CKW, Conjecture 5.9] appear to be incorrect.
Tsuchioka discovered in 2010 that the type B canonical bases considered in [B2] fail
to satisfy appropriate positivity properties, so that the conjecture is certainly false.
After [CKW] appeared, Tsuchioka also pointed out similar issues with the type C
canonical bases studied in [CKW], so that conjecture is likely incorrect, as well.
Despite the fact that there are problems with the original type C conjecture,
many of the arguments used to prove the type A conjecture can be modified to
study the type C blocks. Chapters 3 and 4 of this dissertation develop these ideas.
In particular, we prove a modified version of the Cheng-Kwon-Wang conjecture for
4
type C blocks, where one replaces Lusztig’s canonical basis with Webster’s “orthodox
basis” arising from the indecomposable projective modules of the tensor product
algebras [W1, §4]. This modified conjecture was proposed independently by Cheng,
Kwon and Wang in a revision of their article ([CKW, Conjecture 5.10]). It is not as
satisfactory as the situation for type A blocks, however, since there is no elementary
algorithm to compute Webster’s basis explicitly (unlike the canonical basis).
We will review our results for type A blocks in more detail in Section 1.4.1 below.
In Section 1.4.2 we will discuss our new results for the type C blocks, and Section
1.4.3 will say a little more about the type B blocks.
1.3. Super background
To formulate our results in more detail, we need to briefly recall some basic
notions of supercategories and superalgebra.
1.3.1. Supercategories
Let k be a ground field which is algebraically closed of characteristic zero, and
fix for eternity a choice of
√−1 ∈ k. We adopt the language of [BE1, Definition 1.1]:
– A supercategory is a category enriched in the monoidal category of vector
superspaces (i.e., Z /2Z-graded vector spaces over k with morphisms that are
parity-preserving linear maps).
– Any morphism in a supercategory decomposes uniquely into an even and an
odd morphism as f = f0¯ + f1¯. If f is homogeneous we write |f | ∈ Z /2Z for its
parity. A superfunctor between supercategories means a k-linear functor which
preserves the parities of morphisms.
5
– Given superfunctors F,G : C → D, a supernatural transformation η : F ⇒ G is
a family of morphisms ηM = ηM,0¯ + ηM,1¯ : FM → GM for each M ∈ ob C, such
that ηN,p◦Ff = (−1)|f |pGf ◦ηM,p for every homogeneous morphism f : M → N
in C and each p ∈ Z /2Z. It is even (resp. odd) if ηM = ηM,0¯ (resp. ηM = ηM,1¯)
for all M .
For example, suppose that A is a locally unital superalgebra, i.e. an associative
superalgebra A = A0¯ ⊕ A1¯ equipped with a distinguished collection {1x | x ∈ X} of
mutually orthogonal even idempotents such that A =
⊕
x,y∈X 1xA1y. Then there is a
supercategory A -smod consisting of finite dimensional left A-supermodules M which
are locally unital in the sense that M =
⊕
x∈X 1xM . Even morphisms in A -smod
are parity-preserving linear maps such that f(av) = af(v) for all a ∈ A, v ∈ M ;
odd morphisms are parity-reversing linear maps such that f(av) = (−1)|a|af(v) for
homogeneous a.
For any supercategory C, the Clifford twist CCT is the supercategory whose objects
are pairs (X,φ) where X ∈ ob C and φ ∈ EndC(X) is an odd involution. A morphism
f : (X,φ) → (X ′, φ′) in CCT is a morphism f : X → X ′ in C such that fp ◦ φ =
(−1)pφ′ ◦ fp for each p ∈ Z /2Z. One can also take Clifford twists of superfunctors
and supernatural transformations, so that CT is actually a 2-superfunctor from the
2-supercategory of supercategories to itself in the sense of [BE1, Definition 2.2]. The
following lemma is a variation on [KKT, Lemma 2.3].
Lemma 1.3.1. Suppose C is an additive supercategory in which all even idempotents
split. Also assume that for every X ∈ ob C, there exists another object ΠX ∈ ob C,
and an odd isomorphism ζX : ΠX
∼→ X. Then, the supercategories C and (CCT)CT are
superequivalent.
6
Proof. Note that objects in the supercategory (CCT)CT consist of triples (X,φ, ψ) for
X ∈ ob C and φ, ψ ∈ EndC(X)1¯ such that φ2 = ψ2 = id and φ ◦ ψ = −ψ ◦ φ.
Morphisms f : (X,φ, ψ) → (X ′, φ′, ψ′) in (CCT)CT are morphisms f : X → X ′ in C
such that fp ◦ φ = (−1)pφ′ ◦ fp, and fp ◦ ψ = (−1)pψ′ ◦ fp for each p ∈ Z /2.
Define a superfunctor F : C → (CCT)CT as follows. On an object X ∈ ob C, let
FX := (X ⊕ ΠX,φ, ψ) where
φ =
 0 ζX
ζ−1X 0
 , ψ =
 0 −√−1ζX√−1ζ−1X 0
 .
On a homogeneous morphism f : X → X ′, we let Ff : FX → FX ′ be the morphism
defined by the matrix
 f 0
0 Πf
, where Πf : ΠX → ΠX ′ denotes (−1)|f |ζ−1X′ ◦
f ◦ ζX . We show that F is a superequivalence by checking that it is full, faithful
and evenly dense (see [BE1]). It is obviously faithful. To see that it is full, take an
arbitrary homogeneous morphism f : FX → FX ′ in (CCT)CT. Viewing f as a 2 × 2
matrix
 f11 f12
f21 f22
 of morphisms in C, we need to show that f12 = f21 = 0 and
f22 = Πf11. This follows easily on considering the matrix entries in the identities
φ′ ◦ f = (−1)|f |f ◦ φ and ψ′ ◦ f = (−1)|f |f ◦ ψ.
Finally, to check that F is evenly dense, we take any object (X,φ, ψ) ∈ ob(CCT)CT,
and must show that it is evenly isomorphic to an object in the image of F . Let
e1 :=
1−√−1φ ◦ ψ
2
, e2 :=
1 +
√−1φ ◦ ψ
2
.
These are mutually orthogonal idempotents summing to the identity in EndC(X)0¯.
Hence, we may decompose X as X = X1 ⊕ X2 with Xi being the image of ei. We
7
then have that φ = e2 ◦ φ ◦ e1 + e1 ◦ φ ◦ e2, and similarly for ψ. Now we observe that
e2 ◦ ψ ◦ e1 = e2 ◦ ψ = ψ +
√−1φ
2
=
√−1φ ◦ e1 =
√−1 e2 ◦ φ ◦ e1.
Similarly, e1◦ψ◦e2 = −
√−1 e1◦φ◦e2. The map e2◦φ◦e1◦ζX1 is an even isomorphism
ΠX1
∼→ X2, hence, X = X1 ⊕ X2 ∼= X1 ⊕ ΠX1 = FX1. Under this isomorphism,
φ = e2 ◦ φ ◦ e1 + e1 ◦ φ ◦ e2 corresponds to the matrix
 0 ζX1
ζ−1X1 0
. Similarly,
ψ =
√−1 e2 ◦φ ◦ e1−
√−1 e1 ◦φ ◦ e2 corresponds to
 0 −√−1ζX1√−1ζ−1X1 0
. This
verifies that (X,φ, ψ) is evenly isomorphic to FX1.
For example, if A is a locally unital superalgebra, then there is an obvious
isomorphism between the Clifford twist A -smodCT of this supercategory and the
supercategory A ⊗ C1 -smod, where C1 denotes the rank one Clifford superalgebra
generated by an odd involution c, and A ⊗ C1 is the usual braided tensor product
of superalgebras. Hence, (A -smodCT)CT is isomorphic to A ⊗ C2 -smod where C2 :=
C1⊗C1 is the rank two Clifford superalgebra generated by c1 := c⊗1 and c2 := 1⊗c.
In this situation, the above lemma is obvious as A⊗ C2 is isomorphic to the matrix
superalgebra M1|1(A), which is Morita superequivalent to A.
For another construction of a supercategory, suppose that C is any k-linear
category. Then we let C ⊕ ΠC be the supercategory whose objects are formal direct
sums V1 ⊕ ΠV2 for V1, V2 ∈ ob C, with morphisms V1 ⊕ ΠV2 → W1 ⊕ ΠW2 being
matrices of the form f =
 f11 f12
f21 f22
 for fij ∈ HomC(Vj,Wi). The Z /2Z-grading
is defined so that f0¯ =
 f11 0
0 f22
 and f1¯ =
 0 f12
f21 0
.
8
For example, if B is any locally unital algebra (no super!) and C = B -mod is
the category of finite-dimensional locally unital B-modules, then the supercategory
C ⊕ ΠC may be identified with the category B -smod, where we view B as a purely
even superalgebra.
1.3.2. Lie superalgebras
A Lie superalgebra g is a superspace g = g0¯ ⊕ g1¯, equipped with an even linear
map [·, ·] : g ⊗ g → g which satisfies the following conditions for every homogeneous
x, y, z ∈ g:
– Super skew-symmetry: [x, y] = −(−1)|x||y|[y, x].
– Super Jacobi identity: [x, [y, z]] = [[x, y], z] + (−1)|x||y|[y, [x, z]].
These axioms imply that the restriction of the bracket makes the subspace g0¯ into a
Lie algebra in the ordinary sense.
The basic example is the general linear Lie superalgebra glm|n(k). The elements
of this Lie algebra are (m+ n)× (m+ n) matrices of the form
 A B
C D
 (1.2)
where A is an m × m matrix, D is an n × n matrix, etc. The grading is given by
declaring that the even subspace glm|n(k)0¯ consists of all such matrices with B and
C being zero, while the odd subspace consists of those matrices for which A and D
are zero. The bracket on glm|n(k) is given by the matrix supercommutator:
[x, y] := xy − (−1)|x||y|yx.
9
While this equation only makes sense when x and y are homogeneous, it may be
extended to non-homogeneous elements of glm|n(k) by linearity.
1.4. Statement of results
Now fix n ≥ 1 and let q = q0¯ ⊕ q1¯ be the queer Lie superalgebra qn(k). This is
the subalgebra of gln|n(k) consisting of all matrices of block form
 A B
B A
 . (1.3)
Let b (resp. h) denote the standard Borel (resp. Cartan) subalgebra of q consisting
matrices of the form (1.3) in which A and B are upper triangular (resp. diagonal).
Let t := h0¯. We let δ1, . . . , δn denote the basis for t
∗ such that δi picks out the ith
diagonal entry of the matrix A.
Given λ ∈ t∗, write λ = ∑nr=1 λrδr. It will be convenient to define certain subsets
of t∗ as follows:
– Given some z ∈ k with 2z /∈ Z, and some sign sequence σ = (σ1, . . . , σn) with
each σr = ±, we let Λσz denote the collection of all λ ∈ t∗ such that each λr is
in the set σr(z + Z).
– Let Λ0 denote the collection of all λ ∈ t∗ where each λr is an integer. These are
the integral weights.
– Let Λ 1
2
denote the collection of all λ ∈ t∗ with each λr ∈ 12 + Z. These are the
half-integer weights.
Fixing a choice of symbols X ∈ {σz, 0, 1
2
}, we let sOX denote the supercategory
of all q-supermodules M such that:
10
– M is finitely generated as a q-supermodule;
– M is locally finite-dimensional over b;
– M is semisimple over t with weights in ΛX .
Morphisms in sOX are arbitrary (not necessarily even) q-supermodule homomorphisms,
so sOX is indeed a supercategory.
1.4.1. Type A blocks
Letting σ and z vary, the blocks of the categories sOσz are the type A blocks.
This dissertation does not contain a detailed study of these blocks, but we give an
overview here.
Recent work by Cheng-Kwon-Wang [CKW] observed that the combinatorics
of the type A blocks can be described in terms of the Kac-Moody algebra sl∞
associated to the Dynkin diagram A∞, and its quantization Uq(sl∞). Building off
their observations, joint work with Brundan [BD2] proved the following:
Main Theorem (Type A). When n is even (resp. odd), the supercategory sOσz
(resp. sOCTσz) splits a direct sum O ⊕ ΠO, for some k-linear category O. Moreover,
the category O admits the structure of a tensor product categorification of V ⊗σ :=
V σ1 ⊗ · · ·V σn, where V + is the natural module associated to the Lie algebra sl∞-
module, and V − is its dual.
When n is odd, the theorem actually describes the Clifford twists of the type A
blocks. We can recover the type A blocks by Clifford twisting again, since (sOCTσz)CT
is equivalent to sOσz, by Lemma 1.3.1.
11
The detailed definition of tensor product categorifications will be given in
Chapter 2. The assertion that O is a tensor product categorification of V ⊗σ roughly
means that:
– The category O is a highest weight category in the sense of Cline, Parshall,
and Scott [CPS], with irreducible objects {L(b) | b ∈ B} labeled by the set
B = Zn, which is partially ordered using the Bruhat ordering. This is a certain
canonically defined partial ordering implicit in Lusztig’s work on based modules
[Lu]. For b ∈ B, we let P (b) denote a fixed projective cover of L(b), and
M(b) the corresponding standard object. The category O∆ denotes the exact
subcategory of objects with a filtration by standard modules.
– There are biadjoint functors Fi, Ei : O → O for every i ∈ Z which preserve
O∆, and induce operators on the Grothendieck group such that he linear
isomorphism [O∆] → V ⊗σ given by [M(b)] 7→ vb is an isomorphism of sl∞-
modules. In particular, [O∆] is an integrable representation of sl∞, so there is
a weak categorical action of sl∞ on O. Here {vb | b ∈ B} is the monomial basis
for V ⊗σ.
– The compositions of the functors Fi and Ei admit certain natural
transformations which upgrade the weak categorical action of sl∞ on O to a
strong categorical action.
Combining the above theorem with a powerful result for the uniqueness of tensor
product categorifications of V ⊗σ [BLW, Theorem 2.12], we see that O is equivalent to
the sum of the integral blocks in the BGG category O associated to a general linear
Lie (super)algebra. Results from [CLW, BLW] about general linear Lie superalgebras
12
further imply that
(P (a) : M(b)) = [M(b) : L(a)] = da,b(1)
where the polynomials da,b(q) are the entries of the transition matrix used to express
Lusztig’s canonical basis for the corresponding quantum deformation of V ⊗σ in terms
of the monomial basis. An algorithm to compute the canonical bases, hence the
polynomials da,b(q), can be found in [BD2, §8]. This demonstrates the combinatorics
in the type A blocks are governed by combinatorics associated to the quantum group
Uq(sl∞), which is completely explicit.
1.4.2. Type C blocks
This dissertation is devoted to the study of the type C blocks which are the
blocks in the category sO 1
2
. Let sp∞ denote the Kac-Moody algebra associated to
the Dynkin diagram C∞ with V its natural module. The main result of this thesis
can be summarized as follows:
Main Theorem (Type C). When n is even (resp. odd) the supercategory sO 1
2
(resp.
sOCT1
2
) splits as a direct sum O ⊕ ΠO, where O is a k-linear category. Moreover, O
admits the structure of a tensor product categorification of V ⊗n, where V is the natural
sp∞-module.
As in the type A situation above, this theorem actually describes the Clifford
twists of the type C blocks when n is odd. We may recover the type C blocks by
Clifford twisting a second time and applying Lemma 1.3.1.
Combining this theorem with the uniqueness of tensor product categorifications,
we also demonstrate that the combinatorics in the type C blocks can be expressed in
13
terms of Webster’s orthodox bases associated to the quantum group Uq(sp∞). This
was independently conjectured by Cheng, Kwon, and Wang in the updated version of
their paper ([CKW, Conjecture 5.11]). Although this does not provide as complete a
description of the combinatorics as we have for the type A blocks, it still has significant
consequences for our example. For example, we will also use our type C theorem to
determine the associated crystal underlying the type C blocks, and then classify the
indecomposable projective-injective (or prinjective) objects of O.
1.4.3. Type B blocks
The type B blocks are the blocks of the category sO0. These are the most
interesting blocks of all, but due to time constraints in the preparation of this
dissertation, a study of these blocks will not be included. However, it should be
mentioned that, from the standpoint of higher representation theory, the behavior
of the type B blocks differs substantially from the types A and C blocks. Indeed,
in the type A and C cases, the supercategory sO splits as a direct sum O ⊕ ΠO,
for a k-linear category O, i.e., we can “de-superize” the theory, thereby fitting the
types A and C cases into the existing framework of categorical actions. In contrast,
the behavior exhibited in the type B blocks is genuinely “super.” Because of this,
the detailed study of these blocks requires a notion of supercategorical actions. This
theory is still under-developed, but recent work by Brundan and Ellis [BE2] on super
Kac-Moody 2-categories has laid the foundations for the subject.
We let sOQ0 denote the Serre subcategory of sO0 generated by the type Q
irreducibles, i.e., those which admit an odd involution. Similarly, we let sOM0 denote
the Serre subcategory of sO0 generated by the type M irreducibles, i.e., those which
14
have no odd involution. Despite the fact that the axiomatic definitions have not been
fixed, some version of the following conjecture appears to be true:
Main Conjecture (Type B). The supercategory sOQ0⊕(sOM0)CT admits the structure
of a tensor product supercategorification of V ⊗n, where V is the natural module for
the Lie algebra so∞ associated to the Dynkin diagram B∞, where all long simple roots
are even but the short simple root is odd.
In future work, I plan to make these definitions more explicit and prove this
conjecture. Just as Chuang-Rouquier [CR] used categorical actions to prove Broue´’s
Abelian defect conjecture for the symmetric group, studying these supercategorical
actions may provide insights towards a proof of Broue´’s conjecture for the spin
symmetric group.
1.5. Organization
This dissertation is organized as follows:
– Chapter 2 will provide the necessary background material on categorical actions
and tensor product categorifications. It will recall the proof of the well-known
result that the integral blocks in the BGG category O for gln(k) admits the
structure of a tensor product categorification of (V +)⊗n, where V + is as above.
Having recalled the required definitions, the chapter concludes with a brief
discussion of the type A blocks.
– Chapter 3 is dedicated to the proof of the Main Theorem for type C blocks
which we described in the previous section. Many of results in this section have
proofs which are nearly identical to analogous results proved in [BD2]. Because
of this, the chapter contains many excerpts from coauthored material.
15
– Chapter 4 will apply the results of Chapter 3 to study the structure of the type C
blocks. In particular, it will relate the combinatorics of the block with Webster’s
type C orthodox bases from [W1]. It will also give a concrete description of the
canonically defined associated crystal for the type C blocks, and use it to classify
the prinjective objects in these blocks.
16
CHAPTER II
CATEGORICAL ACTIONS
This chapter gives an overview of the theory of categorical actions of Kac-Moody
algebras. We will use these categorical actions in chapters 3 and 4 to prove results
about the type C blocks in the BGG category O associated to the Lie superalgebra
qn(k).
2.1. Schurian and highest weight categories
Before giving the formal definition of categorical actions of Kac-Moody algebras,
we must review some basic notions.
2.1.1. Schurian categories
We begin by recalling the notion of a Schurian category from [BLW, §2]:
Definition 2.1.1. A Schurian category is a k-linear, Abelian category C such that:
– C has enough projectives and injectives.
– Every object of C has finite length.
– The endomorphism ring of any irreducible object is one-dimensional.
We can view Schurian categories as generalization of the categories of finite
dimensional modules over a finite dimensional algebra. To make this more precise,
suppose that A is a locally unital algebra, i.e., A is equipped with a system of
orthogonal idempotents {1x | x ∈ X} such that A =
⊕
x,y,∈X 1xA1y, and let
mod-A denote the category of finite dimensional, right A-modules M for which
17
M =
⊕
x∈XM1x. We have the following characterization of Schurian categories,
the proof of which is outlined in [BLW, §2.1].
Proposition 2.1.2. A category C is Schurian if and only if it is equivalent to mod-A,
where A is a locally unital algebra for which the one-sided ideals 1xA and A1x are
finite dimensional for every x ∈ X.
If C is a Schurian category, we let pC denote the additive category of projective
objects of C, and write [C] for the complexified split Grothendieck group C⊗ZK0(pC).
2.1.2. Highest weight categories
We also need the notion of a highest weight category, which was first introduced
by Cline, Parshall, and Scott in [CPS].
Definition 2.1.3. A highest weight category (C,B,) is the data of:
– A Schurian category C.
– A set B which indexes a complete set of non-isomorphic irreducible objects
{L(b) | b ∈ B} in C.
– A partial order  on B.
For each b ∈ B, fix a projective cover P (b) of L(b). Define the standard object ∆(b) as
the maximal quotient of P (b) whose composition multiplicities satisfy the properties
[∆(b) : L(b)] = 1 and [∆(b) : L(c)] = 0 unless c  b.
18
In order for (C,B,) to be a highest weight category, we require that each projective
P (b) has a finite filtration
0 = P0 ⊂ P1 ⊂ · · · ⊂ Pd = P (b)
such that Pd/Pd−1 ∼= ∆(b), and for every 1 ≤ r < d, there is some cr  b such that
Pr/Pr−1 ∼= ∆(cr).
If C is a highest weight category, any filtration of an object of C with subquotients
isomorphic to standard objects is called a ∆-flag. We let C∆ denote the exact
subcategory of all objects of C with a ∆-flag, and let [C∆] denote its corresponding
complexified Grothendieck group. The classes {[∆(b)] | b ∈ B} form a basis for [C∆],
and our condition on the ∆-flags of projectives in C implies that there is an inclusion
[C] ↪→ [C∆]. This is an isomorphism when B is finite, or, more generally, when every
b ∈ B is comparable to only finitely many c ∈ B.
2.1.3. Serre subcategories and Serre quotients
We next recall the well-known definitions of Serre subcategories and Serre
quotients. These will play an important role in chapter 4.
Assume that C is a Schurian category with its irreducible objects labeled by a
set B, and suppose B′ ⊂ B. Let C ′ denote the Serre subcategory of C generated by
the irreducible objects
{L(b) | b ∈ B′}.
This means that C ′ is the full subcategory of C whose objects are those M ∈ ob C
satisfying the condition that whenever [M : L(b)] 6= 0, then b ∈ B′. Note that C ′ is
also Schurian.
19
Let C ′′ = C/C ′ denote the corresponding Serre quotient category. This is a
Schurian category, too. By definition, the objects of the category C ′′ are the same
as the objects of C. The morphisms in C ′′ are constructed as follows. Given objects
M,N ∈ ob C ′′ = ob C, let Ω(M,N) denote the collection of all pairs (M ′, N ′), where
M ′ ⊂M and N ′ ⊂ N satisfy M/M ′, N ′ ∈ ob C ′. The set Ω(M,N) is partially ordered
by (M ′, N ′) ≤ (M ′′, N ′′) if and only if M ′′ ⊂ M ′ and N ′ ⊂ N ′′. When this happens,
composing with the inclusionM ′′ ↪→M ′ and the quotient mapN/N ′  N/N ′′ induces
a linear map HomC(M ′, N/N ′)→ HomC(M ′′, N/N ′′). Define
HomC′′(M,N) := lim−→
HomC(M ′, N/N ′)
where the colimit is taken over all pairs (M ′, N ′) ∈ Ω(M,N).
Let pi : C → C ′′ be the obvious exact quotient functor. The definition of
morphisms in C ′′ imply that whenever M is an object of C ′, then piM ∼= 0 in C ′′.
In fact, the category C ′′ and the functor pi are universal with respect to this property:
whenever there is a Schurian category E and an exact functor G : C → E for which
G(M) ∼= 0 for every M ∈ ob C ′, then there exists a unique exact functor G¯ : C ′′ → D
such that G¯ ◦ pi = G.
Next, suppose that C is a highest weight category, so B is equipped with a partial
order . An ideal or lower set in B is a subset B′ ⊂ B which satisfies the property
that whenever b ∈ B′ and a  b, then a ∈ B′, too. Assume that B′ is an ideal, and
set B′′ = B \B′.
The assumption that B′ is an ideal implies that that the subcategory C ′ defined
above inherits a highest weight structure from C, with poset (B′,). The irreducibles
and standards in C ′ are precisely those L(b) and ∆(b) with b ∈ B′. The projective
cover of L(b) is the maximal quotient P ′(b) of P (b) which lies in C ′. The category
20
C ′′ inherits a highest weights structure, with poset (B′′,). The irreducibles,
projective indecomposables, and standard objects in C ′′ are given by applying pi to
the corresponding objects in C.
We record the following lemma for future use:
Lemma 2.1.4 ([BD1, Lemma 2.13]). Assume M,N ∈ ob C are such that all
irreducible constituents of the head of V and the the socle of W are of the form
L(b) for b ∈ B′′. The quotient functor pi induces an isomorphism
HomC(V,W )
∼→ HomC′′(piV, piW )
2.2. Type A and C combinatorics
The types A and C blocks for qn(k) described in the introduction give rise to
highest weight categories, where the labeling set B = Zn is partially ordered by the
respective types A and C Bruhat order. These are particular instances of the “inverse
dominance ordering” of [LW, Definition 3.2], which appears implicitly in Lusztig’s
work on tensor products of based modules in [Lu, §27.3]. The goal of this section is
to introduce the necessary combinatorics to define these orders.
Let A∞ denote the Dynkin diagram
c c c c c−2 −1 0 1 2
which has nodes indexed by I = Z. We denote the associated Kac-Moody algebra by
sl∞, which we identify with the Lie algebra of finitely-supported complex matrices
whose rows and columns are indexed by I. It is generated by the matrix units
fi := ei+1,i and ei := ei,i+1 for i ∈ I. The natural representation V + of sl∞ is the
21
module of column vectors with standard basis {v+i | i ∈ I}. We also need the dual
natural representation V − with basis {v−i | i ∈ I}. The action of the Chevalley
generators on these bases is given by
eiv
+
j = δi+1,jv
+
i , eiv
−
j = δi,jv
−
i+1, (2.1)
fiv
+
j = δi,jv
+
i+1, fiv
−
j = δi+1,jv
−
i . (2.2)
Fix a sign sequence σ ∈ {±}n. We stress that the subsequent notation depends
implicitly on this choice! Define
V ⊗σ := V σ1 ⊗ · · · ⊗ V σn
which has monomial basis {vb | b ∈ B} defined from vb := vσ1b1 ⊗ · · · ⊗ vσnbn . For any
i ∈ I, we define the i-signature of b ∈ B by i-sig(b) = (i-sig(b)1, . . . , i-sig(b)n) ∈
{f, e, •}n where
i-sig(b)t :=

f if either σt = + and bt = i, or σt = − and bt = i+ 1,
e if either σt = + and bt = i+ 1, or σt = − and bt = i,
• otherwise.
(2.3)
For 1 ≤ t ≤ n, let dt denote the element of B with σt1 in the t-th entry, and zero
everywhere else. Then, the Chevalley generators act on the monomial basis of V ⊗σ
by
fivb =
∑
1≤t≤n
i-sig(b)t= f
vb+dt , eivb =
∑
1≤t≤n
i-sig(b)t= e
vb−dt . (2.4)
22
The root system of sl∞ has weight lattice P :=
⊕
i∈I Zωi where ωi is the ith
fundamental weight. For i ∈ I, we set
εi := ωi − ωi−1, αi := εi − εi+1.
We identify εi with the weight of the vector v
+
i in the sl∞-module V
+. Then, v−i ∈ V −
is of weight −εi, and vb ∈ V ⊗σ has weight
wt(b) :=
n∑
r=1
σrεbr ∈ P. (2.5)
Let E denote the dominance order on P defined from β E γ if and only if γ − β ∈⊕
i∈I Nαi. For 1 ≤ s ≤ n, it will also be convenient to define wts(b) :=
∑
1≤r≤s σrεbr ,
so that wtn(b) = wt(b).
Definition 2.2.1. The type A Bruhat order  on B associated to σ is the partial
order defined by declaring that b  a if and only if wts(b) D wts(a) for all s =
1, . . . , n, with equality when s = n. In particular, b and a are comparable only when
wt(b) = wt(a).
Next, we mimic the same constructions, replacing sl∞ with the Kac-Moody
algebra sp∞ associated to the Dynkin diagram C∞
c c c>0 1 2
with nodes indexed by I = N.
23
Denote the Chevalley generators of sp∞ by {ei, fi | i ∈ I}. The natural sp∞-
module V has basis {vj | j ∈ Z} and action defined from
fivj =
 vj+1 if j = ±i0 otherwise , eivj =
 vj−1 if j = 1± i0 otherwise .
Note that sp∞ preserves the non-degenerate symplectic form
(·, ·) : V ⊗ V → k (vj, vk) = sgn(j − k)δj,1−k,
where sgn is the sign function. Hence, V is isomorphic to its dual representation, in
contrast to the type A setting.
We redefine the i-signature i-sig(b) for the type C case as
i-sig(b)t :=

f if b = ±i
e if b = 1± i
• otherwise
(2.6)
Also, let dt denote the element of B with a 1 in the t-th entry, and zero everywhere
else. The nth tensor power V ⊗n of the natural module has basis {vb := vb1⊗· · ·⊗vbn |
b ∈ B} and the action of sp∞ on this basis is given by
fivb =
∑
1≤t≤n
i-sig(b)t=f
vb+dt , eivb =
∑
1≤t≤n
i-sig(b)t=e
vb−dt . (2.7)
In the weight lattice P :=
⊕
i∈I Z εi, we have the simple roots α0 := −2ε0 and αi :=
εi−1 − εi for i > 0, where vi is of weight εi. We denote the corresponding dominance
24
order on P by E. For 1 ≤ s ≤ n, define
wts(b) :=
∑
1≤r≤s
br≥1
εbr−1 −
∑
1≤r≤s
br≤0
ε−br .
The vector vb ∈ V ⊗n is of weight wt(b) := wtn(b).
Definition 2.2.2. The type C Bruhat order  on B by is given by b  a if and only
if wts(b) D wts(a) for every s in 1, . . . , n, with equality when s = n.
2.3. Quiver Hecke categories
In the the next three sections, we are going to introduce quiver Hecke categories
and categorical actions associated to sl∞ and sp∞. In order to unify our treatement
of the two cases, we set s = sl∞ or sp∞, and make the following assumptions:
– If s = sl∞ let I = Z index its simple roots, let P denote its weight lattice. Given
our fixed choice of σ as above, let T denote the mixed tensor space T = V ⊗σ.
We make B = Zn into a poset using the type A Bruhat order  associated to
σ.
– If s = sp∞, let I = N index its simple roots, let P denote its weight lattice, and
let T denote the tensor space T = V ⊗n. We partially order B = Zn with the
type C Bruhat order .
Next, we use the string calculus of [KL1] to define the quiver Hecke category QH
associated to s. An expository account of the string calculus can be found in [BD1,
§3].
Definition 2.3.1. The quiver Hecke category QH associated to s is the strict k-
linear monoidal category generated by objects I and morphisms •
i
: i → i and
25
i2 i1
: i2 ⊗ i1 → i1 ⊗ i2 subject to the following relations:
i2 i1
• −
i2 i1
• =
i2 i1
• −
i2 i1
•
=
 i2 i1
if i1 = i2,
0 if i1 6= i2;
i2 i1
=

0 if i1 = i2,
i2 i1
if |i1 − i2| > 1,
−
i2 i1
• +
i2 i1
•• if s = sp∞, i1 = 0, i2 = 1,
i2 i1
•• −
i2 i1
• if s = sp∞, i1 = 1, i2 = 0,
(i1 − i2)
i2 i1
• + (i2 − i1)
i2 i1
• otherwise;
i3 i2 i1
−
i3 i2 i1
=

i3 i2 i1
• +
i3 i2 i1
• if s = sp∞, i1 = i3 = 1, i2 = 0,
(i1 − i2)
i3 i2 i1
if i1 = i3, |i1 − i2| = 1 and,
when s = sp∞, i2 6= 0,
0 otherwise.
Let Id denote the set of words i = id · · · i1 of length d in the alphabet I, and
identify i ∈ Id with the object id⊗· · ·⊗ i1 ∈ obQH. Then, the locally unital algebra
QHd :=
⊕
i,i′∈Id
HomQH(i, i
′) (2.8)
26
is the quiver Hecke algebra associated to s. These algebras were originally defined by
Khovanov and Lauda [KL1] and Rouquier [R].
2.4. Categorical actions
In this section, we give axioms for (integrable) categorical actions of the Kac-
Moody algebra s. While we are specializing to the case where s = sl∞ or sp∞,
the definition has obvious analogs for other Kac-Moody algebras. Recall that a linear
representation M of s is integrable if it decomposes into weight spaces M =
⊕
λ∈P Mλ,
and the Chevalley generators ei and fi act locally nilpotently on M .
Definition 2.4.1. A categorical action of s on a Schurian category C is the data of:
(D1) A weight decomposition C = ⊕λ∈P Cλ.
(D2) A strict monoidal functor Φ : QH → End (C), where End (C) is the strict
monoidal category whose objects are k-linear functors and whose morphisms
are natural transformations. For i, i1, i2 ∈ I, let Fi denote the functor Φ(i), and
define natural transformations
ξi := Φ( •
i
) : Fi ⇒ Fi τi2,i1 := Φ(
i2 i1
) : Fi2Fi1 ⇒ Fi1Fi2 .
(D3) A functor Ei for every i ∈ I, plus natural transformations ηi : IdC ⇒ EiFi and
εi : FiEi ⇒ IdC making (Fi, Ei) into an adjoint pair.
This data must satisfy the following conditions:
(D4) The natural transformation ξi is locally nilpotent, i.e., for every object M ∈
ob C, the induced endomorphism ξi,M : FiM → FiM is nilpotent.
(D5) The functor Ei is isomorphic to a left adjoint of Fi.
27
(D6) For every λ ∈ P , the restriction of Fi sends Cλ into Cλ−αi .
(D7) The induced operators fi := [Fi] and ei := [Ei] make the complexified
Grothendieck group [C] into an integrable linear representation of s, where [Cλ]
is the λ-weight space.
If a category C is equipped with a categorical action of s, we will call C a categorical
representation of s.
Remark 2.4.2. This is one of several equivalent definitions of categorical actions of s
found in the literature. Theorem 5.30 in [R] shows that this definition is equivalent to
an integrable 2-representation of Rouquier’s Kac-Moody 2-category U(s). Khovanov
and Lauda originally gave a different definition of U(s), but recent work in [B5] shows
that the 2-categories are isomorphic. My expository paper with J. Brundan [BD1]
contains a detailed account of U(s).
Morphisms between categorical representations are strongly
equivariant functors :
Definition 2.4.3. Suppose that C and C ′ are categorical representations of s. For
clarity, let Φ′, F ′i , E
′
i, etc., denote the data associated with the categorical action on
C ′. We say that a functor G : C → C ′ is strongly equivariant if its restriction to Cλ has
its image in C ′λ, and if there exists natural isomorphisms ζi : F ′iG⇒ GFi such that:
(E1) The natural transformation
E ′iGεi ◦ E ′iζiEi ◦ ηiGEi : GEi → E ′iG
is invertible.
28
(E2) We have the equality of natural transformations
Gξi ◦ ζi = ζi ◦ ξ′iG : F ′iG⇒ GFi
(E3) We have the equality
Gτi2i1 ◦ ζi2Fi1 ◦ F ′i2ζi1 = ζi1Fi2 ◦ F ′i1ζi2 ◦ τ ′i2,i1G,
where these are viewed as natural transformations F ′i2F
′
i1
G⇒ GFi2Fi2
2.5. Tensor product categorifications
Given categorical representations C1 and C2 of s, we would like a notion of a
category C1⊗C2 which serves as the “tensor product” of the categorical representations
C1 and C2. At present, there is no known method to construct a general tensor
product of categorical representations, although it it hoped that a construction will
eventually emerge. In the meantime, Losev and Webster have introduced a notion of
tensor product categorification in [LW], which provides a list of properties allowing
one to recognize a given categorical representation C as a tensor product. In that
definition, they assume that the Grothendieck group of C is a tensor product of
highest weight modules. In our setting, the underlying s-module T is not a tensor
product of integrable highest weight modules, so we need to extend the Losev-Webster
definition. For the case where s = sl∞, the following definition is equivalent to the
one found in [BLW, Definition 2.9]. The s = sp∞ case is the obvious reformulation of
the sl∞ definition.
29
Definition 2.5.1. A tensor product categorification of T is a highest weight category
(C,B,) (B and  as above) equipped with the data (D2) and (D3), satisfying the
conditions (D5) and (D4). We also require that:
(TPC1) The functors Ei and Fi preserve the category C∆.
(TPC2) The linear isomorphism [C∆]→ T defined by [∆(b)] 7→ vb intertwines the actions
of the induced operators [Fi] and [Ei] on [C∆] with the action of the Chevalley
generators ei and fi of s on T .
If C is a tensor product categorification of T , it is automatic that C decomposes
as a direct sum C = ⊕λ∈P Cλ, where Cλ is the Serre subcategory of C generated by the
irreducibles of the form L(b), where wt(b) = λ. This shows that the prescribed highest
weight structure on C induces a decomposition as in (D1). In addition, because [C]
embeds into [C∆] ∼= T , we see that [C] is itself an integrable s-module. This implies
that tensor product categorifications are also categorical representations of s, in the
sense of Definition 2.4.1.
In their paper [LW], Losev and Webster prove a powerful uniqueness result for
tensor product categorifications associated to tensor products of integrable highest
weight modules. In our infinite rank setting, the tensor space T is not a tensor product
of highest weight modules, so the uniqueness theorem from [LW] does not immediately
apply. Instead we need the following theorem, which extends the uniqueness in the
type A setting.
Theorem 2.5.2 ([BLW, Theorem 2.11]). For any σ ∈ {±}n, there exists a tensor
product categorification C of V ⊗σ. Moreover, if C ′ is any other tensor product
categorification of V ⊗σ, there is a equivalence of categories G : C → C ′ which is
strongly equivariant, and satisfies G(L(b)) ∼= L′(b) for every b ∈ B.
30
We remark that the result actually proved in [BLW] is slightly more general,
as it includes tensor products of exterior powers of the modules V + and V −. The
techniques used to prove uniqueness in this theorem should also yield a proof that
tensor product categorifications of the sp∞-module V
⊗n are essentially unique. To
our knowledge, the type C blocks of Chapter 3 are the only known categorification
of V ⊗n, so any uniqueness theorem along these lines has no application at this point,
and we do not prove uniqueness in this dissertation.
Because machinery to construct general tensor product
categorifications is currently unavailable, the existence statement in Theorem 2.5.2
relies on an explicit but ad hoc construction using categories of modules associated
to the general linear Lie (super)algebra.
2.6. An example
To illustrate the theory in this chapter, we recall the construction of the tensor
product categorification of the sl∞-module V ⊗σ found in [BLW] for the special case
where each σr = +. In fact, this construction is the basic model for the approach we
are going to follow in Chapter 3 to construct tensor product categorifications of the
sp∞-module V
⊗n.
For this section only, we will denote V + by V , so we write V ⊗n for V ⊗σ. Set
g = gln(k). Let b denote its standard Borel subalgebra of upper triangular matrices,
and t its Cartan subalgebra of diagonal matrices. Let δ1, ..., δn denote the usual
coordinate functions in t∗. Define the weight lattice P =
⊕
i∈I Z δi, and a weight
dictionary B→ P given by b 7→ λb, where λb :=
∑n
r=1 λb,rδr where λb,r = br − r+ 1.
The presence of the −r + 1 in the definition of λb,r is a sort of “ρ-shift.” We let O
31
denote the category of finitely generated g-modules which are locally finite over b and
semisimple over t, with weights in the set P .
We stress that all of this notation applies only to this section. The notation
g, b, t,O, etc. has different meaning outside of this section. The category O which
we define here is the subcategory of the usual BGG category O associated to g
corresponding to integral weights. The fact that O categorifies V ⊗n was well-known
before [BLW], for example, see [CR, §7.4]. In fact, this was a motivating examples
for Losev and Webster’s definition of tensor product categorification.
2.6.1. Highest weight structure
To construct the irreducible objects in O, first define the Verma module M(b)
by
M(b) := U(g)⊗U(b) V (b)
where V (b) is the one-dimensional U(b) module associated to the t-weight λb. By
standard arguments, M(b) has irreducible head L(b), and every irreducible in O is
isomorphic to some L(b).
It is well-known that the category O is a highest weight category with poset
(B,), where the standard modules correspond to the Vermas. This fact can be
extracted from Chapters 1, 3, and 5 of [H].
2.6.2. Functors Fi and Ei
Let U denote the natural g-module of column vectors, with standard basis
u1, ..., un, and U
∗ its dual representation, with dual basis φ1, ..., φn.
If M ∈ obO, then U ⊗ M and U∗ ⊗ M are also objects of O, so we have
functors F := U ⊗ − and E := U∗ ⊗ − : O → O. There are canonical g-module
32
homomorphisms U ⊗U∗ → k and k→ U∗⊗U which induce natural transformations
ε : FE ⇒ IdO and η : IdO ⇒ EF
making the (F,E) into an adjoint pair. The symmetric braiding U ⊗ U∗ → U∗ ⊗ U
induces a natural isomorphism FE ⇒ EF , so there are also natural transformations
ε′ : EF ⇒ IdO and η′ : IdO ⇒ FE
making making (E,F ) into an adjoint pair.
For 1 ≤ r, s,≤ n, let er,s denote the corresponding matrix unit in g. The trace
form
κ : g⊗ g→ g, er,s ⊗ es′,r′ 7→ δr,r′δs,s′
defines an invariant element of the dual module (g ⊗ g)∗ = g∗ ⊗ g∗. Using the trace
form to identify g∗ with g, we see that κ becomes identified with the Casimir tensor
Ω =
∑
r,s
er,s ⊗ es,r.
Hence, Ω is invariant under the adjoint action of g on g⊗g. Working in the associative
algebra U(g) ⊗ U(g), it follows that Ω commutes with the image of the coproduct
∆ : U(g)→ U(g)⊗ U(g).
Define a natural transformation x : F ⇒ F given by letting xM : FM → FM
be endomorphism FM = U ⊗M induced by multiplication by Ω. This is a g-module
homomorphism because Ω commutes with all coproducts. Similarly, define a natural
transformation x∗ : E ⇒ E given by letting x∗M be the endomorphism of EM induced
by multiplication by −Ω. The following theorem can be extracted from [CR, §7.4.3].
33
Theorem 2.6.1. 1. For every b ∈ B, the object M := FM(b) has a filtration
0 = M0 ⊂M1 ⊂ · · · ⊂Mn = M
where Mt/Mt−1 ∼= M(b + dt) for 1 ≤ t ≤ n. Moreover, the endomorphism xM
preserves this filtration, and acts on the subquotient Mt/Mt−1 as multiplication
by the scalar bt.
2. For every b ∈ B, the object M = EM(b) has a filtration
0 = Mn ⊂Mn−1 ⊂ · · ·M0 = M
where M t−1/M t ∼= M(b− dt) for 1 ≤ t ≤ n. The endomorphism x∗M preserves
this filtration, and acts on the subquotient M t−1/M t as multiplication by the
scalar bt − 1.
Proof. (1) The existence of the filtration uses the tensor identity:
FM = U ⊗k
(
U(g)⊗U(b) V (b)
) ∼= U(g)⊗U(b) (U ⊗k V (λb)) .
As a b-module, U has an obvious filtration 0 = U0 ⊂ · · · ⊂ Un, where Ut is the
span of the vectors u1, ..., ut ∈ U . It follows that Ut/Ut−1 is the one-dimensional
b-module of weight δt. Because λb + δt = λb+dt , the b-module U ⊗ V (b) has a
filtration with subquotients of the form V (b + dt). The exactness of the functor
U(g)⊗U(b) − : U(b) -mod→ U(g) -mod implies that M = U ⊗M(b) has the desired
filtration.
34
The fact that xM preserves the filtration comes from the observation that
Ω =
1
2
(∆(C)− 1⊗ C − C ⊗ 1) (2.9)
where C =
∑
r,s er,ses,r is the central Casimir element of U(g), and ∆ is the usual
coproduct. The eigenvalues for the induced operator on Mt/Mt−1 can be calculated
using (2.9), along with the fact that C acts on any highest weight vector of weight
λ =
∑n
r=1 λrδr as multiplication by the scalar cλ :=
∑n
r=1 λ
2
r +
∑
1≤r≤s≤n(λr − λs).
Hence, xM acts on any subquotient of M isomorphic to M(b+ dt) as multiplication
by the scalar 1
2
(cλb+dt − cdt − cλb). This magically simplifies to bt.
The proof of (2) is similar.
Corollary 2.6.2. For any M ∈ obO, the object FM (resp. EM) is a finite direct
sum of generalized eigenspaces with respect to the operator xM (resp. x
∗
M). The
eigenvalues of xM and x
∗
M lie in the set I.
Proof. The theorem implies this immediately when M is a Verma module. Using
exactness of E and F , the corollary also holds for every irreducible M , and hence for
any M .
Using the corollary, the functor F decomposes as F =
⊕
i∈I Fi, where FiM is
defined to be the i-generalized eigenspace for xM : FM → FM . We have a similar
decomposition of the functor E =
⊕
i∈I Ei, where the decomposition is done by taking
eigenvalues with respect to the natural transformation x∗.
The natural transformations η and ε making (F,E) into an adjoint pair induce
and adjunction making (Fi, Ei) into an adjoint pair for any i ∈ I. Indeed, a
straightforward check shows that x∗ : E ⇒ E is the right mate to x : F ⇒ F ,
35
with respect to the adjunction (F,E). In other words, the composition
E
ηE⇒ EFE ExE⇒ EFE Eε⇒ E
is equal to x∗ : E ⇒ E. Elementary facts about adjunctions imply that ε and η
induce an adjunction making (Fi, Ei) and adjoint pair for every i ∈ I. Similarly, x∗ is
the left mate for x with respect to the adjunction (E,F ), so (Ei, Fi) is also an adjoint
pair.
2.6.3. Grothendieck group relations
Given any b ∈ B and any i ∈ I, Theorem 2.6.1 implies that the object FiM(b)
has a filtration with subquotients of the form M(b + dt), where t ranges over all
indices such that bt = i. Similarly, EiM(b) has a filtration with subquotients of the
form M(b−dt), where now t ranges over all indices with bt = i+ 1. Because they are
exact, it follows that these functors preserve the category O∆ of objects of O with a
Verma flag, and (TPC1) is satisfied.
Recall the definition of the i-signature from (2.3) for the special case where each
σr = +. The observations from the previous paragraph demonstrate that
[FiM(b)] =
∑
i-sig(b)t=f
[M(b+ dt)], [EiM(b)] =
∑
i-sig(b)t=e
[M(b− dt)].
Comparing this with (2.4), the linear isomorphism [O∆] ∼→ (V +)⊗n given by [M(b)] 7→
vb can be upgraded to an sl∞-module isomorphism, where the Chevalley generators
ei, fi of sl∞ act on [O∆] as [Ei] and [Fi], respectively. Hence, we have also checked
(TPC2).
36
2.6.4. Affine Hecke Category
The rest of this section will be devoted to the construction of a strict monoidal
functor Φ : QH → End (C), sending i ∈ obQH to Fi, for which the natural
transformation ξi : Fi ⇒ Fi is locally nilpotent, as required by (D2) and (D4). The
construction of this functor is quite subtle. To define it, we need to pass through an
intermediate category.
Definition 2.6.3. The (degenerate) affine Hecke category AH is the strict monoidal
category with generating object 1, and generating morphisms • : 1 → 1 and :
1⊗ 1→ 1⊗ 1, satisfying the relations:
= , = , • − • = .
Define AHd := HomAH(1⊗d, 1⊗d). This is is the well-known (degenerate) affine
Hecke algebra.
In contrast to the category QH, the following theorem demonstrates that the
affine Hecke category AH appears easily in our setting. It was first observed by
Arakawa-Suzuki in Theorem 2.2.2 of [AS]. Its proof is an elementary verification of
relations.
Theorem 2.6.4. There is a strict monoidal functor Ψ : AH → End (O) given by
Ψ(1) = F Ψ( • ) = x Ψ( ) = t.
Here t : F 2 ⇒ F 2 is the natural endomorphism of F 2 = U ⊗U ⊗− given by swapping
the two factors of U .
37
2.6.5. Cyclotomic quotients
In order to pass from the affine Hecke category AH to the quiver Hecke category
QH, we will need to exploit an isomophism between cyclotomic quotients of the
algebtras AHd and QHd. As this is analogous to the situation in Section 3.5 below,
we review the details here.
To define the quotients, we label strings of diagrams from right to left. Given any
1 ≤ r ≤ d, let xr denote the diagram in AHd with a dot on the rth string. Similarly,
given any i ∈ Id, let ξr1i denote the diagram in 1iQHd1i = HomQH(i, i) with a dot
on the r-th string.
Fix µ =
∑
i∈I µiεi ∈ P , and define the cyclotomic quotient AHd(µ) (resp.
QHd(µ)) to be the quotient of AHd (resp. QHd) by the two-sided ideal generated
by the the polynomial
∏
i∈I(x1 − i)µi (resp. the elements {ξµi1 1i | i ∈ Id}). By
abuse, we let xr and ξr1i denote the image of these elements in AHd(µ) and QHd(µ),
respectively.
The cyclotomic quotients are finite-dimensional algebras. By [K, Lemma 7.1.4],
the minimum polynomial of each xr (calculated in AHd(µ)) has its roots in I.
Therefore, the commutative subalgebra of AHd(µ) generated by x1, ..., xd contains a
set of mutually orthogonal idempotents {1i | i ∈ Id} projecting any AHd(µ)-module
onto its i-word space:
1iM = {m ∈M |(xr − ir)Nm = 0 for N  0}
A striking theorem of Brundan and Kleshchev [BK3], which was also noted by
Rouquier [R, Proposition 3.15], shows that there is an isomorphism of locally unital
38
algebras QHd(µ)
∼→ AHd(µ) given by
1i 7→ 1i and ξr1i 7→ (xr − ir)1i.
There is also an explicit formula for crossings which is a bit more complicated. We
will not need the explicit formula here!
We remark that whenever µ, µ′ ∈ P satisfy µi ≤ µ′i for each i, we have surjections
QHd(µ
′) QHd(µ) and AHd(µ′) AHd(µ). Hence, the sets of cyclotomic quotients
{QHd(µ) | µ ∈ P} and {AHd(µ) | µ ∈ P} each form an inverse system of locally unital
algebras with idempotents indexed by Id. Taking the inverse limit of each system,
we obtain the completions
Q̂Hd := lim←
QHd(µ), ÂHd = lim←
ÂHd.
As noted by Webster [W3], the isomorphisms of cyclotomic quotients QHd(µ)
∼→
AHd(µ) induce an isomorphism of completions Q̂Hd
∼→ ÂHd.
2.6.6. From AH to QH
Recall the monoidal functor Ψ : AH → End (O) defined above. Given any
d > 0, Ψ induces an algebra homomorphism ψd : AHd → NTd, the algebra
of natural transformations F d → F d. For any M ∈ obO, there is an induced
algebra homomorphism ψd,M : AHd → EndO(F dM). Corollary 2.6.2 shows that
ψd,M(x1) = F
d−1x acts locally finitely with eigenvalues in I. It follows that there is
some µ ∈ P such that ψd,M factors through the cyclotomic quotient AHd(µ), where
the idempotent 1i projects F
d onto the summand FiM := Fid · · ·Fi1M . Because we
have such a cyclotomic quotient for any M , it follows that there is a locally unital
39
algebra homomorphism
ψˆd : ÂHd → NTd(F )
where the algebra of natural transformations NTd is made into a locally unital
by equipping it with the distinguished idempotents 1i which project F
d onto the
summand Fi.
Pulling back along the maps QHd ↪→ Q̂Hd ∼→ ÂHd, we obtain a locally unital
algebra homomorphism ϕd : QHd → NTd. The algebra homorphism ϕd is the data
of a map between between morphisms spaces in QH and End (O). It is compatible
with the monoidal structure in these categories, so we have a monoidal functor Φ :
QH → End (O) defined on the objects of QH by Φ(i) = Fi. On a general morphism
f ∈ QHd, we define Φ(f) := ϕd(f). Hence, we have constructed the data of (D2).
The fact that dotted strings act locally nilpotently is a consequence of our formula
for the image of dotted strings under the isomorphism of cyclotomic quotients. This
completes the proof that O is a tensor product categorification of V ⊗n.
Remark 2.6.5. In early definitions of categorical actions of type A Kac-Moody
algebras (e.g. [CR]), the data of (D2) satisfying (D4) was instead replaced by the
data of:
(D2∗) A strict monoidal functor Ψ : AH → End (C) with
F := Ψ(1) x := Ψ( • ) : F ⇒ F t := Ψ( ) : F 2 ⇒ F 2.
which satisfies:
(D4∗) The functor F decomposes as F =
⊕
i∈I Fi, where Fi is defined so that FiM is
the i-generalized eigenspace for xM : FiM → FiM .
40
Hence, given the functor Ψ : AH → End (O) from theorem, we have satisfied the
axioms for a type A categorical action in the sense of Chuang-Rouquier, without
needing the additional complexity of passing to QH. While it might be more
convenient to reformulate our definitions in terms of the affine Hecke category, the
approach using the category AH and the algebras AHd has two main disadvantages:
– There is no obvious graded structure, which is essential when studying
categorical actions of quantum groups.
– There is no adaptation of AH to Kac-Moody algebras outside of type A. In
particular, we could not define categorical action of C∞ using AH.
In contrast, the category QH and its associated algebras QHd do not possess these
deficiencies.
2.7. Type A blocks revisited
Having introduced the required definitions, we revisit the classification of type
A blocks in [BD2]. Fix some σ ∈ {±}n, and z ∈ k with 2z /∈ Z. Recall the main type
A theorem of Section 1.4.1, which says that the supercategory
sO =
 sOσz if n is evensOCTσz if n is odd
decomposes as O = O ⊕ ΠO where O is a k-linear category admitting the structure
of a tensor product categorification of V ⊗σ.
To construct a second tensor product categorification of V ⊗σ, set p = #{r |
σr = +} and q = n − p. Let sO′ denote the subcategory of the BGG supercategory
O associated to glp|q(k) whose objects have integer weights. Arguments in [BLW]
41
demonstrate that the supercategory sO′ splits as O′ ⊕ ΠO′, where O′ is a k-linear
category. Using techniques similar to those employed in Section 2.6 above, Section 3
in [BLW] demonstrates that the category O′ admits the structure of a tensor product
categorification of V ⊗n. Actually, in the case where each σr = +, the category O′ is
precisely the category of gln|0(k) = gln(k) modules studied in Section 2.6.
Applying Theorem 2.5.2, there is strongly equivariant equivalence O ∼→ O′.
Hence, the categories sO and sO′ are superequivalent. In the case where n is even,
this demonstrates that every type A block is equivalent to a block in the category
sO′. In the case where n is odd, we Clifford twist and apply Lemma 1.3.1 to show
that every type A block is equivalent to the Clifford twist of a block in the category
sO′.
Because the blocks in the category sO′ have been studied extensively (e.g. [B1,
CLW, BLW]) this result has strong implications for the type A blocks. As mentioned
in Section 1.4.1, it shows that the combinatorics of the composition multiplicities
of Verma modules is controlled by the computable combinatorics of canonical bases
associated to the quantum group Uq(sl∞).
We also remark that, while the number z was important in defining sOσz, it
is irrelevant for defining the category sO′. Therefore, for any z, z′ ∈ k for which
2z, 2z′ /∈ Z, we have a superequivalence sOσz ∼→ sOσz′ .
Example 2.7.1. Suppose n = 2 and fix some σ ∈ {±}n. The set B labels the
irreducible objects in the category O ⊂ sOσz, where the irreducible L(b) has highest
weight
λb := σ1(b1 + z)δ1 + σ2(b2 + z)δ2 ∈ t∗.
Here δ1, δ2 ∈ t∗ are the coordinate functions on the diagonal matrices in q2(k), see
Section 1.4. Using standard facts about the BGG category O for gl2(k) (when σ1 =
42
σ2) and gl1|1(k) (when σ1 6= σ2), we can completely describe the type A blocks as
follows. For any type A block A, one of the following three cases must apply.
Case 1. Suppose that either:
– σ1 = σ2, and A contains an irreducible of the form L(b), where b1 = b2.
– σ1 6= σ2, and A contains an irreducible of the form L(b), where b1 +b2 6= 0.
Then, A is equivalent to the category of finite-dimensional vector spaces over
k. In particular A is semisimple and contains one irreducible object.
Case 2. Suppose that σ1 = σ2, and A contains an irreducible of the form L(b), where
b1 6= b2. Then, A is equivalent to a regular block in the BGG category O for
gl2(k). Hence, A also contains the irreducible L(b′), where b′ = wb is the tuple
obtained from b by interchanging its entries. Without loss of generality, assume
b1 > b2. Then, we have equalities
P (b) = ∆(b) and ∆(b′) = L(b′).
Moreover, the multiplicities of standard objects in projectives in A satisfies:
(P (b) : ∆(b)) = 1 (P (b) : ∆(b′)) = 0
(P (b′) : ∆(b)) = 1 (P (b′) : ∆(b′)) = 1
From these numbers, we may calculate the composition multiplicities of
standard modules using BGG reciprocity. We have the following relations in
the Grothendieck group, which we identify with V ⊗σ by [∆(b)]↔ vb:
[P (b) = vb1 ⊗ vb2 and [P (b′)] = vb2 ⊗ vb1 + vb1 ⊗ vb2 .
43
These define elements of the canonical basis of V ⊗σ. In addition the block A
is equivalent to the category mod-P of finite dimensional modules P , the path
algebra of the quiver
• •
x
y
modulo the relation xy = 0.
Case 3. Suppose that σ1 6= σ2, and A contains an irreducible of the form L(b), where
b1 + b2 = 0. This is perhaps the most interesting case of all. In this case,
A contains all irreducibles of the form L(b), where b1 = −b2. The non-zero
multiplicities of standard objects inside the projectives are given by
[P (b,−b) : ∆(b,−b)] = 1 [P (b) : ∆(b+ 1,−(b+ 1))] = 1
Again, identifying the the Grothendieck group with V ⊗σ, this means that
[P (b,−b)] = vb ⊗ v−b + vb+1 ⊗ v−(b+1)
Again, these are elements of the canonical basis.
The block A is equivalent to the category mod-Q of finite dimensional modules
over Q, the path algebra of the quiver
· · · • • • •
xi−1 xi
yi−1
xi+1
yi yi+1
· · ·
with vertex set Z, modulo the relations xiyi = −yi+1xi+1 and xi+1xi = yiyi+1 = 0
for all i ∈ Z.
44
CHAPTER III
TYPE C BLOCKS
Recall from the Chapter 1 that the type C blocks for qn(k) are the blocks of the
category sO 1
2
of qn(k)-supermodules whose weights are in the set Λ 1
2
of half-integer
weights. The goal of this chapter is to demonstrate that sum of these blocks admits
the structure of a tensor product categorification of the sp∞-module V
⊗n. Because
the results in this chapter are analogous to results for type A blocks from [BD2],
the structure of this chapter closely follows that paper, and many excerpts are taken
directly from the coauthored material. Because J. Brundan and I worked closely, it
would be impossible to separate our contributions to that paper. The results in this
chapter will be formulated into the coauthored paper [BD3].
3.1. The supercategory sO
3.1.1. Choice of square roots
Recall from Chapter 1 that k is an algebraically closed field of characteristic 0
with a fixed choice of
√−1 ∈ k. Here we also fix a choice of
√
i+ 1
2
∈ k, for each
non-negative integer i. Next, define
√
−i− 1
2
:= (−1)i√−1
√
i+ 1
2
. Thus, we have
a fixed choice of a square root of every element of Z+1
2
, and our choices satisfy the
equation √
i+
1
2
·
√
i− 1
2
=
√
−i− 1
2
·
√
−i+ 1
2
(3.1)
for every i ∈ Z. We write write
I := N, J :=
{
±
√
i+
1
2
√
i− 1
2
∣∣∣ i ∈ I} . (3.2)
45
We follow the convention that 0 ∈ N.
3.1.2. General linear superalgebras revisited
Fix some n ≥ 1, and set m := dn/2e, so that n = 2m or 2m − 1. Let ĝ denote
the Lie superalgebra gl2m|2m(k) of 4m × 4m matrices. Its natural representation of
column vectors Û has basis u1, ..., u2m of Û0¯ and u2m+1, ..., u4m of Û1¯. Write xr,s for
the rs-matrix unit in ĝ, so xr,sut = δs,tur. For 1 ≤ r, s ≤ 2m, we define
er,s := xr,s + x2m+r,2m+s, e
′
r,s := x2m+r,s + xr,2m+s, (3.3)
fr,s := xr,s − x2m+r,2m+s, f ′r,s := x2m+r,s − xr,2m+s. (3.4)
Also let
hr := er,r, h
′
r := e
′
r,r. (3.5)
Note that the elements er,s, fr,s and hr define even elements of ĝ, while the elements
e′r,s, f
′
r,s, and h
′
r are odd.
We record how these distinguished elements of ĝ act on the natural module Û .
For the sake of simplicity, write u′r for u2m+r, so that Û1¯ has basis u
′
1, ..., u
′
2m. We
have that
er,sut = δs,tur, e
′
r,sut = δs,tu
′
r, er,su
′
t = δs,tu
′
r, e
′
r,su
′
t = δs,tur, (3.6)
fr,sut = δs,tur, f
′
r,sut = δs,tu
′
r, fr,su
′
t = −δs,tu′r, f ′r,su′t = −δs,tur. (3.7)
Finally let Û∗ be the dual supermodule to Û , with basis φ1, . . . , φ2m, φ′1, . . . , φ
′
2m that
is dual to the basis u1, . . . , u2m, u
′
1, . . . , u
′
2m. The action of the distinguished elements
46
of ĝ is given by
er,sφt = −δr,tφs, er,sφ′t = −δr,tφ′s, e′r,sφt = −δr,tφ′s, e′r,sφ′t = δr,tφs, (3.8)
fr,sφt = −δr,tφs, fr,sφ′t = δr,tφ′s, f ′r,sφt = δr,tφ′s, f ′r,sφ′t = δr,tφs. (3.9)
3.1.3. Updated definitions
As implied by the statement of the main type C theorem in section 1.4.2, the type
C blocks behave differently depending on whether n is even or odd. We define a new
Lie superalgebra g to unify the two cases. When n = 2m is even, we set g = qn(k),
with standard Borel subalgebra b and Cartan subalgebra h as defined in the Chapter
1. In particular, g is the subalgebra of ĝ spanned by {er,s, e′r,s | 1 ≤ r, s ≤ n}, while
h has basis {hr, h′r | 1 ≤ r ≤ 2m}.
When n = 2m − 1 is odd, we let g denotes the Lie superalgebra qn(k) ⊕ q1(k),
which we identify with the subalgebra of ĝ spanned by {er,s, e′r,s | 1 ≤ r, s ≤ n} unionsq
{h2m, h′2m}. In this case, we must change our definitions of b and h from Chapter 1
to the following:
– b is the Borel subalgebra of g spanned by {er,s, e′r,s | 1 ≤ r ≤ s ≤ n}unionsq{h2m, h′2m};
– h is the Cartan subalgebra spanned by {hr, h′r | 1 ≤ r ≤ 2m}.
In both the even and the odd cases, the subspaces U ⊆ Û and U∗ ⊆ Û∗ spanned
by u1, . . . , un, u
′
1, . . . , u
′
n and φ1, . . . , φn, φ
′
1, . . . , φ
′
n, respectively, may be viewed as g-
supermodules. Also set t := h0¯ and let δ1, . . . , δ2m be the basis for t
∗ that is dual to
the basis h1, . . . , h2m for t. Again, when n is odd, these definitions differs from the
ones given in Chapter 1.
47
It will be convenient to index the subset of t∗ corresponding to half-integer
weights with n-tuples of integers. Set B = Zn as in Chapter 2. Given b ∈ B
and 1 ≤ r ≤ n, define λb ∈ t∗ by
λb :=

∑n
r=1 λb,rδr if n = 2m is even∑n
r=1 λb,rδr + δ2m if n = 2m− 1 is odd
(3.10)
where we write λb,r = br − 12 . Recall also from the type C combinatorics in Chapter
2 that dr denotes the element of B with a 1 in the r-th entry, and zero everywhere
else, and note that
λb±dr = λb ± δr. (3.11)
Let Λ denote the collection of elements of t∗ of the form {λb | b ∈ B}. When n is
even, Λ is the set Λ 1
2
from Chapter 1. In the odd case this is a new definition.
We let sO denote the category of g-supermodules which are:
– Finitely generated over g.
– Locally finite over b.
– Semisimple t-modules, with weights in the set Λ ⊂ t∗.
Hence, when n is even, sO is the category sO 1
2
defined in Chapter 1. When n is
odd, sO is equivalent to the supercategory (sO 1
2
)CT. Indeed, if M is a supermodule in
sO, the restriction of M to the subalgebra qn(k), equipped with the odd involution
defined by the action of h′2m, gives an object of the Clifford twist of sO 12 .
48
3.1.4. Construction of irreducibles
In contrast to the situation for reductive Lie algebras, the Cartan subalgebra h of
g is supercommutative rather than commutative. Hence, the irreducible modules over
the Cartan are not necessarily one-dimensional. We proceed to define some irreducible
h-supermodules {V (b) | b ∈ B}. Let C2 be the rank 2 Clifford superalgebra with
odd generators c1, c2 subject to the relations c
2
1 = c
2
2 = 1, c1c2 = −c2c1. Let V be the
irreducible C2-supermodule on basis v, v
′ with v even and v′ odd, and action defined
by
c1v = v
′, c1v′ = v, c2v =
√−1v′, c2v′ = −
√−1v.
Then, for b ∈ B, we set V (b) := V ⊗m. For 1 ≤ r ≤ n, we let hr act by the scalar λb,r
and h′r act by left multiplication by
√
λb,r id
⊗(s−1)⊗cr+1−2s⊗id(m−s) where s := br/2c
(and we are using the usual superalgebra sign rules). In the odd case, we also need
to define the actions of h2m and h
′
2m: these act as the identity and the odd involution
id⊗(m−1)⊗c2, respectively. In all cases, V (b) is an irreducible h-supermodule of type
M, and its t-weight is λb. Moreover, by construction, h
′
1 · · ·h′2m acts on any even (resp.
odd) vector in V (b) as cb (resp. −cb), where
cb := (
√−1)m√λb,1 · · ·√λb,n. (3.12)
The sign here distinguishes V (b) from its parity flip.
Lemma 3.1.1. For b ∈ B, any h-supermodule that is semisimple of weight λb over t
decomposes as a direct sum of copies of the supermodules V (b) and ΠV (b).
Proof. We can identify h-supermodules that are semisimple of weight λb over t with
supermodules over the Clifford superalgebra C2m := C
⊗m
2 , so that h
′
r (r = 1, . . . , n)
49
acts in the same way as
√
λb,r id
⊗(s−1)⊗cr+1−2s ⊗ id(m−s) where s := br/2c, and in
the odd case h′2m acts as id
⊗(m−1)⊗c2. The lemma then follows since C2m is simple,
indeed, it is isomorphic to the matrix superalgebra M2n−1|2n−1(k).
Let sO denote the underlying category consisting of the same objects as sO
but only the even morphisms. This is obviously an Abelian category. In order to
parametrize its irreducible objects explicitly, we introduce the Verma supermodule
M(b) for b ∈ B by setting
M(b) := U(g)⊗U(b) V (b), (3.13)
where we are viewing V (b) as a b-supermodule by inflating along the surjection b h.
The weight λb is the highest weight of M(b) in the usual dominance order on t
∗, i.e.
λ ≤ µ if and only if µ − λ ∈ ⊕n−1r=1 N(δr − δr+1). Note also that we can distinguish
M(b) from its parity flip in the same way as for V (b): the element h′1 · · ·h′2m acts on
any even (resp. odd) vector in the highest weight space M(b)λb as the scalar cb (resp.
−cb).
As usual, the Verma supermodule M(b) has a unique irreducible quotient
denoted L(b). Thus, L(b) is an irreducible g-supermodule of highest weight λb, and
the action of h′1 · · ·h′2m on its highest weight space distinguishes it from its parity flip.
The irreducible supermodules {L(b),ΠL(b) | b ∈ B} give a complete set of pairwise
inequivalent irreducible supermodules in sO. The endomorphism algebras of these
objects are all one-dimensional, so they are irreducibles of type M. Moreover, by a
standard argument involving restricting to the underlying even Lie algebra as in [B3,
Lemma 7.3], we get that sO is a Schurian category in the sense of Definition 2.1.1.
50
3.1.5. Duality on sO
Let xT denote the usual transpose of a matrix x ∈ ĝ. This induces an
antiautomorphism of g, i.e., we have that [x, y]T = [yT , xT ]. Given any M ∈ ob sO,
the definition of sO implies that M has finite dimensional weight spaces. We view the
direct sum
⊕
b∈BM
∗
λb
of the linear duals of the weight spaces of M as a g-supermodule
with action defined by (xf)(v) := f(xTv). Let M? be the object of sO obtained from
this by applying also the parity switching functor Πm. Making the obvious definition
on morphisms, this gives us a contravariant superequivalence ? : sO → sO. We have
incorporated the parity flip into this definition in order to get the following lemma.
Lemma 3.1.2. For b ∈ B, we have that L(b)? ∼= L(b) via an even isomorphism.
Proof. By weight considerations, we either have that L(b)? is evenly isomorphic to
L(b) or to ΠL(b). To show that the former holds, take an even highest weight vector
f ∈ L(b)?. We must show that h′1 · · ·h′2mf = cbf (rather than −cbf). Remembering
the twist by Πm in our definition of ?, there is a highest weight vector v ∈ L(b) of
parity m (mod 2) such that f(v) = 1. Then we get that
(h′1 · · ·h′2mf)(v) = f(h′2m · · ·h′1v) = (−1)mf(h′1 · · ·h′2mv) = cbf(v).
Hence, h′1 · · ·h′2mf = cbf .
Let P (b) be a projective cover of L(b) in sO. There are even epimorphisms
P (b)  M(b)  L(b). Applying ?, we deduce that there are even monomorphisms
L(b) ↪→ M(b)? ↪→ P (b)?. The supermodule P (b)? is an injective hull of L(b), while
M(b)? is the dual Verma supermodule.
51
3.1.6. Verma flags
The following lemma is well known; it follows from central character
considerations (e.g. see [CW, Theorem 2.48]) plus the universal property of Verma
supermodules.
Lemma 3.1.3. Suppose that λb is dominant and typical, i.e., whenever 1 ≤ r < s ≤
n, we have λb,r > λb,s and λb,r + λb,s 6= 0. Then P (b) = M(b).
Remark 3.1.4. The condition λb,r > λb,s is equivalent to br > bs, while the condition
λb,r + λb,s 6= 0 is equivalent to br + bs 6= 1.
Let sO∆ be the full subcategory of sO consisting of all supermodules possessing
a Verma flag, i.e., for which there is a finite filtration 0 = M0 ⊂ · · · ⊂ Ml = M with
subquotients Mk/Mk−1 evenly isomorphic to M(b)’s or ΠM(b)’s for b ∈ B. Since the
classes of all M(b) and ΠM(b) are linearly independent in the Grothendieck group
of the underlying category sO, the multiplicities (M : M(b)) and (M : ΠM(b)) of
M(b) and ΠM(b) in a Verma flag of M are independent of the particular choice of
flag. The following lemma is quite standard.
Lemma 3.1.5. For M ∈ ob sO∆ and b ∈ B, we have that
(M : M(b)) = dim HomsO(M,M(b)?)0¯,
(M : ΠM(b)) = dim HomsO(M,M(b)?)1¯.
Also, any direct summand of M ∈ ob sO∆ possesses a Verma flag.
Proof. The first part of the lemma follows by induction on the length of the Verma
flag, using the following two observations: for all a, b ∈ B we have that
52
– HomsO(M(a),M(b)?) is zero if a 6= b, and it is one-dimensional of even parity
if a = b;
– Ext1sO(M(a),M(b)
?) = 0.
To check these, for the first one, we use the universal property of M(a) to see that
HomsO(M(a),M(b)?) is zero unless λa ≤ λb. Similarly, on applying ?, it is zero unless
λb ≤ λa. Hence, we may assume that a = b. Finally, due to weight considerations,
any non-zero homomorphism M(a) → M(a)? must send the head to the socle, so
HomsO(M(a),M(a)?) is evenly isomorphic to HomsO(L(a), L(a)?), which is one-
dimensional and even thanks to Lemma 3.1.2. For the second property, we must
show that all short exact sequences in sO of the form
0→M(a)? →M →M(b)→ 0 or 0→ ΠM(a)? →M →M(b)→ 0
split. Either λa or λb is a maximal weight of M . In the latter case, using also
Lemma 3.1.1, we can use the universal property of M(b) to construct a splitting of
M M(b). In the former case, we apply ?, the resulting short exact sequence splits
as before, and then we dualize again.
The final statement of the lemma may be proved by mimicking the argument for
semisimple Lie algebras from [H, §3.2].
3.2. Special projective superfunctors
Next, we investigate the superfunctors U ⊗ − and U∗ ⊗ − defined by tensoring
with the g-supermodules U and U∗ introduced in the previous section. They clearly
preserve the properties of being finitely generated over g, locally finite-dimensional
over b, and semisimple over t. Since the t-weights of U and U∗ are δ1, . . . , δn and
53
−δ1, . . . ,−δn, respectively, and using (3.11), we get for each M ∈ ob sO that all
weights of U⊗M and U∗⊗M are of the form λb for b ∈ B. Hence, these superfunctors
send objects of sO to objects of sO, i.e. we have defined
sF := U ⊗− : sO → sO, sE := U∗ ⊗− : sO → sO. (3.14)
Let
ω :=
n∑
r,s=1
(
fr,s ⊗ es,r − f ′r,s ⊗ e′s,r
) ∈ U(ĝ)⊗ U(g). (3.15)
Left multiplication by ω (resp. by −ω) defines a linear map xM : U ⊗M → U ⊗M
(resp. x∗M : U
∗ ⊗M → U∗ ⊗M) for each g-supermodule M . In view of the next
lemma, these maps define a pair of even supernatural transformations
x : sF ⇒ sF, x∗ : sE ⇒ sE. (3.16)
Lemma 3.2.1. The linear maps xM and x
∗
M just defined are even g-supermodule
homomorphisms.
Proof. This is straightforward to verify directly, but we give a more conceptual
argument which better explains the origin of these maps. The odd element
f ′ :=
n∑
t=1
f ′t,t ∈ U(ĝ) (3.17)
supercommutes with the elements of U(g). Hence, f ′ ⊗ 1 ∈ U(ĝ) ⊗ U(g)
supercommutes with the image of the comultiplication ∆ : U(g) → U(g) ⊗ U(g) ⊂
54
U(ĝ)⊗ U(g). The odd Casimir tensor
Ω′ :=
n∑
r,s=1
(
er,s ⊗ e′s,r − e′r,s ⊗ es,r
) ∈ U(g)⊗ U(g)
also supercommutes with the image of ∆. Hence, the even tensor
Ω := Ω′(f ′ ⊗ 1) = −
n∑
r,s,t=1
(
er,sf
′
t,t ⊗ e′s,r + e′r,sf ′t,t ⊗ es,r
) ∈ U(ĝ)⊗ U(g)
commutes with the image of ∆. Consequently, left multiplication by Ω defines even
g-supermodule endomorphisms xM : U ⊗M → U ⊗M and x∗M : U∗⊗M → U∗⊗M .
It remains to observe that these endomorphisms agree with the linear maps defined
by left multiplication by ω and −ω, respectively. Indeed, by a calculation using (3.6)–
(3.9), the elements er,sf
′
t,t and e
′
r,sf
′
t,t of U(ĝ) act on vectors in U (resp. U
∗) in the
same way as δs,tf
′
r,s and −δs,tfr,s (resp. −δr,tf ′r,s and δr,tfr,s), respectively.
Lemma 3.2.2. Suppose that b ∈ B and let M := M(b).
1. There is a filtration
0 = M0 ⊂M1 ⊂ · · · ⊂Mn = U ⊗M
with Mt/Mt−1 ∼= M(b + dt) ⊕ ΠM(b + dt) for each t = 1, . . . , n. The
endomorphism xM preserves this filtration, and the induced endomorphism
of the subquotient Mt/Mt−1 is diagonalizable with exactly two eigenvalues
±√λb,t√λb,t + 1. Its √λb,t√λb,t + 1-eigenspace is evenly isomorphic to M(b+
dt), while the other eigenspace is evenly isomorphic to ΠM(b+ dt).
55
2. There is a filtration
0 = Mn ⊂ · · · ⊂M1 ⊂M0 = U∗ ⊗M
with M t−1/M t ∼= M(b − dt) ⊕ ΠM(b − dt) for each t = 1, . . . , n. The
endomorphism x∗M preserves this filtration, and the induced endomorphism
of the subquotient M t−1/M t is diagonalizable with exactly two eigenvalues
±√λb,t√λb,t − 1. Its √λb,t√λb,t − 1-eigenspace is evenly isomorphic to M(b−
dt), while the other eigenspace is evenly isomorphic to ΠM(b− dt).
Proof. (1) The filtration is constructed in [B2, Lemma 4.3.7], as follows. By the
tensor identity
U ⊗M = U ⊗ (U(g)⊗U(b) V (b)) ∼= U(g)⊗U(b) (U ⊗ V (b)).
As a b-supermodule, U has a filtration 0 = U0 ⊂ U1 ⊂ · · · ⊂ Un = U in which the
section Ut/Ut−1 is spanned by the images of ut and u′t. Let Mt be the submodule of
U ⊗M that maps to U(g)⊗U(b) (Ut ⊗ V (λ)) under this isomorphism.
Now fix t ∈ {1, . . . , n}. Let v1, . . . , vk be a basis for the even highest weight space
M(b)λb,0¯, so that h
′
tv1, . . . , h
′
tvk is a basis for M(b)λb,1¯. The subquotient Mt/Mt−1 ∼=
U(g) ⊗U(b) (Ut/Ut−1 ⊗ V (b)) is generated by the images of the vectors {ut ⊗ vi, u′t ⊗
vi, ut ⊗ h′tvi, u′t ⊗ h′tvi | i = 1, . . . , k}, which by weight considerations span a b-
supermodule isomorphic to V (b+ dt)⊕ ΠV (b+ dt). Hence,
Mt/Mt−1 ∼= M(b+ dt)⊕ ΠM(b+ dt).
56
The action of fr,s⊗es,r−f ′r,s⊗e′s,r on any of ut⊗vi, u′t⊗vi, ut⊗h′tvi or u′t⊗h′tvi is zero
unless r ≤ s = t, and if r < s = t then it sends these vectors into Mt−1. Therefore,
xM preserves the filtration. Moreover, this argument shows that it acts on the highest
weight space of the quotient Mt/Mt−1 in the same way as xt := ft,t ⊗ ht − f ′t,t ⊗ h′t.
Now consider the purely even subspace Si,t of Mt/Mt−1 with basis given by the
images of ut ⊗ vi, u′t ⊗ h′tvi. Recalling that ht acts on vi and on h′tvi by λb,t, and that
(h′t)
2 = ht, it is straightforward to check that the matrix of the endomorphism xt of
Si,t in the given basis is equal to
A :=
 λb,t λb,t
1 −λb,t
 .
Also recall from our construction of V (b) that h′1 · · ·h′2m acts on vi as the scalar cb
from (3.12), and it acts on h′tvi as −cb. Using this, another calculation shows that
h′1 · · ·h′2m acts on Si,t as the matrix cbλb,tA. Similarly, on the purely odd subspace S ′i,t
with basis given by the images of u′t ⊗ vi, ut ⊗ h′tvi, xt has matrix −A and h′1 · · ·h′2m
has matrix − cb
λb,t
A.
Since the matrix A has eigenvalues ±√λb,t√λb,t + 1, the calculation made in
the previous paragraph implies that xt is diagonalizable on Mt/Mt−1 with exactly
these eigenvalues. Moreover on any even highest weight vector in its
√
λb,t
√
λb,t + 1-
eigenspace, we get that h′1 · · ·h′2m acts as
cb
λb,t
√
λb,t
√
λb,t + 1 = cb+dt .
This implies that the
√
λb,t
√
λb,t + 1-eigenspace is evenly isomorphic to M(b + dt).
Similarly, the −√λb,t√λb,t + 1-eigenspace is evenly isomorphic to ΠM(b+ dt).
57
(2) Similar.
Corollary 3.2.3. For M ∈ ob sO, all roots of the minimal polynomials of xM and x∗M
(computed in the finite dimensional superalgebras EndsO(sF M) and EndsO(sEM))
belong to the set J from (3.2).
Proof. This is immediate from the theorem in case M is a Verma supermodule. We
may then deduce that it is true for all irreducibles, hence, for any M ∈ ob sO.
Corollary 3.2.3 implies that we can decompose
sF =
⊕
j∈J
sFj, sE =
⊕
j∈J
sEj, (3.18)
where sFj (resp. sEj) is the subfunctor of sF (resp. sE) defined by letting sFjM
(resp. sEjM) be the generalized j-eigenspace of xM (resp. x
∗
M) for each M ∈ ob sO.
For i ∈ I, recall the definition of i-sig(b) from (2.6). The following theorem relates
the combinatorics for the Verma subquotients of sFjM(b) with the summands of
fivb ∈ V ⊗n, see (2.7).
Theorem 3.2.4. Given b ∈ B and i ∈ I, let j :=
√
i− 1
2
√
i+ 1
2
. Then:
(1) sFjM(b) has a multiplicity-free filtration with sections that are evenly
isomorphic to the Verma supermodules
{M(b+ dt) | for 1 ≤ t ≤ n such that i-sig(b)t = f}.
(2) sEjM(b) has a multiplicity-free filtration with sections that are evenly
isomorphic to the Verma supermodules
{M(b− dt) | for 1 ≤ t ≤ n such that i-sig(b)t = e}.
58
Proof. (1) It is immediate from Lemma 3.2.2 that sFjM(b) has a multiplicity-free
filtration with sections that are evenly isomorphic to the supermodules M(b+dt) for
t = 1, . . . , n such that
√
λb,t
√
λb,t + 1 = j. Squaring both sides, this equation implies
that (λb,t + 12)
2 = i2. Hence,
λb,t = bt − 1
2
= −1
2
± i
We deduce that bt = ±i. Since we squared our original equation, it remains to check
that we do indeed get solutions to that in both cases. This follows from 3.1.
(2) Similar.
The superfunctors sF and sE are both left and right adjoint to each other via
some canonical (even) adjunctions. The adjunction making (sE, sF ) into an adjoint
pair is induced by the linear maps
ε : U∗ ⊗ U → k, φ⊗ u 7→ φ(u), η : k→ U ⊗ U∗, 1 7→
n∑
r=1
(ur ⊗ φr + u′r ⊗ φ′r).
Thus, the unit of adunction c : 1⇒ sF sE is defined on supermodule M by the map
cM : M
can−→ k⊗M η⊗id−→ U ⊗ U∗ ⊗M , and the counit of adjunction d : sE sF ⇒ 1 is
defined by dM : U
∗ ⊗ U ⊗M ε⊗id−→ k⊗M can−→ M . Similarly, the adjunction making
(sF, sE) into an adjoint pair is induced by the linear maps
U ⊗ U∗ → k, u⊗ φ 7→ (−1)|φ||u|φ(u), k→ U∗ ⊗ U, 1 7→
n∑
r=1
(φr ⊗ ur − φ′r ⊗ u′r).
The following lemma implies that these adjunctions restrict to adjunctions making
(sFj, sEj) and (sEj, sFj) into adjoint pairs for each j ∈ J . It follows that all of these
59
superfunctors send projectives to projectives, and they are all exact, i.e. they preserve
short exact sequences in sO.
Lemma 3.2.5. The supernatural transformation x∗ : sE ⇒ sE is both the left and
right mate of x : sF ⇒ sF with respect to the canonical adjunctions defined above.
Proof. We just explain how to check that x∗ is the left mate of x with respect to the
adjunction (sE, sF ); the argument for right mate is similar. We need to show for
each M ∈ ob sO that the composition
U∗ ⊗M id⊗cM−→ U∗ ⊗ U ⊗ U∗ ⊗M id⊗xU∗⊗M−→ U∗ ⊗ U ⊗ U∗ ⊗M dU∗⊗M−→ U∗ ⊗M
is equal to x∗M : U
∗ ⊗ M → U∗ ⊗ M . Recall for this that x∗M is defined by
left multiplication by
∑n
r,s=1
(
f ′r,s ⊗ e′s,r − fr,s ⊗ es,r
)
, while xU∗⊗M is defined by left
multiplication by
∑n
r,s=1(fr,s⊗ es,r⊗ 1 + fr,s⊗ 1⊗ es,r− f ′r,s⊗ e′s,r⊗ 1− f ′r,s⊗ 1⊗ e′s,r).
Now one computes the effect of both maps on homogeneous vectors of the form φt⊗v
and φ′t ⊗ v using (3.6)–(3.9).
3.3. Bruhat order revisited
In this section we describe the irreducible subquotients of M(b) in terms of the
Bruhat order. Recall the type C combinatorics from Section 2.2. In particular, for the
remainder of this dissertation, we let P denote the weight lattice for sp∞, containing
elements {εi | i ∈ I} and simple roots α0 = −2ε0 and αi = εi−1 − εi for i > 0. We
also have the dominance ordering E on P and the type C Bruhat ordering  on the
set B.
60
We begin with several technical lemmas to make the ordering more concrete.
Using these lemmas, we can prove in particular that whenever a  b, it is also true
that λa ≤ λb in the usual dominance ordering on t∗.
Lemma 3.3.1. Let β =
∑
i∈I βiεi and γ =
∑
i∈I γiεi be elements of P . Then, β E γ
if and only if the sum
∑
i≥j (βi − γi) is positive for every j ∈ I, and in addition, when
j = 0, the sum is even.
Proof. First suppose that β E γ, so that γ − β = ∑i∈I(γi − βi)εi is a (finite) sum∑
i∈I hiαi where each hi ≥ 0. Writing each αi in terms of the εj’s, we see that
γi − βi =
 h1 − 2h0 if i = 0hi+1 − hi if i > 0 , hence
∑
i≥j
(βi − γi) =
 2h0 if j = 0hj if j > 0 .
This proves the sums are positive for any j ∈ I, and even when j = 0. The other
implication is proved by reversing these steps.
Lemma 3.3.2. For a ∈ B, i ∈ I and 1 ≤ s ≤ n, define
N[1,s](a, i) := #{1 ≤ r ≤ s | ar > i} −#{1 ≤ r ≤ s | ar ≤ −i}.
Then a  b if and only if
– N[1,n](a, i) = N[1,n](b, i) for all i ∈ I;
– N[1,s](a, 0) ≡ N[1,s](b, 0) (mod 2) for each s = 1, . . . , n− 1;
– N[1,s](a, i) ≥ N[1,s](b, i) for all i ∈ I and s = 1, . . . , n− 1.
Proof. Given a ∈ B and 1 ≤ s ≤ n, write wts(a) =
∑
i∈I βiεi. For j ∈ I, the number
N[1,s](a, j) is precisely the sum
∑
j≥i βi. The lemma then follows from the definition
of , along with Lemma 3.3.1.
61
Recall from (2.7) the Chevalley generators of sp∞ act on the monomial basis of
V ⊗n by
fivb =
∑
i-sig(b)t= f
vb+dt .
The following lemmas explores the interaction between the action of fi and the Bruhat
ordering.
Lemma 3.3.3. Suppose that a  b and i-sig(a)r = i-sig(b)n = f for some i ∈ I and
1 ≤ r ≤ n. Then a+ dr  b+ dn, with equality if and only if a = b and r = n.
Proof. We use the conditions from Lemma 3.3.2. For either j 6= i and 1 ≤ s ≤ n, or
j = i and 1 ≤ s < r, we have that
N[1,s](a+ dr, j) = N[1,s](a, j) ≥ N[1,s](b, j) = N[1,s](b+ dn, j).
When j = i and r ≤ s < n, we have that:
N[1,s](a+ dr, i) = N[1,s](a, i) + 2
δi,0 ≥ N[1,s](b, i) + 2δi,0 > N[1,s](b, i)
= N[1,s](b+ dn, i).
Finally,
N[1,n](a+ dr, i) = N[1,n](a, i) + 1 = N[1,n](b, i) + 1
= N[1,n](b+ dn, i).
62
Lemma 3.3.4. Suppose we are given b ∈ B. Define a ∈ B by setting a1 := b1, and
then inductively define each as for s = 2, . . . , n to be the greatest integer such that
as ≤ bs and the following hold for all 1 ≤ r < s:
– The entry as < ar;
– The entry as ≤ −br.
Define a monomial X = Xn · · ·X2 in the Chevalley generators {fi | i ∈ I} by setting
Xr :=

fbr−1fbr−2 · · · far+1far if ar ≥ 0,
f1−brf2−br · · · f−ar if br ≤ 0,
fbr−1 · · · f1f0f1 · · · f−ar if br > 0 > ar,
for each r = 1, . . . , n. Then, working in the sp∞-module V
⊗n, we have that
Xva = vb + (a sum of vc’s for c  b),
Proof. We proceed by induction on n, where the base case n = 1 is trivial. When
n > 1, define b¯ = (b1, ..., bn−1), a¯ = (a1, ..., an−1), X¯ = Xn−1 · · ·X2. Applying the
induction hypothesis in the sp∞-module V
⊗n−1, we can write
X¯va¯ = vb¯ + (a sum of vc¯’s for c¯  b¯)
We observe that whenever fi appears as a factor of some Xr with r < n, then
fivan = 0. To prove this, we need to demonstrate that an 6= ±i. This follows from
the definition of the monomials Xr:
– When ar ≥ 0, we have ar ≤ i ≤ br − 1, or equivalently, 1 − br ≤ −i ≤ ar ≤ 0.
Because an ≤ −br, it follows that an < −i.
63
– When, br ≤ 0, we have 1 − br ≤ i ≤ −ar, or equivalently, ar ≤ −i ≤ br − 1.
Because an < ar, it follows that an < −i.
– When ar < 0 < br, we have i ≤ max{−ar, br − 1}, or equivalently, −i ≥
min{ar,−br + 1}. Because an < ar and −br + 1, it follows that an < −i.
Therefore, if b˜ = (b1, ..., bn−1, an), then
X¯va = vb˜ + (a sum of vc where c  b˜)
Lastly, we act with Xn, which sends van to vbn , and apply Lemma 3.3.3.
Example 3.3.5. If b = (2, 0,−1, 0, 2, 0,−1) then a = (2,−2,−3,−4,−5,−6,−7)
and X = (f2f3f4f5f6f7)(f1f2f3f4f5f6)(f1f0f1f2f3f4f5)(f1f2f3f4)(f2f3)(f1f2).
Theorem 3.3.6. For every b ∈ B, the indecomposable projective supermodule P (b)
has a Verma flag with top section evenly isomorphic to M(b) and other sections evenly
isomorphic to M(c)’s for c ∈ B with c  b.
Proof. Let notation be as in the statement of Lemma 3.3.4. Let i1, . . . , il ∈ I be
defined so that X is the monomial fil · · · fi2fi1 . Let jk :=
√
ik − 12
√
ik +
1
2
for each k
and consider the supermodule
P := sFjl · · · sFj2sFj1M(a).
For each 1 ≤ r < s ≤ n, we have that ar > as. In addition, as ≤ −br ≤ −ar, so
as + ar < 1. This implies that the weight λa is typical and dominant, hence M(a)
is projective by Lemma 3.1.3. Since each sFj is left adjoint to the exact functor
sEj, it sends projectives to projectives, and we deduce that P is projective. The
64
combinatorics for how the Chevalley generators fi act on the elements vb from (2.7)
matches that of Theorem 3.2.4, we can reinterpret Lemma 3.3.4 as saying that P has a
Verma flag with M(b) as a subquotient, and all other subquotients evenly isomorphic
to M(c)’s for c  b. Actually, the order of induction from Lemma 3.3.4 constructs
M(b) as a quotient of P , hence P (b) is evenly isomorphic to a summand of P , and
it just remains to apply Lemma 3.1.5.
Corollary 3.3.7. For b ∈ B, we have that [M(b) : L(b)] = 1. All other composition
factors of M(b) are evenly isomorphic to L(c)’s for c ≺ b.
Proof. This follows immediately from Theorem 3.3.6 and the following analog of BGG
reciprocity :
[M(b) : L(c)] = HomsO(P (c),M(b)?)0¯ = (P (c) : M(b))
[M(b) : ΠL(c)] = HomsO(P (c),M(b)?)1¯ = (P (c) : ΠM(c))
These equalities are given by Lemma 3.1.2 and 3.1.5.
Corollary 3.3.8. For any b ∈ B, every irreducible subquotient of the indecomposable
projective P (b) is evenly isomorphic to L(a) for a ∈ B with wt(a) = wt(b).
Proof. By Theorem 3.3.6, P (b) has a Verma flag with sections M(c) for c  b. By
Corollary 3.3.7, the composition factors of M(c) are L(a)’s for a  c. Hence, every
irreducible subquotient of P (b) is evenly isomorphic to L(a) for a ∈ B such that
a  c  b for some c. This condition implies that wt(a) = wt(b).
65
3.4. Weak categorical action
Let O be the Serre subcategory of sO generated by the supermodules
{L(b) | b ∈ B},
i.e. it is the full subcategory of sO consisting of all supermodules whose composition
factors are evenly isomorphic to L(b)’s for b ∈ B (and not ΠL(b)’s). Since each L(b)
is of type M, there are no non-zero odd morphisms between objects of O. Because of
this, we forget the super and view O as a k-linear category.
Theorem 3.4.1. We have that sO = O ⊕ ΠO in the sense of Chapter 1.
Proof. Let ΠO be the Serre subcategory of sO generated by the supermodules
{ΠL(a) | a ∈ B}. By Corollary 3.3.8, all even extensions between ΠL(a) and L(b)
are split. Hence, every supermodule in sO decomposes uniquely as a direct sum of
an object of O and an object of ΠO. The result follows.
Theorem 3.4.2. The category O is a highest weight category in the sense of
Definition 2.1.3, with weight poset (B,). Its standard objects are the Verma
supermodules {M(b) | b ∈ B}.
Proof. It is clear that O is a Schurian category with isomorphism classes of irreducible
objects represented by {L(b) | b ∈ B}. By Theorem 3.3.6, P (b) has a Verma flag
with M(b) at the top and other sections that are evenly isomorphic to M(c)’s for
c  b. It just remains to observe that the Verma supermodules M(b) coincide
with the standard objects ∆(b). This follows using the filtration just described plus
Corollary 3.3.7.
66
Remark 3.4.3. By Lemma 3.1.2, the duality ? on sO restricts to a duality ? : O → O
fixing isomorphism classes of irreducible objects.
Next, take i ∈ I and set j :=
√
i− 1
2
·
√
i+ 1
2
. Theorem 3.2.4 implies that the
exact functors sFj and sEj send the standard objects in O to objects of O with a
∆-flag. Using this, we deduce that they send irreducibles in O to irreducibles in O,
and then that they send arbitrary object of O to objects in O. Thus, their restrictions
define exact endofunctors
Fi := sFj|O : O → O, Ei := sEj|O : O → O. (3.19)
Again, these functors are biadjoint. Let O∆ be the full subcategory of O consisting
of all objects possessing a Verma flag. This is an exact subcategory of O. Its
complexified Grothendieck group C⊗ZK0(O∆) has basis {[M(b)] | b ∈ B}.
Theorem 3.4.4. For each i ∈ I, the functors Fi and Ei are exact endofunctors of
O∆. Moreover, if we identify C⊗ZK0(O∆) with V ⊗n so that [M(b)] ↔ vb for each
b ∈ B, then the induced endomorphisms [Fi] and [Ei] of the Grothendieck group act
in the same way as the Chevalley generators fi and ei of sp∞.
Proof. Compare Theorem 3.2.4 with (2.7).
Thus, we have constructed a highest weight category O with weight poset , and
equipped it with biadjoint endofunctors Ei and Fi for every i ∈ I, which induce an
action of sp∞ on the Grothendieck group. This gives us the data of a weak categorical
action of sp∞ in the sense of [CR, R].
67
3.5. Strong categorical action
In this section, we upgrade the weak categorical action of sp∞ on O constructed
so far to a strong categorical action. Recalling (D2) and (D4) from the definition of
categorical actions (Definition 2.4.1), we must prove the following:
Theorem 3.5.1. There is a strict monoidal functor Φ : QH → End (O) sending the
generating objects i ∈ I to the endofunctors Fi from (3.19). Moreover, for all M ∈
obO and i ∈ I, the endomorphism FiM → FiM defined by the natural transformation
Φ
(
•
i
)
is nilpotent.
Here, the construction of Φ is similar to the construction of the monoidal functor
Φ : QH → End (O) in Subsection (2.6.4)–(2.6.6). In those subsections O was a
category of gln(k)-modules, and Φ was constructed by first defining a “obvious”
monoidal functor Ψ : AH → End (O′) and then exploiting some isomorphism theorems
to pass to QH.
In our present situation, the construction of Φ : QH → End (O) is analogous, but
even more subtle. We begin by producing an easily-defined monoidal superfunctor
Ψ : AHC → End (sO), where AHC is the affine Hecke-Clifford supercategory (defined
below), and End (sO) is the monoidal supercategory whose objects are superfunctors
sO → sO, and whose morphisms are supernatural transformations. From there,
we exploit a remarkable isomorphism theorem of [KKT] to produce a monoidal
superfunctorQHC → End (sO), whereQHC is the quiver Hecke-Clifford supercategory
(also defined below). Lastly, we realize QH as a full subcategory of QHC with only
even morphisms to obtain the functor Φ required by Theorem 3.5.1.
68
3.5.1. Intermediate categories
Both AHC and QHC are examples of (strict) monoidal supercategories, meaning
that they are supercategories equipped with a monoidal product in an appropriate
enriched sense. We refer the reader to the introduction of [BE1] for the precise
definition, just recalling that morphisms in a monoidal supercategory satisfy the super
interchange law rather than the usual interchange law of a monoidal category: in terms
of the string calculus as in [BE1] we have that
g
f = gf = (−1)|f ||g| gf (3.20)
for homogeneous morphisms f and g of parities |f | and |g|, respectively.
Definition 3.5.2. The (degenerate) affine Hecke-Clifford supercategory AHC is the
strict monoidal supercategory with a single generating object 1, even generating
morphisms • : 1 → 1 and : 1 ⊗ 1 → 1 ⊗ 1, and an odd generating morphism
◦ : 1→ 1. These are subject to the following relations:
◦• = − •◦ , ◦◦ = , = ,
◦ = ◦
• − • = − ◦◦ , = .
Denoting the object 1⊗d ∈ obAHC simply by d, the (degenerate) affine Hecke-Clifford
superalgebra is the superalgebra
AHCd := EndAHC(d). (3.21)
69
This was introduced originally by Jones and Nazarov [JN].
For a supercategory C, we write End (C) for the strict monoidal supercategory
consisting of superfunctors and supernatural transformations.
Theorem 3.5.3. There is a strict monoidal superfunctor Ψ : AHC → End (sO)
sending the generating object 1 to the endofunctor sF = U ⊗− from (3.14), and the
generating morphisms • , ◦ and to the supernatural transformations x, c and t
which are defined on M ∈ ob sO as follows:
– xM : U ⊗M → U ⊗M is left multiplication by the tensor ω from (3.15);
– cM : U ⊗M → U ⊗M is left multiplication by
√−1 f ′ ⊗ 1 for f ′ as in (3.17);
– tM : U ⊗ U ⊗M → U ⊗ U ⊗M sends u⊗ v ⊗m 7→ (−1)|u||v|v ⊗ u⊗m.
Proof. This an elementary check of relations, similar to the one made in the proof of
[HKS, Theorem 7.4.1].
Definition 3.5.4. The quiver Hecke-Clifford supercategory of type sp∞ is the
monoidal supercategory QHC with objects generated by the set J , even generating
morphisms •
j1
: j1 → j1 and
j2 j1
: j2⊗ j1 → j1⊗ j2, and odd generating morphisms
◦
j1
: j1 → −j1, for all j1, j2 ∈ J . These are subject to the following relations:
•◦
j1
= − ◦•
j1
, ◦◦
j1
=
j1
, ◦
j2 j1
= ◦
j2 j1
, ◦
j2 j1
= ◦
j2 j1
,
70
j2 j1
• −
j2 j1
•
=

j2 j1
if j1 = j2,
◦ ◦
j2 j1
if j1 = −j2,
0 otherwise;
j2 j1
• −
j2 j1
• =

j2 j1
if j1 = j2,
− ◦ ◦
j2 j1
if j1 = −j2,
0 otherwise;
j2 j1
=

0 if i1 = i2,
j2 j1
if |i1 − i2| > 1;
−κ1
j2 j1
• + κ2
j2 j1
•• if i1 = 0, i2 = 1,
κ1
j2 j1
•• − κ2
j2 j1
• if i1 = 1, i2 = 0
κ1(i1 − i2)
j2 j1
• + κ2(i2 − i1)
j2 j1
• if |i1 − i2| = 1, i1, i2 6= 0,
71
j3 j2 j1
−
j3 j2 j1
=

κ1
j3 j2 j1
• + κ1
j3 j2 i1
• if j1 = j3, i1 = 1, i2 = 0,
κ2
j3 j2 j1
•◦ ◦ + κ1
j3 j2 i1
◦ ◦• if j1 = −j3, i1 = 1, i2 = 0,
κ1(i1 − i2)
j3 j2 j1
if j1 = j3, |i1 − i2| = 1, i2 6= 0,
κ1(i2 − i1) ◦ ◦
j3 j2 j1
if j1 = −j3, |i1 − i2| = 1, i2 6= 0,
0 otherwise.
In the above, we have adopted the convention given jr ∈ J that ir ∈ I and κr ∈ {±1}
are defined from jr = κr
√
ir − 12
√
ir +
1
2
. Identifying the word j = jd · · · j1 ∈ Jd with
jd ⊗ · · · ⊗ j1 ∈ obQHC, the quiver Hecke-Clifford superalgebra is the locally unital
algebra
QHCd :=
⊕
j,j′∈Jd
HomQHC(j, j
′). (3.22)
This is exactly as in [KKT, Definition 3.5] in the special case of the sp∞-quiver.
3.5.2. Isomorphisms of completions
As stated at the beginning of the section, we are going to exploit a remarkable
isomorphism theorem between certain completions ÂHCd and Q̂HCd of the
superalgebras superalgebras AHCd and QHCd from (3.21) and (3.22), which was
constructed in [KKT]. To define these, we need some further notation.
Numbering strands of a diagram by 1, . . . , d from right to left, AHCd is generated
by its elements xr, cr (1 ≤ r ≤ d) and tr (1 ≤ r < d) corresponding to the closed dot on
the rth strand, the open dot on the rth strand, and the crossing of the rth and (r+1)th
strands, respectively. Let HCd := Sd n Cd be the Sergeev superalgebra, that is, the
smash product of the symmetric group Sd with basic transpositions t1, . . . , td−1 acting
72
on the Clifford superalgebra Cd on generators c1, . . . , cd. Let Ad denote the purely even
polynomial superalgebra k[x1, . . . , xd]. Then the natural multiplication map gives a
superspace isomorphism HCd ⊗ Ad ∼→ AHCd. Transporting the multiplication on
AHCd to HCd ⊗ Ad via this isomorphism, the following describe how to commute a
polynomial f ∈ Ad past the generators of HCd:
(1⊗ f)(cr ⊗ 1) = cr ⊗ cr(f), (3.23)
(1⊗ f)(tr ⊗ 1) = tr ⊗ tr(f) + 1⊗ ∂r(f) + crcr+1 ⊗ ∂˜r(f), (3.24)
for operators cr, tr, ∂r, ∂˜r : Ad → Ad such that
– tr is the automorphism that interchanges xr and xr+1 and fixes all other
generators;
– cr is the automorphism that sends xr 7→ −xr and fixes all other generators;
– ∂r is the Demazure operator ∂r(f) :=
tr(f)−f
xr−xr+1 ;
– ∂˜r is the twisted Demazure operator cr+1 ◦ ∂r ◦ cr, so ∂˜r(f) = tr(f)−cr+1(cr(f))xr+xr+1 .
Given a tuple µ = (µi)i∈I of non-negative integers all but finitely many of which are
zero, the quotient superalgebra
AHCd(µ) := AHCd
/〈∏
i∈I
(
x21 − i2 +
1
4
)µi 〉
(3.25)
is a (degenerate) cyclotomic Hecke-Clifford superalgebra in the sense of [BK2, §3.e].
It is finite dimensional. Moreover, all roots of the minimal polynomials of all xr ∈
AHCd(µ) belong to the set J . It follows for each j = jd · · · j1 in the set Jd of
words of length d in letters J that there is an idempotent 1j ∈ AHCd(µ) defined
73
by the projection onto the simultaneous generalized eigenspaces for x1, . . . , xd with
eigenvalues j1, . . . , jd, respectively. Moreover, we have that
AHCd(µ) =
⊕
j,j′∈Jd
1j′AHCd(µ)1j .
If µ ≤ µ′, i.e. µi ≤ µ′i for all i, there is a canonical surjection AHCd(µ′) AHCd(µ)
sending xr, cr, tr, 1j ∈ AHCd(µ′) to the elements of AHCd(µ) with the same names.
Let
ÂHCd := lim←−
µ
AHCd(µ) (3.26)
be the inverse limit of this system of superalgebras taken in the category of locally
unital superalgebras with distinguished idempotents indexed by Jd. Using the basis
theorem for the cyclotomic quotients AHCd(µ) from [BK2, §3-e], one can identify
ÂHCd with the completion defined in [KKT, Definition 5.3]
1. In particular, letting
Âd :=
⊕
j∈Jd
k[[x1 − j1, . . . , xd − jd]]1j ,
there is a superspace isomorphism HCd ⊗ Âd ∼→ ÂHCd induced by the obvious
multiplication maps HCd⊗Âd  AHCd(µ) for all µ. The multiplication on HCd⊗Âd
corresponding to the one on ÂHCd via this isomorphism has the following properties
1Note there is a sign error in [KKT, (5.5)]: it should read −CaCa+1 . . . .
74
for all f ∈ Âd:
(1⊗ f1j)(cr ⊗ 1j′) = cr ⊗ cr(f)1cr(j)1j′ , (3.27)
(1⊗ f1j)(tr ⊗ 1j′) = tr ⊗ tr(f)1tr(j)1′j + 1⊗
tr(f)1tr(j) − f1j
xr − xr+1 1j
′
+ crcr+1 ⊗ tr(f)1tr(j) − cr+1(cr(f))1cr+1(cr(j))
xr + xr+1
1j′ . (3.28)
The fractions on the right hand side of (3.28) make sense: in the first, (xr−xr+1)1j′ is
invertible unless j′r = j
′
r+1, in which case the expression equals ∂r(f)1j1j′ ; the second
is fine when j′r 6= −j′r+1 as then (xr + xr+1)1j′ is invertible, while if j′r = −j′r+1 it
equals ∂˜r(f)1tr(j)1j′ .
Similarly, there is a completion Q̂HCd of QHCd. To introduce this, we denote
the elements of QHCd1j defined by an open dot on the rth strand, a closed dot on
the rth strand and a crossing of the rth and (r+ 1)th strands by γr1j , ξr1j and τr1j ,
respectively. For µ = (µi)i∈I as above, we define the cyclotomic quiver Hecke-Clifford
superalgebra
QHCd(µ) := QHCd
/〈
ξ2µi1 1j
∣∣∣ j ∈ Jd, i ∈ I with j21 = (z + i)(z + i+ 1)〉 . (3.29)
Using the relations, it is easy to see that the images of all ξr1j are nilpotent in
QHCd(µ). Then we set
Q̂HCd := lim←−
µ
QHCd(µ), (3.30)
taking the inverse limit once again in the category of locally unital superalgebras
with distinguished idempotents indexed by Jd. The obvious locally unital
homomorphisms QHCd ⊗k[ξ1,...,ξd] k[[ξ1, . . . , ξd]]  QHCd(µ) for each µ induce a
75
surjective homomorphism
QHCd ⊗k[ξ1,...,ξd] k[[ξ1, . . . , ξd]]→ Q̂HCd.
This map is actually an isomorphism, as may be deduced using the basis theorem
for QHCd from [KKT, Corollary 3.9] plus the observation that the image of any
non-zero element u ∈ QHCd is non-zero in QHCd(µ) for sufficiently large µ; the
latter assertion follows by elementary considerations involving the natural Z-grading
on QHCd. Consequently, Q̂HCd is isomorphic to the completion introduced in a
slightly different way in [KKT, Definition 3.16]. Moreover, there is a locally unital
embedding QHCd ↪→ Q̂HCd.
At last, we are ready to state the crucial theorem from [KKT]. We need this only
in the special situation of [KKT, §5.2(i)(c)], but emphasize that the results obtained
in [KKT] are substantially more general. In particular, for us, all elements of the set
I are even in the sense of [KKT, §3.5], so that we do not need the more general quiver
Hecke superalgebras of [KKT].
Theorem 3.5.5. There is a superalgebra isomorphism Q̂HCd
∼→ ÂHCd such that
1j 7→ 1j , γr1j 7→ cr1j , ξr1j 7→ yr1j , τr1j 7→ trgr1j + fr1j + crcr+1f˜r1j ,
76
for all j ∈ Jd and r. Here, yr ∈ k[[xr − jr]] and gr, fr, f˜r ∈ k[[xr − jr, xr+1 − jr+1]]
are the power series determined uniquely by the following:
jr = κr
√
i− 1
2
√
i+
1
2
for ir ∈ I and κr ∈ {±},
yr =
 κr
(√
i2r + 2jr(xr − jr) + (xr − jr)2 − ir
)
∈ (xr − jr) if ir 6= 0
κr ((xr − jr)2 + 2jr(xr − jr)) if ir = 0
pr =
(x2r − x2r+1)2
2(x2r + x
2
r+1)− (x2r − x2r+1)2
,
gr =

−1 if ir < ir+1,
pr (κryr − κr+1yr+1) if ir = ir+1 + 1,
pr if ir > ir+1 + 1,
√
pr
yr−yr+1 ∈
xr−xr+1
yr−yr+1 + (xr − xr+1) if jr = jr+1,
√
pr
yr+yr+1
∈ xr+xr+1
yr+yr+1
+ (xr + xr+1) if jr = −jr+1;
fr =
gr
xr − xr+1 −
δjr,jr+1
yr − yr+1 , f˜r =
gr
xr + xr+1
− δjr,−jr+1
yr + yr+1
.
(All of this notation depends implicitly on j.)
Proof. This is a special case of [KKT, Theorem 5.4]. To help the reader to translate
between our notation and that of [KKT], we note that the set J in [KKT] is the same
as our set J , but the set I there is J2 = {j2 | j ∈ J}. We have made various other
choices as stipulated in [KKT] in order to produce concrete formulae:
– We have taken the functions ε : J → {0, 1} and h : J2 → k from [KKT, (5.7)]
so that ε(j) = (1− κ)/2 and λ(j) := h(j2) = i for j = κ
√
i− 1
2
√
i+ 1
2
;
– For [KKT, (5.11)] we took Gjr,jr+1 (our gr) to be −1 when ir < ir+1.
We remark that the existence of well-defined fractions gr, fr, and f˜r ∈ k[[xr −
jr, xr+1 − jr+1]] is justified by [KKT, Lemma 5.5]. In the case where jr = ±jr+1,
77
the choice of
√
pr used in gr is uniquely determined by the specified containment.
A different choice of square root would have forced gr to be an element of the set
−xr−xr+1
yr−yr+1 + (xr − xr+1) or −
xr+xr+1
yr+yr+1
+ (xr + xr+1), respectively. Similarly, the square
root defining yr when ir 6= 0 is uniquely determined by the specified containment
yr ∈ (xr − jr), as a different choice of square root would have a constant term in the
power series expansion formula for yr in terms of xr − jr.
The formula for yr can be extracted from Subsection 5.3.2 of [KKT] as follows.
In that section, λr ∈ k[[xr − jr]] satisfies
λ2r = x
2
r +
1
4
= (xr − jr)2 − 2jr(xr − jr) + j2r .
In our C∞ situation, there are two cases depending on ir, which is the same as their
λ(jr):
– When ir = 0, yr = λ
2
r.
– When ir > 0, yr = λr − ir.
In each of these cases, there is a unique choice of square root allowing us to write yr
as a power series in the ideal generated by (xr − jr).
The formula defining the image of τr1j is a bit more complicated. First, as in
[KKT, (5.12)], define s˜r1j ∈ ÂHCd by
s˜r1j := ϕrgr1j = trgr1j +
gr
x1 − x2 1j + c1c2
gr
x1 + x2
1j ,
78
where ϕr = tr1j +
1
x1−x2 1j + c1c2
1
x1+x2
1j is the intertwiner for AHCd given in [KKT,
(5.1)]. Next, using [KKT, Theorem 3.8], define
σr1j =

s˜r1j if jr 6= ±jr+1
s˜r1j − 1yr−yr+1 1j if jr = jr+1
s˜r1j − 1yr+yr+1 1j if jr = −jr+1
Then, σr1j is precisely the element trgr1j + fr1j + crcr+1f˜r1j from the statement of
the theorem.
3.5.3. Construction of Φ
We are ready to prove Theorem 3.5.1 by constructing Φ : QH → End (O):
Proof of Theorem 3.5.1. For i = id · · · i1 ∈ Id, let Fi := Fid · · ·Fi1 : O → O. The
usual composition of natural transformations makes the vector space
NTd :=
⊕
i,i′∈Id
Hom(Fi, Fi′)
into a locally unital algebra with distinguished idempotents {1i | i ∈ Id} arising
from the identity endomorphisms of each Fi. Also horizontal composition of natural
transformations defines homomorphisms ad2,d1 : NTd2 ⊗ NTd1 → NTd2+d1 for all
d1, d2 ≥ 0. Recalling (2.8), the data of a strict monoidal functor Φ : QH → End (O)
sending i to Fi is just the same as a family of locally unital algebra homomorphisms
Φd : QHd → NTd for all d ≥ 0, such that 1i 7→ 1i for each i ∈ Id and
ad2,d1 ◦ Φd2 ⊗ Φd1 = Φd2+d1 ◦ bd2,d1 (3.31)
79
for all d1, d2 ≥ 0, where bd2,d1 : QHd2 ⊗ QHd1 → QHd2+d1 is the obvious embedding
defined by horizontal concatenation of diagrams.
To construct Φd, we start from the monoidal superfunctor Ψ from Theorem 3.5.3.
This induces superalgebra homomorphisms Ψd : AHCd → End(sF d) for all d ≥ 0,
where End(sF d) denotes supernatural endomorphisms of sF d : sO → sO. For each
M ∈ ob sO, Corollary 3.2.3 implies that evM ◦Ψd : AHCd → EndsO(sF dM) factors
through all sufficiently large cyclotomic quotients AHCd(µ). Hence, Ψd extends
uniquely to a locally unital superalgebra homomorphism Ψ̂d : ÂHCd → SNTd, where
SNTd :=
⊕
j,j′∈Jd
Hom(sFj , sFj′) ⊂ End(sF d)
and sFj := sFjd · · · sFj1 . Composing Ψ̂d with the isomorphism from Theorem 3.5.5
and the inclusion QHCd ↪→ Q̂HCd, we obtain a locally unital superalgebra
homomorphism Θd : QHCd → SNTd. It is obvious from Definitions 2.3.1 and 3.5.4
that there is a locally unital algebra homomorphism in : QHd → (QHCd)0¯ sending
the idempotent 1i to 1j for j with jr :=
√
ir − 12
√
ir +
1
2
, and taking the elements of
QHd1i defined by the dot on the rth strand and the crossing of the rth and (r+ 1)th
strands to ξr1j and τr1j , respectively. Also, recalling (3.19), restriction from sO to
O defines a homomorphism
pr :
⊕
j,j′∈Jd+
1j′(SNTd)0¯1j → NTd
where J+ :=
{√
i− 1
2
√
i+ 1
2
∣∣ i ∈ I} ⊂ J . Then the composition pr ◦ Θd ◦ in gives
us the desired locally unital homomorphism Φd : QHd → NTd sending 1i 7→ 1i for
each i ∈ Id. It just remains to observe that the property (3.31) is satisfied, and that
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Φd(xr1i)M is nilpotent for each r, i ∈ Id and M ∈ obO. These things follow from
the explicit formulae in Theorems 3.5.3 and 3.5.5 plus Corollary 3.2.3 once again.
3.6. Proof of main theorem (type C)
Recall the following theorem from Chapter 1:
Main Theorem (Type C). When n is even, the supercategory sO 1
2
splits as a direct
sum O 1
2
⊕ΠO 1
2
, where O 1
2
is a k-linear category. Moreover, O 1
2
admits the structure
of a tensor product categorification of V ⊗n, where V is the natural sp∞-module. When
n is odd, analogous results hold where the supercategory sO 1
2
is replaced by sOCT1
2
.
Proof. Theorem 3.4.1 shows that there is a decomposition sO = O ⊕ ΠO, where sO
denotes the category sO 1
2
from Chapter 1 when n is even, and the category sOCT1
2
when n is odd. We have also checked the following:
– Theorem 3.4.2: The category O is highest weight, with B labeling its irreducible
objects, partially ordered by  .
– Theorem 3.5.1: There is a strict monoidal functor Φ : QH → End(O), such that
Φ(i) = Fi. The natural transformation Φ
(
•
i
)
: Fi ⇒ Fi is locally nilpotent.
This gives us the data of (D2) satisfying (D4).
– The functor Ei is both left and right adjoint to Fi, as implied by Lemma 3.2.5.
This gives us the data of (D3) satisfying axiom (D5).
Theorem 3.4.4 shows that the exact functors Fi and Ei preserve O∆, and induced
linear maps give an action of sp∞ on [O∆]. Further, the map [O∆] → V ⊗n given
by [M(b)] 7→ vb induces an isomorphism of sp∞-modules, so axioms (TPC1) and
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(TPC2) hold. Thus, we have verified that O admits the structure of a tensor product
categorification of V ⊗n.
82
CHAPTER IV
APPLICATIONS
In this chapter we continue with all notation set up in Chapter 3, including the
category O defined in Section 3.4, functors Ei, Fi : O → O, etc.
In their paper [LW], Losev and Webster prove several theorems about tensor
product categorifications for tensor products of highest weight modules, including
a uniqueness theorem [LW, Theorem 6.1] and an explicit description of underlying
crystals [LW, Theorem 7.2]. Because the sp∞-module V
⊗n is not a tensor product of
highest weight modules, these theorems cannot be directly applied to our category
O from Chapter 3. To invoke them, for k > 0, we pass to the Serre subquotient
Ok of O defined below. This category admits the structure of a tensor product
categorification of V ⊗nk , where Vk is the natural spk-module. In particular, Vk is a
highest weight module. By applying Losev and Webster’s theorems to the category
Ok and then transporting these results back to O, we achieve the following:
– We prove that the combinatorics of the type C blocks are determined by
Webster’s orthodox basis.
– We calculate the underlying crystal for the category O.
– We classify all projective-injective (or prinjective) indecomposable objects of O.
4.1. The root system Ck
Recall that the nodes of the Dynkin diagram C∞ are labeled by the index set
I = N, where 0 ∈ I indexes the long root in the corresponding root system. Given
k ≥ 0, let Ik ⊂ I denote the set of all natural numbers between 0 and k − 1, and let
83
Ck denote the subdiagram of C∞ with nodes indexed by Ik. Denote its associated
Kac-Moody algebra by spk, which we realize as the subalgebra of sp∞ with Chevalley
generators {ei, fi | i ∈ Ik}. Note that spk inherits many notions from sp∞, including
a weight lattice
⊕
i∈Ik Z εi, set of simple roots {αi | i ∈ Ik}, etc.
Restricting the natural sp∞-module V to spk, let Vk denote the natural spk-
module spanned by the vectors {vb | −k < b ≤ k} ⊂ V . This is a 2k-dimensional
irreducible module of highest weight −εk−1. The tensor space V ⊗nk ⊂ V ⊗n has basis
{vb | b ∈ Bk}, where
Bk = {b ∈ B | −k < bs ≤ k for all 1 ≤ s ≤ n}.
Also, the type C Bruhat ordering  on B restricts to a partial ordering on Bk.
Because V ⊗nk is a tensor product of highest weight modules, [LW, Definition 3.2]
applies. To rephrase their definition in terms similar to Definition 2.5.1, we replace
B in Definition 2.5.1 with Bk, I with Ik, and the monoidal category QH with the full
subcategory QHk generated as a monoidal category by the objects i ∈ Ik.
4.2. The category Ok
Let B≤k denote the set of all b ∈ B such that N[1,s](b, k) ≤ 0 for all s = 1, . . . , n.
Next, define B<k as the set of all b ∈ B≤k such that N[1,s](b, k) < 0 for at least one s.
Lemma 3.3.2 implies immediately that B≤k and B<k are ideals in the sense of Section
2.1.3.
Observe that the set Bk is the difference B≤k \B<k. Indeed, the elements b of
B≤k \B<k satisfy N[1,s](b, k) = 0 for every 1 ≤ s ≤ n. A straight-forward induction
starting with s = 1 then shows that −k < bs ≤ k for every s.
84
We are precisely in the situation described in Section 2.1.3. Let O≤k denote the
Serre subcategory of O generated by the irreducible supermodules {L(b) | b ∈ B≤k},
and similarly define O<k. Because B≤k and B<k are ideals, the categories O≤k and
O<k inherit a highest weight structure from O. Hence, the quotient category Ok :=
O≤k/O<k has an induced highest weight structure with poset (Bk,).
Let pi : O≤k → Ok be the quotient functor. For b ∈ Bk, define Lk(b) = piL(b)
and Mk(b) = piM(b). The object Lk(b) is irreducible in Ok, and Mk(b) is a standard
module in the highest weight structure.
Lemma 4.2.1. Given any a, b ∈ B, let k be large enough so that a, b ∈ Bk. Then,
[M(b) : L(a)] = [Mk(b) : Lk(a)].
Proof. This is immediate from the exactness of the quotient functor pi.
4.3. Categorical actions on Ok
Lemma 4.3.1. For every i ∈ Ik, the functors Fi and Ei : O → O preserve the
subcategories O≤k and O<k.
Proof. We will check this for Fi. The proof for Ei is similar. As in [BLW, Lemma
2.18], it suffices to check that FiM(b) is an object of O≤k (resp. O≤k) whenever
b ∈ B≤k (resp. B<k).
Both cases follow from the observation that if i ∈ Ik, and i-sig(b)t = f,
N[1,s](b, k) = N[1,s](b+ dt, k). (4.1)
85
Indeed, if i-sig(b)t = f, then bt = ±i. From the restriction i ∈ Ik, it follows that
−k + 1 ≤ bt ≤ k − 1, so −k + 2 ≤ bt + 1 ≤ k. Hence,
N[1,s](b, k) = N[1,s](b+ dt, k)
for any 1 ≤ s ≤ n, because the t-th entry does not contribute to either sum in 4.1.
Because FiM(b) is filtered by Vermas of the form M(b+dt) where i-sig(b)t = f,
this observation proves that FiM(b) is in O≤k when b ∈ B≤k, and similarly for O<k
and B<k.
Theorem 4.3.2. The category Ok admits the structure of a tensor product
categorification of V ⊗nk .
Proof. Using the lemma, we see that Fi and Ei restrict to biadjoint, exact
endofunctors of O≤k and O<k. Hence, they induce biadjoint, exact endofunctors
F¯i, E¯i of the quotient category Ok which satisfy piFi = F¯ipi, and piEi = E¯ipi.
Recall from Theorem 3.5.1 that there is a strict monoidal functor Φ : QH →
End (O) sending the generating object i ∈ I ⊂ obQH to Fi. Restricting to the
subcategory QHk of QH generated by i ∈ Ik, and restricting the functors Fi to O≤k
and O<k, we see that there are well-defined monoidal functors:
QHk → End (O≤k) and QHk → End (O<k)
sending the generating object i ∈ Ik to Fi. Hence, there is an induced monoidal
functor Φk : QHk → End (Ok). The fact that Φk
(
•
i
)
is locally nilpotent follows
from the fact that the corresponding natural transformation is locally nilpotent in
O≤k.
86
Hence, we have equipped the highest weight category Ok with the spk-analogs
of (D2) and (D3), satisfying (D4) and (D5). Whenever i ∈ Ik, the fact that the
functors F¯i and E¯i satisfy the spk versions of the axioms (TPC1) and (TPC2) follows
immediately from the corresponding statements for the functors Fi and Ei on O.
Webster’s general theory from [W1] gives another construction of an spk-tensor
product categorification of V ⊗nk in terms of the category T
n
k -mod of finite dimensional
modules over the tensor product algebra T nk of [W1, §4] associated to the spk-module
V ⊗nk . See also [LW, Proposition 3.11].
Theorem 4.3.3. The category Ok is strongly equivariantly equivalent to T nk -mod.
Proof. We are now in finite rank, so this follows from the uniqueness of tensor product
categorifications exactly as established in [LW, Theorem 6.1].
As another application, by choosing k sufficiently large, we combine Theorem
4.3.3 with Lemma 4.2.1 to see that the composition multiplicities of the Verma
supermodules in O are the same as the corresponding composition multiplicities of
the standard objects in T nk -mod. This was conjectured independently in [CKW,
Conjecture 5.11].
Along similar lines, we have isomorphisms of Grothendieck groups
[T nk -mod
∆] ∼= [O∆k ] ∼= V ⊗nk
given by identifying the classes of standard objects in T kn -mod and Ok with the
monomial basis in V ⊗nk . In [W1, W2], Webster defines the orthodox basis of V
⊗n
k
to be the basis arising from the classes of indecomposable projectives in T nk -mod.
Taking k sufficiently large so that P (b) is an object in O≤k and hence Ok, we see that
the class [P (b)] corresponds to an orthodox basis element of V ⊗nk .
87
Webster’s general theory constructs tensor product categorifications for any
tensor product of highest weight modules over a symmetrizable Kac-Moody algebra.
In the case where the Kac-Moody algebra is of finite type A, D, E, the orthodox
bases described here are precisely Lusztig’s canonical basis for the tensor space.
Outside of those types, in particular, for the type Ck case described here, there is no
known elementary algorithm to compute these bases, and hence, the combinatorics
underlying the type C blocks.
This contrasts with the situation in the type A blocks, where the composition
multiplicities of Verma supermodules can be computed in terms of certain canonical
bases associated to the quantum group of type A∞, which can can be constructed
using elementary methods. See Section 1 of [BD2].
4.4. Crystals
As another application of the results of this dissertation, we identify the
associated crystal for O with the crystal underlying the sp∞-module V ⊗n. Crystals
were originally developed by Kashiwara [Ka]. We recall the definition for sp∞ here,
although the definition adapts easily to any other Kac-Moody algebra.
Definition 4.4.1. A normal sp∞ crystal is a set B with a decomposition B =⊔
λ∈P Bλ, plus crystal operators e˜i, f˜i : B → Bunionsq{0} for each i ∈ I satisfying the
following axioms:
(C1) for every λ ∈ P , the crystal operator e˜i restricts to a map Bλ → Bλ+αi unionsq{0};
(C2) for b ∈ B, we have that e˜i(b) = b′ 6= 0 if and only if f˜i(b′) = b 6= 0;
(C3) for every b ∈ B, there is an r ∈ N such that e˜ri (b) = f˜ ri (b) = 0.
88
For each i, define functions εi, ϕi : B→ N by
εi(b) = max{r ∈ N | e˜ri (b) 6= 0}, ϕi(b) = max{r ∈ N | f˜ ri (b) 6= 0}.
Then we also require that
(C4) ϕi(b)− εi(b) = 〈hi, λ〉 for each b ∈ Bλ and i ∈ I.
There is an sp∞-crystal structure on B defined using the combinatorics
underlying the sp∞-module V
⊗n. The decomposition B =
⊔
λ∈P Bλ is given by setting
Bλ = {b ∈ B | wt(b) = λ}. The crystal operators are defined using Kashiwara’s
tensor product rule, described as follows. Pick any b ∈ B. Starting from the i-
signature i-sig(b) from (2.6), we inductively define the reduced i-signature by replacing
pairs of entries of the form ef (possibly separated only by •’s) with •’s. This process
iterates until all e entries appear to the right of the entries f. Then define f˜ib to be
b+dr if the rightmost f in the reduced i-signature appears in position r, or 0 if there
are no f’s remaining in the reduced i-signature. Similarly, define e˜ib to be b − ds if
the leftmost e in the reduced i-signature appears in position s, or 0 if there are no
e’s present. This defines our crystal operators e˜i and f˜i. A routine check shows that
this data makes B into a normal sp∞ crystal. By restricting to the crystal operators
e˜i and f˜i for i ∈ Ik, we can make the set Bk ⊂ B into an spk crystal.
Example 4.4.2. Take b = (1, 2,−1, 4,−2,−2, 3, 1,−1). The 2-signature of b is the
tuple (•, f, e, •, f, f, e, •, e). Reducing the 2-signature replaces all ef pairs (possibly
separated by •’s) with •’s, so the reduced 2-signature is (•, f, •, •, •, f, e, •, e). Because
the rightmost f occurs in the sixth entry, we have
f˜2b = b+ d6 = (1, 2,−1, 4,−2,−1, 3, 1,−1).
89
Similarly, the leftmost e occurs in the seventh entry, and it follows that
e˜2b = b− d7 = (1, 2,−1, 4,−2,−2, 2,−1).
The following theorem demonstrates that the crystal structure on B can be
induced by applying the functors Fi and Ei to irreducibles in O. In this way, the
irreducibles objects in O categorify the crystal B.
Theorem 4.4.3. If f˜ib 6= 0, then both the the head and socle of FiL(b) are isomorphic
to L(f˜ib); otherwise, FiL(b) = 0. A similar statement holds with Fi and f˜i replaced
by Ei and e˜i.
Proof. Let k > i be sufficiently large so that b ∈ Bk, and all of the composition
factors FiL(b) are labeled by elements of Bk. Arguing in Ok, Theorem 7.2 from [LW]
demonstrate that F¯iLk(b) is non-zero if and only if f˜ib 6= 0, in which case the head
and socle of F¯iLk(b) is isomorphic to Lk(f˜ib).
Lemma 2.1.4 and the fact that piFi ∼= F¯ipi imply that HomOk(Lk(c), F¯iLk(b)) ∼=
HomO≤k(L(c), FiL(b)). Hence, Lk(c) is in the socle of F¯iLk(b) if and only if L(c) is
in the socle of FiL(b), and similarly for the heads. This proves the version of the
theorem involving Fi. The proof for Ei is similar.
4.5. Classification of Prinjectives
An object P in a category C is said to be prinjective if it is both projective and
injective. In this section, we use the expicit description of the associated crystal for O
from Theorem 4.4.3 to classify the prinjectives in O. We prove the following analog
of Theorem 2.24 in [BLW]:
Theorem 4.5.1. Given b ∈ B, the following are equivalent:
90
1. The label b is antidominant, i.e., b1 ≤ b2 ≤ · · · ≤ bn.
2. The projective indecomposable P (b) is prinjective.
3. We have an isomorphism P (b) ∼= P (b)?.
4. There exists some a ∈ B for which L(b) is in the socle of M(a).
We can easily prove a weaker version of this theorem in the case where the entries
of b are constant:
Lemma 4.5.2. Suppose that b ∈ B satisfies b1 = · · · = bn. Then, L(b) = M(b) =
P (b). Because L(b) ∼= L(b)?, it follows that P (b) ∼= P (b)? is prinjective.
Proof. Let b denote the common value of the entries bt. Then,
wt(b) =
 nεb−1 if b > 0,−nε−b if b ≤ 0,
From this, it is clear that there is no other c ∈ B with wt(c) = wt(b), so b is not
Bruhat-comparable to any other element of B. The lemma then follows from the
observation that P (b) can have no Verma subquotients other than M(b), and M(b)
can have no composition factors other than L(b).
Let B◦ denote the collection all elements of B which can be obtained from
z = (0, . . . , 0) ∈ B by applying a sequence of crystal operators. In other words, B◦ is
the connected component of B containing z.
Lemma 4.5.3. We have b ∈ B◦ if and only if b is antidominant.
Proof. For the first implication, we show that any b which can be obtained from the
antidominant z ∈ B using crystal operators is also antidominant. Hence, it suffices
to show that whenever a ∈ B is antidominant, then so are f˜ia and e˜ia.
91
To check that the entries in f˜ia are in increasing order, we recall that f˜ia = a+ds
where s is the maximal index for which the reduced i-signature of a contains an f.
Then, f˜ia is antidominant only if bs < bs+1. On the contrary, if they were equal, then
we would have i-sig(a)s = i-sig(a)s+1 = f. Because we cancel ef pairs (and not fe!)
it would then follow that the reduced i-signature of a contains a f in its (s+1) entry,
which contradicts our assumption about s. The proof of the antidominance of e˜ia is
similar, and it follows that the elements of B◦ are antidominant.
To prove the reverse implication, suppose that b1 ≤ · · · ≤ bn. For every index s,
define a monomial
X˜s :=
 f˜bs−1f˜bs−2 · · · f˜0 if bs ≥ 0e˜−bs e˜−bs+1 · · · e˜1 if bs < 0
Note that, by definition X˜s is an empty monomial if bs = 0. Let t denote the maximal
index for which bt < 0, and define
X˜ := X˜tX˜t−1 · · · X˜1X˜t+1X˜t+2 · · · X˜n.
A straightforward calculation shows that Xz = b, so b ∈ B◦. We emphasize that this
calculation relies heavily on the fact that the bt are in increasing order.
Proof of Theorem 4.5.1. The proof is similar to that of [BLW, Theorem 2.24].
(1) =⇒ (2). Fix a antidominant b ∈ B. Let X˜ be the corresponding monomial
in the crystal operators {e˜i, f˜i | i ∈ I} defined in the proof of Lemma 4.5.3. By
replacing each e˜i with Ei and each f˜i with Fi, we obtain a monomial X in the
functors {Ei, Fi | i ∈ I}. Using Lemma 4.5.2, the irreducible L(z) is prinjective, and
because the functors Fi and Ei are right and left adjoint to exact functors, it follows
that T := XL(z) is prinjective, too. Using exactness of the Ei and Fi and iterating
92
Theorem 4.4.3 demonstrates that L(b) is in the head and socle of T . Therefore, the
projective indecomposable P (b) is a summand of T . Because P (b) is a summand of
an injective object, it follows that P (b) is injective, too.
(2) =⇒ (3). If P (b) is prinjective, then it must be isomorphic to an
indecomposable injective P (c)? for some c ∈ B. We need to verify that c = b.
Because ? is exact and preserves irreducibles, we see that the indecomposable
projectives P (b) and P (c) have the same composition multiplicities. Because the
classes of indecompsable projectives are linearly independent in the Grothendieck
group of O, it follows that b = c.
(3) =⇒ (4). If P (b) ∼= P (b)?, then the socle of P (b) is isomorphic to L(b).
Using the filtration of P (b) by Vermas, there is some a  b with L(b) ↪→M(a).
(4) =⇒ (1). Suppose that L(b) ↪→ M(a). Pick k large enough that all
composition factors of ∆(a) are labeled by elements of Bk. Passing to Ok, the
irreducible Lk(b) is in the socle of Mk(a). Because κ = (1 − k, ..., 1 − k) ∈ Bk
labels the vector in the maximal weight space of V ⊗nk , the weight wt(κ) = −nεk−1 is
maximal among the weights of elements of Bk. In particular,
wt(κ)− wt(b) =
∑
i∈Ik
hiαi (hi ≥ 0)
is a sum of simple roots for spk. We let h :=
∑
i∈Ik hi denote the height of wt(κ) −
wt(b). We prove that b is antidominant by induction on h.
When h = 0, we have b = κ, so b is weakly decreasing. When h > 0, we
apply Proposition 5.2 of [LW] in the quotient category Ok to deduce that there is
some i ∈ Ik with E¯iLk(b) non-zero. Theorem 7.2 from [LW] shows that Lk(e˜ib) ↪→
F¯iLk(b) ↪→ F¯iMk(a). Because F¯iMk(a) is filtered by standard objects, it follows that
93
Lk(e˜ib) embeds into a standard object in Ok. The height of wt(κ)−wt(e˜ib) is h− 1,
so the induction shows that e˜ib is antidominant. Because f˜ie˜ib = b, arguments in the
proof of Lemma 4.5.3 shows that b is antidominant.
94
REFERENCES CITED
[AS] T. Arakawa and T. Suzuki, Duality between sln(C) and the degenerate
affine Hecke algebra, J. of Alg. 209 (1998), 288–304. MR1652134
(99h:17005)
[A] S. Ariki, On the decomposition numbers of the Hecke algebra of G(m, 1, n),
J. Math. Kyoto Univ. 36 (1996), 789–808. MR1443748 (98h:20012)
[BFK] J. Bernstein, I. Frenkel and M. Khovanov, A categorification of the
Temperley-Lieb algebra and Schur quotients of Uq(sl2) via projective
and Zuckerman functors, Selecta Math. 5 (1999), 199–241.
MR1714141 (2000i:17009)
[B1] J. Brundan, Kazhdan-Lusztig polynomials and character formulae for the
Lie superalgebra gl(m|n), J. Amer. Math. Soc. 16 (2003), 185–231.
MR1937204 (2003k:17007)
[B2] J. Brundan, Kazhdan-Lusztig polynomials and character formulae for the
Lie superalgebra q(n), Advances Math. 182 (2004), 28–77.
MR2028496 (2004m:17018)
[B3] J. Brundan, Tilting modules for Lie superalgebras, Commun. Alg. 32
(2004), 2251–2268. MR2100468 (2005g:17014)
[B4] J. Brundan, Representations of the general linear Lie superalgebra in the
BGG category O, in: “Developments and Retrospectives in Lie
Theory: Algebraic Methods,” eds. G. Mason et al., Developments in
Mathematics 38, Springer, 2014, pp. 71–98. MR3308778
[B5] J. Brundan, On the definition Kac-Moody 2-Category, Math. Ann. 365
(2016), 353 – 372. MR3451390
[BD1] J. Brundan and N. Davidson, Categorical actions and crystals, to appear
in Contemp. Math.; arXiv:1603.08938.
[BD2] J. Brundan and N. Davidson, Type A blocks in super category O;
arXiv:1606.05775.
[BD3] J. Brundan and N. Davidson, Type C blocks in super category O; In
preparation.
[BE1] J. Brundan and A. Ellis, Monoidal supercategories; arXiv:1603.05928.
95
[BE2] J. Brundan and A. Ellis, Super Kac-Moody 2-categories, in preparation.
[BK1] J. Brundan and A. Kleshchev, On translation functors for general linear
and symmetric groups, Proc. London Math. Soc. 80 (2000), 75–106.
MR1719176 (2000j:20080)
[BK2] J. Brundan and A. Kleshchev, Hecke-Clifford superalgebras, crystals of
type A
(2)
2` and modular branching rules for Ŝn, Represent. Theory. 5
(2001), 317–403. MR1870595 (2002j:17024)
[BK3] J. Brundan and A. Kleshchev, Blocks of cyclotomic Hecke algebras and
Khovanov-Lauda algebras; Invent. Math. 178 (2009), 451–484.
MR2551762 (2010k:20010)
[BLW] J. Brundan, I. Losev, and B. Webster, Tensor product categorifications
and the super Kazhdan-Lusztig conjecture, to appear in IMRN;
arXiv:1310.0349.
[BGS] A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in
representation theory, J. Amer. Math. Soc. 9 (1996), 473–527.
MR1322847 (96k:17010)
[CL] B. Cao and N. Lam, An inversion formula for some Fock spaces;
arXiv:1512.00577. MR3497973
[C] C.-W. Chen, Reduction method for representations of queer Lie
superalgebras; arXiv:1601.03924.
[CC] C.-W. Chen and S.-J. Cheng, Quantum group of type A and
representations of queer Lie superalgebra; arXiv:1602.04311.
[CKW] S.-J. Cheng, J.-H. Kwon, and W. Wang, Character formulae for queer Lie
superalgebras and canonical bases of type C; arXiv:1512.00116.
[CLW] S.-J. Cheng, N. Lam and W. Wang, Brundan-Kazhdan-Lusztig conjecture
for general linear Lie superalgebras, Duke Math. J. 164 (2015),
617–695. MR3322307
[CW] S.-J. Cheng and W. Wang, Dualities and Representations of Lie
Superalgebras, Graduate Studies in Mathematics 144, AMS, 2012.
MR3012224
[CR] J. Chuang and R. Rouquier, Derived equivalences for symmetric groups
and sl2-categorification, Ann. of Math. 167 (2008), 245–298.
MR2373155 (2008m:20011)
96
[CPS] E. Cline, B. Parshall and L. Scott, Finite dimensional algebras and highest
weight categories, J. Reine Angew. Math. 391 (1988), 85–99.
MR0961165 (90d:18005)
[F] A. Frisk, Typical blocks of the category O for the queer Lie superalgebra,
J. Algebra Appl. 6 (2007), 731–778. MR2355618 (2008g:17013)
[G] I. Grojnowski, Affine slp controls the representation theory of the
symmetric group and related Hecke algebras;
arXiv:math.RT/9907129.
[HKS] D. Hill, J. Kujawa and J. Sussan, Degenerate affine Hecke-Clifford algebras
and type Q Lie superalgebras, Math. Zeit. 268 (2011), 1091–1158.
MR2818745 (2012i:20009)
[H] J. Humphreys, Representations of Semisimple Lie Algebras in the BGG
Category O, Graduate Studies in Mathematics 94, AMS, 2008.
MR2428237 (2009f:17013)
[JN] A. Jones and M. Nazarov, Affine Sergeev algebra and q-analogues of the
Young symmetrizers for projective representations of the symmetric
group, Proc. London Math. Soc. 86 (2003), 29–69. MR1674836
(2000a:20021)
[Ka] M. Kashiwara, On crystal bases, in: “Representations of Groups (Banff
1994),” CMS Conf. Proc. 16 (1995), 155–197. MR1357199 (97a:17016)
[KKT] S.-J. Kang, M. Kashiwara and S. Tsuchioka, Quiver Hecke superalgebras,
J. Reine Angew. Math. 711 (2016), 1–54. MR3456757
[KL1] M. Khovanov and A. Lauda, A diagrammatic approach to categorification
of quantum groups I, Represent. Theory 13 (2009), 309–347.
MR2525917 (2010i:17023)
[KL2] M. Khovanov and A. Lauda, A categorification of quantum sl(n),
Quantum Top. 1 (2010), 1–92. MR2628852 (2011g:17028)
[K] A. Kleshchev, Linear and Projective Representations of Symmetric
Groups, Cambridge University Press, Cambridge, 2005. MR2165457
(2007b:20022)
[LLT] A. Lascoux, B. Leclerc, and J.-Y. Thibon, Hecke algebras at roots of unity
and crystal bases of quantum affine algebras, Commun. Math. Phys.
181 (1996), 205-263. MR1410572 (97k:17019)
97
[LW] I. Losev and B. Webster, On uniqueness of tensor products of irreducible
categorifications, Selecta Math. 21 (2015), 345–377. MR3338680
[Lu] G. Lusztig, Introduction to Quantum Groups, Birkha¨user, 1993.
MR1227098 (94m:17016)
[M] V. Mazorchuk, Parabolic category O for classical Lie superalgebras, in:
Advances in Lie Superalgebras, M. Gorelik and P. Papi (eds.),
Springer INdAM Series 7, Springer, 2014, pp. 149–166. MR3205086
[R] R. Rouquier, 2-Kac-Moody algebras; arXiv:0812.5023.
[W1] B. Webster, Knot invariants and higher representation theory, to appear in
Mem. Amer. Math. Soc.; arXiv:1309.3796.
[W2] B. Webster, Canonical bases and higher representation theory, Compositio
Math. 151 (2016), 121-166. MR3305310
[W3] B. Webster, A note on isomorphism between Hecke algebras;
arXiv:1305.0599v3.
98