Reduction method for representations of queer Lie superalgebras

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by  Chih-Whi Chen

REDUCTION METHOD FOR REPRESENTATIONS OF QUEER LIE
SUPERALGEBRAS
6
1
0 CHIH-WHICHEN
2

b Abstract. We develop a reduction procedure which provides an equivalence from
e
an arbitrary block of the BGG category for the queer Lie superalgebra q(n) to a
F
”Z±s-weights” (s ∈ C) block of a BGG category for finite direct sum of queer Lie

7 superalgebras. Wegivedescriptionsofblocks. Wealsoestablishequivalencesbetween
1
certainmaximalparabolicsubcategories forq(n)andblocksofatypicality-oneofthe

category of finite-dimensional modules for gl(ℓ|n−ℓ).
]
T
R
.
h
t 1. Introduction
a
m
1.1. The character problem for finite-dimensional irreducible modules over queer Lie
[
superalgebrasq(n)wasfirstsolvedbyPenkovandSerganova[PS1,PS2]. Theyprovided

2 an algorithm for computing the coefficient a of the character of the irreducible q(n)-
v λµ
4 module L(µ) in the expansion of the character of the associated Euler characteristic
2 E(λ) for given dominant weights λ,µ.
9
In [Br2] Brundan developed a different approach to computing the coefficient a
3 λµ
0 for integer dominant weights λ,µ. Let Fn be the nth exterior power of the natural
.
1 representation of type B quantum group U (b ) with infinite rank (cf. [JMO]). It
q ∞
0
was proved that the transition matrix (a ) is given by the transpose of the transition
6 λµ
1 matrix between canonical basis and the natural monomial basis of Fn at q = 1. This
: gives all irreducible characters of finite-dimensional integer weight modules in terms
v
i of a combinatorial algorithm for computing canonical bases. A new interpretation of
X
the irreducible characters of finite-dimensional half-integer weight modules was given
r
a in [CK] and [CKW] as well.
The celebrated Brundan’s Kazhdan-Lusztig conjecture [Br1] for the BGG category
of integer weight gl(m|n)-modules has beenproved by Cheng,Lam and Wang in [CLW]
(also see [BLW]). Furthermore, in [CMW], by using twisting functors and parabolic
inductionfunctorsCheng,MazorchukandWangreducedtheirreduciblecharacterprob-
lem of an arbitrary weight to the problem of integer weight.
In the present paper, we study the problem analogous to [CMW] for the queer Lie
superalgebra. One of the main goals is to study the (indecomposable) blocks of the
Key words and phrases: Queer Lie superalgebra, general linear Lie superalgebra, BGG category,
block decomposition, maximal parabolic subcategory, equivalence.
2010 Mathematics Subject Classification. Primary 17B10.
1
2 CHEN
BGG category for queer Lie superalgebra. In particular, we will prove equivalence of
categories between certain blocks for q(n) and gl(ℓ|n−ℓ).
Throughout the paper we denote by g the queer Lie superalgebra q(n) with the
standard Cartan subalgebra h for a fixed integer n ≥ 1. Let Og denote the BGG
category (see, e.g., [Fr, Section 3]) of finitely generated g-modules which are locally
finite b-modules and semisimple h -modules. Note that morphisms in Og are even.
¯0
For a finite direct sum of queer Lie superalgebras and reductive Lie algebras, we have
analogous notation of its BGG category. Let m ∈ Z , 0 ≤ ℓ ≤ m and s ∈ C. If m ≥ 1,
+
let Λ (m) ⊂ Cm:
sℓ
(1) λ ≡ s mod Z for 1 ≤ i≤ ℓ,
i
(1.1) Λ (m):= λ = (λ ,...,λ )| .
sℓ 1 m (2) λ ≡ −s mod Z for ℓ+1≤ i ≤ n.
i
(cid:26) (cid:27)
We define q(0) and Λ (0) to be 0 and the empty set, respectively. For each λ ∈ h∗,
sℓ ¯0
we shall assign a specific irreducible module L(λ) of highest weight λ and then define
the corresponding block Og, see the definitions in Section 2.2. The following theorem
λ
is the first main result of this paper.
Theorem 1.1. Let λ ∈ h∗. Then Og is equivalent to a block Ol of a Levi subalgebra
¯0 λ µ
l = q(n )×q(n )×···×q(n )⊆ g with k n = n and the weight µ of the form
1 2 k i=1 i
(1.2) µ ∈ Λ (n )×Λ P(n )×···×Λ (n ),
sℓ1 1 sℓ2 2 sℓk k
1 2 k
such that s 6≡ ±s mod Z for all i 6=j.
i j
Accordingly, the study of blocks of Og is reduced to the study of blocks of the
following three types: (i) (s = 0) a BGG category On,Z of the q(n)-modules of integer
weights, see, e.g., [Br2]. (ii) (s ∈ Z+ 1) a BGG category O of the q(n)-modules
2 n,1+Z
of half-integer weights, see, e.g., [CK], [CKW]. (iii) (s ∈/ Z/2) a2 BGG category O of
n,sℓ
the q(n)-modules of ”±s-weights”, see the definition in Section 4.2.
1.2. Let gl(ℓ|n−ℓ) be the general linear Lie superalgebra with the standard Cartan
subalgebrah for1 ≤ ℓ ≤ n. Anothermainresultofthepresentpaperistoestablish
ℓ|n−ℓ
an equivalence between a block F of certain maximal parabolic categories F for q(n)
λ
andcertain block ofatypicality-one ofthefinite-dimensionalmodulecategory F for
ℓ|n−ℓ
gl(ℓ|n−ℓ), see the definitions in Sections 4.1 and 4.2. Their identical linkage principle
(see Lemma 4.1) is the first piece of evidence to support such an equivalence.
For a weight λ ∈ h∗, or λ ∈ h∗ , we denote by ♯λ the atypicality degree of
¯0 ℓ|n−ℓ
λ (see, e.g. [CW, Definitions 2.29, 2.49]). According to [Ser98, Theorem 2.6] and
[BS12, Theorem 1.1] the blocks (F ) (see Section 4.1) for all ℓ,n − ℓ with the
ℓ|n−ℓ λ
same ♯λ are equivalent. More precisely, the endomorphism ring of projective generator
of (F ) is isomorphic to the opposite ring of the diagram algebra K∞ (see, e.g.,
ℓ|n−ℓ λ ♯λ
[BS12, Introduction]). In particular, K∞ is the path algebra of the infinite quiver
1
REDUCTION METHOD FOR REPRESENTATIONS OF QUEER LIE SUPERALGEBRAS 3
xi−1 xi xi+1
··· •−←→−• −←→− •−←→−• ···
yi−1 yi yi+1
modulothe relations x ·y = y ·x and x ·x =y ·y = 0 for all i ∈Z. We are
i i i−1 i−1 i i+1 i+1 i
now in a position to state the following theorem, which provides a Morita equivalence
between F and (F ) .
λ 1|1 0
Theorem 1.2. Let L(λ)∈ F with ♯λ = 1. Then the endomorphism ring of the projec-
tive generator of F is isomorphic to (K∞)op.
λ 1
1.3. The paper is organized as follows. In Section 2, we recall definitions of queer Lie
superalgebras, general linear Lie superalgebras and their categories of modules.
In Section 3, an approach of reduction similar to [CMW] is established for queer
Lie superalgebras. Equivalences of blocks via twisting functors and parabolic induction
functors are established. In addition, a description of decomposition of blocks of O is
given in Theorem 3.8.
In Section 4, we recall the category of finite-dimensional modules for gl(ℓ|n−ℓ) and
introduce certain maximal parabolic category for q(n). A correspondence preserving
linkage principles between their irreducibles is established. Finally, we compute the
endomorphism ring of projective generator to obtain Theorem 1.2.
Acknowledgments. The author is very grateful to Shun-Jen Cheng for numerous
helpful comments and suggestions.
2. Preliminaries
2.1. Lie superalgebras gl and q. For positive integers m,n ≥ 1, let Cm|n be the
complex superspace of dimension (m|n). Let {v ,...,v } be an ordered basis for the
¯1 m¯
even subspace Cm|0 and {v ,...,v } be an ordered basis for the odd subspace C0|n so
1 n
that the general linear Lie superalgebra gl(m|n) may be realized as (m+n)×(m+n)
complex matrices indexed by I(m|n):= {¯1 < ··· < m¯ < 1 < ··· < n}:
A B
(2.1) ,
C D
(cid:18) (cid:19)
where A,B,C and D are respectively m×m,m×n,n×m,n×n matrices. For m = n,
the subspace
A B
(2.2) g := q(n) = A,B : n×n matrices
B A
(cid:26)(cid:18) (cid:19)(cid:12) (cid:27)
(cid:12)
forms a subalgebra of gl(n|n) called the qu(cid:12)eer Lie superalgebra.
(cid:12)
Let E be the elementary matrix in gl(m|n) with (a,b)-entry 1 and other entries 0,
ab
fora,b ∈I(m|n). Then{e ,e¯ |1 ≤ i,j ≤ n}isalinearbasisforg,wheree = E +E
ij ij ij ¯i¯j ij
and e¯ = E +E . Note that the even subalgebra g is spanned by {e |1 ≤ i,j ≤ n},
ij ¯ij i¯j ¯0 ij
which is isomorphic to the general linear Lie algebra gl(n).
4 CHEN
Let h and h∗ berespectively the standard Cartan subalgebra of gl(m|n) and its
m|n m|n
dual space, with linear bases {E |i∈ I(m|n)} and {δ |i∈ I(m|n)} such that δ (E ) =
ii i i jj
δ .
i,j
Let h = h ⊕h be the standard Cartan subalgebra of g, with linear bases {h :=
¯0 ¯1 i
e |1 ≤ i ≤ n} and {h¯ := e¯ |1 ≤ i ≤ n} of h and h , respectively. Let {ε |1 ≤ i ≤ n}
ii i ii ¯0 ¯1 i
be the basis of h∗ dual to {h |1 ≤ i ≤ n}. We define a symmetric bilinear form (,) on
¯0 i
h∗ by (ε ,ε )= δ , for 1 ≤ i,j ≤ n.
¯0 i j ij
We denote by Φ,Φ ,Φ the sets of roots, even roots and odd roots of g, respectively.
¯0 ¯1
Let Φ+ = Φ+ ⊔ Φ+ be the set of positive roots with respect to its standard Borel
¯0 ¯1
subalgebra b = b ⊕ b , which consists of matrices of the form (2.2) with A and B
¯0 ¯1
upper triangular. Denote the set of negative roots by Φ− := Φ \ Φ+. Ignoring the
parity we have Φ = Φ = {ε −ε |1 ≤ i,j ≤ n} and Φ+ = {ε −ε |1 ≤ i< j ≤ n}. We
¯0 ¯1 i j i j
denote by ≤ the partial order on h∗ defined by using Φ+. The Weyl group W of g is
¯0
defined to be the Weyl group of the reductive Lie algebra g and hence acts naturally
¯0
on h∗ by permutation. We also denote by s the reflection associated to a root α ∈Φ+.
¯0 α
For a given root α = ε −ε ∈ Φ, let α¯ := ε +ε . For each λ ∈ h∗, we have the integral
i j i j ¯0
root system Φ := {α ∈ Φ|(λ,α) ∈ Z} and the integral Weyl group W defined to be
λ λ
the subgroup of W generated by all reflections s , α ∈ Φ .
α λ
2.2. Categories of modules. Let V = V ⊕V be a superspace. For a given homoge-
0 1
nous element v ∈ V (i ∈ Z ), we let v= i denote its parity. Let Π denote the parity
i 2
change functor on the category of superspaces. Let Π0 be the identity functor. For a
g-module M and µ ∈ h∗, let M := {m ∈ M|h·m = µ(h)m, for h ∈ h } denote its
¯0 µ ¯0
µ-weight space. If M has a weight space decomposition M = ⊕µ∈h∗Mµ, its character is
¯0
given as usual by chM = µ∈h∗dimMµeµ, where e is an indeterminate. In particular,
¯0
we have the root space decomposition g = h⊕(⊕ g ) with respect to the adjoint
P α∈Φ α
representation of g.
Let λ = n λ ε ∈ h∗, and consider the symmetric bilinear form on h∗ defined
i=1 i i ¯0 ¯1
by h·,·i := λ([·,·]). Let ℓ(λ) be the number of i’s with λ 6= 0 and δ(λ) = 0 (resp.
λ i
P
δ(λ) = 1) if ℓ(λ) is even (resp. odd). Let 1 ≤ i < i < ··· < i ≤ n such that
1 2 ℓ(λ)
λ ,λ ,...,λ are non-zero. Denote by ⌈·⌉ the ceiling function. Then the space
i1 i2 iℓ(λ)
−λ
(2.3) h′ := ⊕ Ch ⊕ ⊕ℓ(λ)−⌈ℓ(λ)/2⌉C(h + i2k−1h ) ,
¯1 (cid:16) j6=i1,...,iℓ(λ) j(cid:17) k=1 i2k−1 p λi2k i2k !
is a maximal isotropic subspace of h associated to h·,·i . Put h′ =ph ⊕h′. Let Cv be
¯1 λ ¯0 ¯1 λ
the one-dimensional h′-module with v = ¯0, h·v = λ(h)v and h′·v = 0 for h ∈ h ,
λ λ λ λ ¯0
h′ ∈ h′. Then I := Indh Cv is an irreducible h-module of dimension 2⌈ℓ(λ)/2⌉ (see,
¯1 λ h′ λ
g
e.g., [CW, Section 1.5.4]). We let M(λ) := Ind I be the Verma module, where I is
b λ λ
extended to a b-module in a trivial way, and define L(λ) to be the unique irreducible
quotient of M(λ).
REDUCTION METHOD FOR REPRESENTATIONS OF QUEER LIE SUPERALGEBRAS 5
Let Og denote the BGG category (see, e.g., [Fr, Section 3]) of finitely generated g-
moduleswhicharelocallyfiniteoverbandsemisimpleoverh . Notethatthemorphisms
¯0
in Og are even. It is known (see, e.g., [CW, Section 1.5.4]) that L(λ) ∼= ΠL(λ) if and
only if δ(λ) = 1. Therefore we have the following.
Lemma 2.1. {L(λ)|λ ∈ h∗ with δ(λ) = 1}∪{L(λ),ΠL(λ)|λ ∈ h∗ with δ(λ) = 0} is a
¯0 ¯0
complete set of irreducible g-modules in Og up to isomorphism.
We denote by Z(g) the center of U(g). As in the case of Lie algebras, the BGG cate-
gory Og of ghas adecomposition into subcategories correspondingtocentral characters
χ : Z(g) → C for λ ∈ h∗. We have a refined decomposition by the linkage principle
λ ¯0
(see, e.g., [CW, Section 2.3])
(2.4) Og = Og,
λ
λ∈h∗/∼
M¯0
where the equivalence relation ∼ on h∗ is defined by
¯0
(2.5) λ ∼ µ if and only if χ = χ and µ ∈ λ+ZΦ,
λ µ
and Og is the Serre subcategory of Og generated by simple objects with highest weight
λ
µ such that λ ∼ µ. The subcategories Og are decomposable in general.
λ
For a finite direct sum of queer Lie superalgebras and reductive Lie algebras, we
have analogous notation and decomposition of its BGG category. When there is no
confusion, we denote Og by O. For λ ∈ h∗, denote the block of O containing L(λ) by
¯0
O . Namely, itistheSerresubcategorygeneratedbythesetofverticesintheconnected
λ
component of the Ext-quiver for O containing L(λ).
3. Equivalences and Reductions for Blocks of Queer Lie Superalgebra
3.1. Equivalence using twisting functors. For a simple root α ∈ Φ+, we can de-
finedthetwistingfunctorT associatedtoα. Thetwistingfunctorwasoriginallydefined
α
byArkhipovin[Ar]andfurtherinvestigated inmoredetailin[AS],[KM],[CMW],[AL],
[MS], [GG13], [KM]. Recall the precise definition of T as follows. First, fix a non-zero
α
root vector X ∈ (g ) . Since the adjoint action of X on g is nilpotent, by using a
¯0 −α
standard argument (see e.g. [MO00, Lemma 4.2]) we can form the Ore localization
U′ of U(g) with respect to the set of powers of X. Since X is not a zero divisor in
α
U(g), U(g) can be viewed as an associative subalgebra of U′. The quotient U′/U(g)
α α
has the induced structure of a U(g)-U(g)-bimodule. Let ϕ = ϕ be an automorphism
α
of g that maps (g ) to (g ) for all simple root β and i ∈ {¯0,¯1}. Finally, consider
i β i sα(β)
the bimodule ϕU , which is obtained from U by twisting the left action of U(g) by
α α
ϕ. We also have an analogous construction with respect to the subalgebra g to obtain
¯0
the U(g )-U(g )-bimodule ϕU¯0. Now we are in a position to define twisting functors:
¯0 ¯0 α
(3.1) Tα(−) :=ϕ Uα⊗− :Og → Og and Tα¯0(−) :=ϕ Uα¯0⊗− :Og¯0 → Og¯0.
6 CHEN
Then T and T¯0 have right adjoints K and K¯0, respectively (see, e.g., [AS]). Let
α α α α
Db(O) and Db(Og¯0) be the bounded derived categories of O and Og¯0, respectively. It
is not hard to prove that T and T¯0 are right exact functors. Let L T , L T¯0 the
α α i α i α
i-th left derived functors of T , T¯0, respectively. It was proved in [AS] that L T0¯ = 0
α α i α
for i > 1 and L T0¯ is isomorphic to the functor of taking the maximal submodule on
1 α
which the action of g is locally nilpotent. Similarly, we have analogous definition for
−α
right derived endofunctors RiK and RiK¯0 of K and K¯0, respectively. Furthermore,
α α α α
RiK¯0 = 0 for i > 1 and R1K¯0 is isomorphic to the functor of taking the maximal
α α
subquotient on which the action of g is locally nilpotent.
−α
The star action ∗ of s on weights had been introduced in [GG13, Introduction] and
α
[CM]: s ∗λ := s λ if (λ,α¯) 6=0 and s ∗λ := s λ−α if (λ,α¯)= 0. We call the former
α α α α
an α-typical weight and the later an α-atypical weight (also see [GG13, Section 1.2.3]).
The following theorem is inspired by [CM, Proposition 8.6].
Theorem 3.1. Let λ ∈ h∗ and α ∈ Φ+ be a simple root such that (λ,α) ∈/ Z. Then
¯0
Πi ◦ T : O → O is an equivalence with inverse Πj ◦K : O → O for some
α λ sαλ α sαλ λ
i,j ∈ {0,1}.
Proof. We claim that T and K are exact functors on O and O , respectively. To
α α λ sαλ
see this, we first note that L T¯0 and R1K¯0 vanish at each simple g -module of highest
1 α α ¯0
weight µ with (µ,α) 6∈ Z (e.g., [Mar, Chapter 3]). Next we recall that Resg ◦L T =
g¯0 i α
L T¯0 ◦Resg and Resg ◦RiK = RiK¯0 ◦Resg (e.g. [CM, Lemma 5.1]) for all i ≥ 0.
i α g¯0 g¯0 α α g¯0
This means that L T M = RiK M′ = 0 for all M ∈ O ,M′ ∈ O and i ≥ 1. As
i α α λ sαλ
a conclusion, T and K are exact functors on O and O , respectively. For µ ∈ h∗
α α λ sαλ ¯0
with (µ,α) 6∈ Z, it is proved in [CM, Lemma 5.8] that T L(µ) is simple with T2L(µ) ∈
α α
{L(µ),ΠL(µ)}. By a similar argument we can show that K L(µ) is also simple with
α
K2L(µ) ∈ {L(µ),ΠL(µ)}. That is, that T and K preserve simple objects of O and
α α α λ
O , respectively. Finally recall that chT M(µ) = chM(s µ) [CM, Lemma 5.5] for
sαλ α α
all µ ∈ h∗¯0. From this together with the fact that HomO(TαL,L′) = HomO(L,KαL′)
for all simple objects L,L′ ∈ O, we conclude that T sends objects of O to objects of
α λ
O and K sends objects of O to objects of ΠiO , for some i = 0,1. Consequently,
sαλ α sαλ λ
the restrictions of T and K make T : O → ΠiO an equivalence with inverse
α α α λ sαλ
K : O → ΠjO , for some i,j ∈ {0,1}. (cid:3)
α sαλ λ
Remark 3.2. Let λ,α be as in Theorem 3.1. It is worth pointing out that T L(λ) only
α
depends on whether λ is α-typical or α-atypical. That is, it was determined in [CM,
Corollary 8.15]: By the classification of simple q(2)-highest weight modules in [Mar10]
we have [T L(λ) : ΠiL(s ∗λ)] 6= 0, for some i = 0,1. This is also proved in [GG13,
α α
Proposition 4.7.1]. As a consequence, we have T L(λ) = ΠiL(s ∗λ) for some i = 0,1.
α α
Example 3.3. Let n = 3 and λ := (−π,π,−π). Then by Theorem 3.1 and Remark
3.2, Tǫ1−ǫ2 : Oλ → Oλe is an equivalence sending L(λ) to ΠiL(λ−(ǫ1 −ǫ2)) for some
i = 0,1, where λ = (π,−π,−π) .
e
e
REDUCTION METHOD FOR REPRESENTATIONS OF QUEER LIE SUPERALGEBRAS 7
3.2. Equivalence using parabolic induction functor. The goal of this section is
to show that the parabolic induction functors give equivalences of blocks under some
suitable condition. For given integers ℓ,m with 1 ≤ ℓ ≤ m and s ∈ C, recall the set
Λ (m) defined in 1.1. In this section, we consider blocks O with the weight λ ∈ h∗ of
sℓ λ ¯0
the following form
1
(3.2) λ ∈ Λ (n )×···×Λ (n ) such that s = 0, s = and s 6≡ ±s mod Z,
sℓ1 1 sℓk k 1 2 2 i j
1 k
for all i 6= j. We define Φ := {α ∈ Φ|(λ,α¯) ∈Z} Φ and l := h⊕ ⊕ g to be
λ λ λ α∈Φλ α
the Levi subalgebra associated to λ. In this caseS, we denote by uλ t(cid:16)he corresp(cid:17)onding
nilradical. Furthermore, we have isomorphisms
(3.3) W ∼= S ×S ×(S ×S )×···×(S ×S ),
λ n1 n2 ℓ3 n3−ℓ3 ℓk nk−ℓk
and l ∼= q(n ) × q(n ) × ··· × q(n ). In order to prove that the parabolic induc-
λ 1 2 k
tion functors are equivalences in this setting, we first recall the following well-known
characterization of central characters (see, e.g., [CW, Theorem 2.48]).
Lemma 3.4. For λ,µ ∈ h∗, χ = χ if and only if there exist w ∈ W, {k } ⊂ C,
¯0 λ µ j j
and a subset of mutually orthogonal roots {α } such that µ = w(λ − k α ) and
j j j j j
(λ,α )= 0 for all j .
j
P
Definearelation ≈on h∗ as follows. For λ,µ ∈ h∗ welet λ ≈µ if thereexistw ∈ W ,
¯0 ¯0 λ
{k }⊂ Z,andasubsetofmutuallyorthogonalroots{α }suchthatµ = w(λ− k α )
j j j j j
and (λ,α ) = 0 for all j. The following lemma shows that ∼ and ≈ coincide in our
j
P
setting.
Lemma 3.5. Let λ ∈ h∗ be of the form (3.2). Then µ ∼ λ if and only if µ ≈ λ. In
¯0
particular, if ΠiL(µ)∈ O , for some i= 1,2, then µ ≈ λ.
λ
Proof. Since χ = χ we have µ = w(λ − k α ) for some w ∈ W, {k } ⊂ C
λ µ j j j j j
and {α } ⊂ Φ such that (λ,α ) = 0 for all j by Lemma 3.4. Furthermore, we have
j j j
P
λ ∈µ+ZΦ. It follows that w ∈ W and k ∈ Z for all j. This completes the proof. (cid:3)
λ j
The following theorem is inspired by [CMW, Proposition 3.6].
Theorem 3.6. Let λ ∈ h∗ be of the form (3.2). Let l := l ,u := u . Then there
¯0 λ λ
are i,j ∈ {0,1} such that the parabolic induction functor Πi ◦Indg : Ol → O is an
l+u λ λ
equivalence, with inverse equivalence Πj ◦Resg : O → Ol defined by M 7→ Mu, where
l λ λ
Mu is the maximal trivial u-submodule of M.
Proof. As in the proof of [CMW, Propositon 3.6], it suffices to show that Indg L0 is
l+u µ
irreducible for each irreducible l-module L0 ∈Ol of highest weight µ. We first assume
µ λ
that ζ ∈ h∗ is a weight of a non-zero singular vector in Indg L0. Then by Lemma
¯0 l+u µ
3.5 there exist w ∈ W ,{k } ⊂ Z, and a subset of mutually orthogonal roots {α }
µ j j j j
such that ζ = w(µ− k α ) and (µ,α ) = 0 for all j (note that Φ = Φ ). On the
j j j j λ µ
P
8 CHEN
other hand, by consideration of the weights of Indg L0, we have ζ ∈ µ− Z α.
l+u µ α∈Φ+ ≥0
Henceζ ∈ µ− Z αby(3.3). ThismeansthateverysubquotientofIndg L0
α∈Φµ∩Φ+ ≥0 P l+u µ
intersects L0 and so Indg L0 is irreducible. This completes the proof. (cid:3)
µ P l+u µ
Proof of Theorem 1.1. Let λ ∈ h∗. We can first apply a sequence of suitable twisting
¯0
functors (see Theorem 3.1) to Oλ and obtain an equivalent block Oλe such that λ ∈
Λ (n )×Λ (n )×···×Λ (n ) and s 6≡ ±s mod Z for all i 6= j. Next we can
sℓ1 1 sℓ2 2 sℓk k i j
1 2 k e
apply the parabolic induction functor (see Theorem 3.6) to obtain an equivalent block
of the desired Levi subalgebra. This completes the proof. (cid:3)
Example 3.7. Let λ := (1,1,−π, 3,π,−3,−π). Then by applying a sequence of
5 2 2
twisting functors Πiα ◦T with some i ∈ {0,1} in Theorem 3.1 related to α-typical
α α
weights, we may transform λ to the weight λ = (1, 1,3,−3,−π,π,−π), which gives
5 2 2
an equivalence from Oλ to Oλe sending L(λ) to L(λ). Then we apply the twisting
e
functor Πi◦T with some i = 0,1 to obtain the weight λ = (1, 3,−3,1,π,−π,−π)
ε5−ε6 e 2 2 5
and an equivalence Oλ to Oee which sends L(λ) to L(λ −e(ε5 − ε6)). Next we use
λ e
the parabolic induction functors. Define α := (ε5 − eeε6). Note that λ, λ−α ∈
Λ00(1) × Λ11(2) × Λ10(1) × Λπ1(3) and lee = lee ∼= q(1) × q(2) × q(1) × q(3). By
2 5 λ−α λ ee ee
Theorem 3.6, there is an equivalence from O to Ol sending L(λ) to the irreducible
λ ee
λ
l-module with highest weight λ−α.
3.3. Description of blocks.ee
Theorem 3.8. Let λ,µ ∈ h∗. Then ΠiL(µ) ∈O for some i = 0,1 if and only if µ ≈ λ.
¯0 λ
Proof. First assume that λ ∈ h∗ is of the form (3.2). Thanks to Lemma 3.5, it remains
¯0
to show that µ ≈ λ implies ΠiL(µ) ∈ O for some i = 0,1. Recall the fundamental
λ
lemmain[PS2,Proposition2.1]byPenkovandSerganova. ItfollowsfromHom (M(λ−
g
α),ΠjM(λ)) 6= 0for some j = 0,1, for all α ∈ Φ+ with (λ,α) =0 that ΠiL(λ−α) ∈ O
λ
for some i = 0,1. Therefore we may assume that µ is of the form s(λ), for some
reflection s ∈ W corresponding to a simple root ε − ε . In this case, we have
λ i i+1
λ −λ = k ∈ Z. Without loss of generality, assume that k > 0. Let v ∈ M(λ) be a
i i+1 λ
highestweight vector, itisnothardtocomputethate ek+1 v isasingularvector in
i,i+1 i+1,i λ
M(λ) of weight s(λ) (see, e.g., [CW, Lemma 2.39]). This means that ΠiL(µ) ∈ O for
λ
somei = 0,1. Forarbitraryλ ∈ h∗¯0,thereareλ′ ∈ h∗¯0 oftheform(3.2)andT :Oλ → Oλ′
an equivalence constructed by using a sequence of twisting functors in Theorem 3.1.
For ζ,ζ′ ∈h∗ and simple reflection s ∈ W, note that s∗ζ ≈ s∗ζ′ if and only if ζ ≈ ζ′.
¯0
The theorem now follows by Remark 3.2. (cid:3)
Remark 3.9. Ifℓ(λ)isodd,thenO istheSerresubcategorygeneratedby{L(µ)|µ ≈ λ}.
λ
REDUCTION METHOD FOR REPRESENTATIONS OF QUEER LIE SUPERALGEBRAS 9
4. Equivalences of Certain Maximal Parabolic Subcategory
In this section, we fix non-negative integers n,ℓ with n ≥ ℓ and s 6∈ Z/2. The goal
of this section is to establish an equivalence between certain block of atypicality-one of
finite-dimensional category for gl(ℓ|n−ℓ) and some block of certain maximal parabolic
subcategory for q(n).
4.1. Finite-dimensional representations of gl(ℓ|n−ℓ). We denote by F the
ℓ|n−ℓ
category ofintegralweight, finite-dimensionalgl(ℓ|n−ℓ)-moduleswithevenmorphisms.
LetΛa := ⊕n Zδ betheweightlattice. Recallthatthesetofallirreducibleoebjects(up
i=1 i
to parity) of F are parametrized by its highest weight λ in Λa,+ := {λ ∈ Λa| λ ≥
ℓ|n−ℓ i
λ , for 1 ≤ i < ℓ and ℓ ≤ i < n}. We define |λ| := (λ, n δ ) (mod 2). Recall
i+1 i=ℓ+1 i
that for a giveen M ∈ F , there is a decomposition M = M ⊕M of gl(ℓ|n−ℓ)-
ℓ|n−ℓ P + −
modules, where M+ := ⊕µ∈Λa(Mµ)|µ| and M− := ⊕µ∈Λa(Mµ)|µ|+1. This induces a
decomposition F =e F ⊕ ΠF (see, e.g., [Br1, Section 4-e]), where F
ℓ|n−ℓ ℓ|n−ℓ ℓ|n−ℓ ℓ|n−ℓ
(resp. ΠF ) is the full subcategory consisting of all M ∈ F such that M = M
ℓ|n−ℓ ℓ|n−ℓ +
(resp. M = M e). For ζ ∈ Λa,+, denote by (F ) the block of F containing the
− ℓ|n−ℓ ζ ℓ|n−ℓ
(unique) irreduciblemoduleLa of highest weight ζ. Namely, iet is the Serresubcategory
ζ
generated by thesetofvertices intheconnected componentoftheExt-quiver forF
ℓ|n−ℓ
containing La.
ζ
As we mentioned in Section 1, the diagram algebra K∞ is the path algebra of a
1
certain infinite quiver. Therefore we can identify (K∞)op as the associative algebra
1
generated by elements {zi,xj,yk}i,j,k∈Z and relations
z c = cz = y y = x x = 0,
i i i j j i
x y = z , y x = z ,
i i i+1 i i i
for all i,j ∈Z,c ∈ {xs,yt}s,t∈Z.
4.2. Parabolic categories of q(n) and Equivalences. We define a bijection ·a :
Λ (n) → Λa by
sℓ
n ℓ n
(4.1) λ = λ ε ∈ Λ (n)7−→ λa := (λ −s)δ + (λ +s)δ −ρ∈ Λa,
i i sℓ i i i i
i=1 i=1 i=ℓ+1
X X X
where ρ:= ℓ −(ℓ−i+1)δ + n (i−ℓ)δ .
i=1 i i=ℓ+1 i
Let χa be the central character of gl(ℓ|n−ℓ) corresponding to λ ∈ h∗ . We first
λ P P ℓ|n−ℓ
consider the linkage principles under this bijection.
Denote by Ψ := {δ −δ |1 ≤ i 6= j ≤ n} the root system of gl(ℓ|n−ℓ). Recall that
i j
the linkage principle in (2.5) for q(n) defines an equivalence relation ∼. The following
lemma follows from Lemma 3.5 and the proof of [CMW, Proposition 3.3].
Lemma 4.1. Letλ,µ ∈ Λ (n). Thenλ ∼ µ ifand only ifχa = χa and µa ∈ λa+ZΨ.
sℓ λa µa
10 CHEN
We define the set Λ+(n) := {λ ∈ Λ (n)| λ > λ , for 1 ≤ i < ℓ and ℓ ≤ i <
sℓ sℓ i i+1
n}. Note that we have Λa,+ = (Λ+(n))a. For arbitrary λ ∈ Λ (n) we have a Levi
sℓ sℓ
subalgebra l := h⊕(⊕ g )∼= q(ℓ)×q(n−ℓ) and the maximal parabolic subalgebras
α∈Φλ α
p := h⊕ ⊕ g . Let u be the corresponding nilradical of p. We denote by Op
α∈Φλ∪Φ+ α
the maximal parabolic subcategory of (see, e.g., [Mar14, Section 3.1]) O with respect
(cid:0) (cid:1)
to p. Namely, Op is the Serre subcategory of O generated by p-locally finite, and l -
¯0
semisimple g-modules. We define O to be the full subcategory of g-modules in O
n,sℓ
with weights in Λ (n) and F := Op ∩ O its maximal parabolic subcategory. For
sℓ n,sℓ
each M ∈ F, note that M is also l-semisimple since all weights of M are l-typical. As
a conclusion, if ΠiL(µ)∈ F for some i = 0,1 then we have µ ∈ Λ+(n).
sℓ
Let λ ∈ Λ+(n). Note that every irreducible l-module of highest weight λ can be
sℓ
extended to a p-module by letting u act trivially. We define L0(λ) to be the finite-
dimensional irreducible l-module with highest weight space I . Therefore the corre-
λ
sponding parabolic Verma module K(λ) := IndgL0(λ) has the irreducible quotient
p
L(λ). Furthermore, we note that K(λ) is p-locally finite and all the l-weights of K(λ)
are l-typical. Therefore we have K(λ),L(λ) ∈ F. Consequently, F is the Serre subcat-
egory of O generated by {ΠiL(λ)|λ ∈ Λ+(n), i ∈{0,1}}.
sℓ
For λ ∈ Λ+(n). We also denote by P(λ) and U(λ) the projective cover of L(λ) and
sℓ
the tilting module corresponding to λ in Op, respectively. For their definitions and
existences, we refer to [Mar14, Proposition 1,7] and [Mar14, Theorem 2]. Note that
all weights of P(λ),U(λ) are in Λ (n) since they are indecomposable (by definition).
sℓ
That is, P(λ),U(λ) ∈ F.
Let P be the free abelian group on basis {ǫa}a∈Z. Let wt(·) : Λ+sℓ(n) → P be the
weight function defined by (c.f. [Br2, Section 2-c])
ℓ n
(4.2) wt(λ) := ǫ + (−ǫ ).
λi−s −(λi+s)
i=1 i=ℓ+1
X X
ByLemma3.4,wehaveχ = χ ifandonlyifwt(λ) = wt(µ). By(2.4),wehavedecom-
λ µ
position F = ⊕λ∈h∗¯0Fχλ = ⊕γ∈PFγ according to central characters χλ with wt(λ) = γ.
Let Cn|n and (Cn|n)∗ be the standard representation and its dual, respectively. De-
note the projection functor from F to F by pr . We define the translation functors
γ γ
E ,F : F → F as follows
a a
(4.3) E (M) := pr (M ⊗(Cn|n)∗), F (M) := pr (M ⊗Cn|n),
a γ+(ǫa−ǫa+1) a γ−(ǫa−ǫa+1)
for all M ∈ F , γ ∈ P , a ∈ Z. For each a ∈ Z , it is not hard to see that both E and
γ a
F are exact and bi-adjoint to each other. We write λ → µ if λ,µ ∈ Λ+(n) and there
a a sℓ
exists 1 ≤ i ≤ ℓ such that λ = µ −1 = a+s or there exists ℓ+1 ≤ i ≤ n such that
i i
λ = µ −1 = −a−1−s, and in addition, λ = µ for all i 6= j. We have the following
i i j j
lemma.