李超代數與無窮維李代數之表現理論的研究(1/3)

行政院國家科學委員會專題研究計畫 期中進度報告

李超代數與無窮維李代數之表現理論的研究(1/3)

中 華 民 國 94 年 5 月 17 日

SUPERALGEBRAS AND INFINITE-DIMENSIONAL LIE ALGEBRAS (1/3)

SHUN-JEN CHENG

The purpose of this note is to report on [CWZ] on certain connections between the representation theories of Lie superalgebras and Lie algebras funded by NSC-grant 93-2115-M-002-015 of the R.O.C. For more details the interested reader is referred to the paper. We are grateful to the NSC for supporting this work.

1. Introduction

In 1977 Kac classified finite-dimensional complex simple Lie superalgebras [K]. Since then the representation theory of these Lie superalgebras has been studied extensively. Especially the problem of finding a character formula for the finite-dimensional irreducible representations. It turned out that the problem of charac-ter for the most basic Lie superalgebras, e.g. general linear Lie superalgebras and ortho-sympletic Lie superalgebras, is the most challenging.

For the general linear Lie superalgebra many partial results have been obtained. In 1996 Serganova [S] was the first to obtain a complete solution. She defined Kazhdan-Lusztig polynomials [KL], and found an algorithm to compute them. But her method is difficult to implement in practice.

In 2003 Brundan [B] gave a satisfying solution to the problem by reformulating these polynomials as coefficients of a certain transition matrix on a Fock space.

In [CWZ] we offer another different solution to this problem.

2. Combinatorial Character Formula for gl(m|n)

Let gl(m|n) be the general linear superalgebra over C and let Cm|n _{be the}

com-plex superspace of dimension (m|n). Let Tk _{= (C}m|n₎⊗k _{be the k-th tensor power,}

on which gl(m|n) acts and it is known that Tk_{is completely reducible. By [BR, Sv]}

the irreducible modules of gl(m|n) that appear in Tk _{are precisely of the following}

form: Let λ be a partition lying in the (m|n)-hook, i.e. λ1 ≤ n, where

λ = (λ−m, · · · , λ−1; λ1, λ2, · · · ), We write λ = (λ<0_{; λ}>0_{), where}

λ<0 _{= (λ}

−m, · · · , λ−1), λ>0 = (λ1, λ2, · · · ).

Let λ\_{:= (λ}<0_{; (λ}>0₎0_{). Then the irreducible representations of gl(m|n) that appear}

in Tk _{are precisely of highest weights λ}\ _{such that |λ| = k.}

In what follows for µ a highest weight of gl(m|n) we let Ln(µ) denote the

irre-ducible module of highest weight µ. 1

2 SHUN-JEN CHENG

The character of Ln(λ\) [BR, Sv] is given by the Hook Schur polynomial in x−m, · · · , x−1and x1, · · · , xn, which is as follows: Let sλ(x−m, · · · , x−1; x1, · · · , xn, · · · )

be the Schur function associated to the partition λ and regard it as a character of the infinite-dimensional Lie algebra gl(m + ∞). We will write L∞(λ) or simply

just L(λ) for the irreducible representation of gl(m + ∞) of highest weight λ. Let ω+ be the involution defined by

ω+( ∞ Y i=1 1 1 − txi ) = ∞ Y i=1 (1 + txi).

Here t is an indeterminate. We apply ω+ to sλ(x−m, · · · , x−1; x1, · · · ) to obtain the so-called Hook Schur function associated to λ\_{. Now we set the variables} xn+1 = xn+2 = · · · = 0, and the resulting polynomial is precisely the Hook Schur

polynomial associated to the partition λ\_{. That is}

ch_Ln(λ\₎ = ω₊(ch_L(λ))(x_−m, · · · , x₋₁; x₁, · · · , x_n, 0, · · · ). (2.1)

In fact one can let n → ∞ and one thus obtain

chL(λ\₎ = ω₊(ch_L(λ)), (2.2)

where L(λ\_{) denotes the irreducible gl(m|∞)-module of highest weight λ}\_.

The derivation of (2.1) and (2.2) relies heavily the complete reducibility structure of Tk _{and also not all the finite-dimensional highest weights of gl(m|n) are of the}

form λ\_{, where λ lies in the (m|n)-hook.}

In [CWZ] it is shown that (2.1) and (2.2) indeed hold for general finite-dimensional highest weight modules of gl(m|n). But we need to be careful, as one cannot sim-ply apsim-ply ω+ to any character of L(µ), for any µ a gl(m|n)-highest weight. First note that if µ is a finite-dimensional gl(m|n)-highest weight, then writing

µ = (µ−m, · · · , µ−1; µ1, · · · , µn),

one has µi− µi+1 ∈ Z+, for i = −m, · · · , −2, 1, · · · , n − 1. Let h be the standard Cartan subalgebra and let xj = eδj. Here δj is the j-th fundamental weight.

Recall that gl(m|n) has a standard consistent Z-gradation

gl(m|n) = gl(m|n)−1⊕ gl(m|n)0⊕ gl(m|n)+1,

where gl(m|n)0 = gl(m)⊕gl(n). Let L0n(µ) be the irreducible gl(m)⊕gl(n)-module

of highest weight µ and extend it trivially to a gl(m|n)+1⊕ gl(m|n)0-module. The Kac module is

Kn(µ) = Indgl(m|n)_gl(m|n)≥0L0n(µ).

By Kac’s results one knows that Kn(µ) is irreducible if and only if µ is a typical

weight. Since the character of Kn(µ) is easy, the typical case can be ignored.

Now by tensoring with the one-dimensional representation of gl(m|n) of highest weight (α, · · · , α; −α, · · · , −α), α ∈ C, if necessary, we may restrict ourselves to the case when µ is integral. Now tensoring with (−p, · · · , −p; p, · · · , p), p ∈ Z+, if necessary, we may restrict ourselves to the case when µ has the form that (µ1, · · · , µn) is a partition. Now for such a µ it is easy to see that the character

the involution ω+ in the limiting case n → ∞. We now can state the following theorem.

Theorem 2.1. [CWZ] For µ as above we have

chLn(µ)(x−m, · · · , x−1; x1, · · · , xn) = ω+(chL(µ\))(x−m, · · · , x−1; x1, · · · , xn, 0, 0, · · · ),

where we regard µ\ _{as a gl(m + ∞)-highest weight.}

We will explain below how to prove this theorem.

3. Brundan’s Kazhdan-Lusztig Theory of gl(m|n)

Motivated by [LLT] Brundan [B] shows that the Kazhdan-Lusztig polynomials for the finite-dimensional representations of gl(m|n) can be realized as coefficients of a certain transition matrix between the standard monomial basis and the dual canonical basis on certain Fock space. We will recall some of his results that we will need in the sequel.

Let Uq(gl(∞)) be the quantum group of gl(∞) acting on its standard module V , whose standard basis we parameterize by integers. Let W = V∗ _{be the}

corre-sponding restricted dual, on which the quantum group also acts naturally. Letting

va, a ∈ Z, be the standard basis for V we let wa∈ V∗ be defined by wa(vb) = (−q)−aδab.

We can form the module Λm_{(V ) ⊗ Λ}n_{(W ), on which U}

q(gl(∞)) acts (via a

co-multiplication).

Let f : I(m|n) = {−m, · · · , −1; 1, · · · , n} → Z with f (−m) > · · · > f (−1) and f (1) < f (2) < · · · < f (n) and denote the set of such f by Zm|n₊ . Let X₊m|n be the set of finite-dimensional integral highest weights for gl(m|n) and let ρ be the half (super) sum of positive roots of gl(m|n). We have a bijection between

X₊m|n → Zm|n₊ given by

λ → fλ,

where fλ(i) = (λ + ρ|δi). The bilinear form here is the usual super bilinear form

and we take ρ to be

ρ = (m, m − 1, · · · , 1| − 1, −2, −3, · · · , −n).

Thus the super Bruhat ordering on X₊m|n induces a partial ordering < on Zm|n₊ and we can import the notion of degree of atypicality to Zm|n₊ .

One has the standard monomial basis for Λm_{(V ) ⊗ Λ}n_{(W ) given by} Kf = vf (−m)∧ · · · ∧ vf (−1)⊗ wf (1)∧ · · · ∧ wf (n).

The space Λm_{(V ) ⊗ Λ}n_{(W ) (actually a topological completion of it) admits a}

bar-involution compatible with < and hence by the usual arguments going back to Kazhdan and Lusztig the space Λm_{(V ) ⊗ Λ}n_{(W ) has two sets of distinguished}

bar-invariant bases called the canonical and the dual canonical basis, also parameterized by Zm|n₊ . We will denote the canonical basis element corresponding to f by Uf and

4 SHUN-JEN CHENG Theorem 3.1. Lf = X g4f `gfKf, Uf = X g4f ugfKf,

where uf f = 1 and ugf ∈ qZ[q], and where `f f = 1 and `gf ∈ q−1Z[q−1], for g 6= f .

Now let λ ∈ X₊m|n and consider the category Om|n _{of finite-dimensional}

gl(m|n)-modules. Let Un(λ) denote the tilting module (with Kn(λ) at the bottom of its Kac

flag). For an element M in Om|n _{let us denote by [M] the corresponding element}

in the Grothendieck group.

Theorem 3.2. [B] [Ln(λ)] = X µ4λ `µλ(1)[Kn(µ)], [Un(λ)] = X µ4λ uµλ(1)[Kn(µ)],

where here uµλ and `µλ are defined via the bijection.

In fact there is a natural correspondence between the combinatorial picture and the representation theoretical picture given by

Kfλ ←→ Kn(λ), Ufλ ←→ Un(λ), Lfλ ←→ Ln(λ).

Brundan has provided formulas to compute these polynomials, and thus obtains a solution to finding the characters of gl(m|n).

Let X₊₊m|n be the subset of X₊m|n consisting of λ with (λ1, · · · , λn) a partition.

Now let us restrict to a smaller category O₊m|n, which consists of modules whose composition factors are of the form Ln(λ), where λ ∈ X++m|n. This subcategory carries all the information of the category Om|n_{. One observes from Brundan’s}

formulas for ugf and `gf that these polynomials satisfy certain stability.

4. Kazhdan-Lusztig Theory of gl(m + n) We can form the Uq(gl(∞))-module Λm(V ) ⊗ Λn(V ).

Let f : I(m|n) → Z with f (−m) > · · · > f (−1) and f (1) > f (2) > · · · > f (n) and denote the set of such f by Zm+n

+ .

Consider the Lie algebra gl(m + n), which we equip with a Z-gradation, similar to gl(m|n) before:

gl(m + n) = gl(m + n)−1⊕ gl(m + n)0⊕ gl(m + n)+1.

Let gl(m+n)≥0= gl(m+n)0⊕gl(m+n)+1 and let X+m+nbe the set of gl(m+n)≥0

-locally finite integral highest weights for gl(m + n) and let ρc be the half sum of

positive roots of gl(m + n). We have a bijection between Xm+n

+ → Zm+n+ given by

λ → fλ,

where fλ(i) = (λ + ρc|δi)c. Here we take ρc to be

ρc= (m, m − 1, · · · , 1|0, −1, −2, · · · , −n + 1).

Thus the Bruhat ordering on Xm+n

One has the standard monomial basis for Λm_{(V ) ⊗ Λ}n_{(V ) given by}

Kf = vf (−m)∧ · · · ∧ vf (−1)⊗ vf (1)∧ · · · ∧ vf (n).

The space Λm_{(V ) ⊗ Λ}n_{(V ) also admits a bar-involution compatible with ≥ and}

hence Λm_{(V )⊗Λ}n_{(V ) has the canonical and the dual canonical basis, parameterized}

by Zm+n

+ . We will denote the canonical basis element corresponding to f by Uf and

the dual canonical basis element by Lf. We have the following standard theorem.

Theorem 4.1. Lf = X g4f lgfKf, Uf = X g4f ugfKf,

where uf f = 1 and ugf ∈ qZ[q], and where lf f = 1 and lgf ∈ q−1Z[q−1], for g 6= f .

Now let λ ∈ Xm+n

+ and consider the category Om+n of finitely generated gl(m +

n)-modules which are locally finite over gl(m+n)≥0 and semisimple over gl(m+n)0 such that the weight spaces are integral. Let Kn(λ) denote the generalized Verma

module of highest weight λ and let Un(λ) denote the tilting module (with Kn(λ)

at the bottom of its generalized Verma flag). For an element M in Om+n _{let us}

denote by [M] the corresponding element in the Grothendieck group.

Theorem 4.2. [CWZ] [Ln(λ)] = X µ≤λ lµλ(1)[Kn(µ)], [Un(λ)] = X µ≤λ uµλ(1)[Kn(µ)], where here again uµλ and lµλ are defined via the above bijection.

Similarly to the super picture there is a natural correspondence between the combinatorial picture and the representation theoretical picture given by

Kfλ ←→ Kn(λ), Ufλ ←→ Un(λ), Lfλ ←→ Ln(λ).

One can obtain quite explicit formulas for these polynomials quite similar to Brundan’s formulas for gl(m|n).

Let Xm+n

++ be the subset of X+m+n consisting of λ with (λ1, · · · , λn) a partition.

Now let us restrict to a smaller category O₊m+n, which consists of modules whose composition factors are of the form Ln(λ), where λ ∈ X++m+n. Again this subcate-gory carries all the information of the catesubcate-gory Om+n_{. From our explicit formulas}

for ugf and lgf one observes that these polynomials again satisfy certain stability.

5. An isomorphism between Λm_{(V ) ⊗ Λ}∞_{(V ) and Λ}m_{(V ) ⊗ Λ}∞_(V∗₎

Using the stability of the polynomials ugf and `gf one can generalize Brundan’s

results to the n = ∞ situation.

Also using the stability of the polynomials ugf and lgf one can generalize the

results on gl(m + n) to the n = ∞ situation.

The Uq(sl(∞))-module Λ∞(V ) [KMS] has a basis of the form vm1 ∧ vm2 ∧ vm3 ∧ · · · ,

6 SHUN-JEN CHENG

m1 > m2 > m3 > · · · and mi = 1 − i, for i >> 0. (Our Λ∞(V ) actually is the zero

sector of KMS.) It has another basis parameterized by partitions λ = (λ1, λ2, · · · ):

|λ >:= vλ1 ∧ vλ2−1∧ vλ3−2∧ · · · .

Let Zm+∞

+ consist of f : I(m|∞) → Z with f (−m) > · · · > f (−1), f (1) > f (2)) >

· · · and f (i) = 1 − i, for i >> 0. A basis of Λm_{(V ) ⊗ Λ}∞_{(V ) is given by standard}

monomial basis

Kf = vf (−m)∧ · · · vf (−1)⊗ vf (1)∧ vf (2)∧ · · · , f ∈ Zm+∞+ .

As in the finite n case we have canonical basis elements Uf and dual canonical basis

element Lf. These basis elements correspond in a similar fashion to generalized

Verma modules, tilting modules and irreducible module in the category Om+∞

+ , which consists of gl(m + ∞)-modules that are finitely generated, locally finite over

gl(m + ∞)≥0, semisimple over gl(m + ∞)0 and whose composition factors are L(λ), with λ ∈ X₊₊m+∞. That is Theorem 4.2 holds for n = ∞ as well. We want to mention that the n = ∞ version of the theorem implies the finite n theorem.

On the super side the Uq(sl(∞))-module Λ∞(W ) has a basis of the form wm1 ∧ wm2 ∧ wm3 ∧ · · · ,

m1 < m2 < m3 < · · · and mi = i, for i >> 0. It has another basis parameterized

by partitions λ = (λ1, λ2, · · · ) as well:

|λ0

∗ >:= w1−λ0

1 ∧ w2−λ02 ∧ w3−λ03 ∧ · · · .

Let Zm|∞₊ consist of f : I(m|∞) → Z with f (−m) > · · · > f (−1), f (1) < f (2)) <

· · · and f (i) = i, for i >> 0. A basis of Λm_{(V ) ⊗ Λ}∞_{(W ) is given by standard}

monomial basis

Kf = vf (−m)∧ · · · vf (−1)⊗ wf (1)∧ wf (2)∧ · · · , f ∈ Zm|∞+ .

We have canonical basis elements Uf and dual canonical basis element Lf. These

basis elements correspond to Kac modules, tilting modules and irreducible mod-ule in the category Om|∞₊ , which consists of gl(m|∞)-modules that are finitely generated, locally finite over gl(m|∞)≥0, semisimple over gl(m|∞)0 and whose composition factors are L(λ), with λ ∈ X₊₊m|∞. That is, Brundan’s Theorem 3.2 holds for n = ∞ as well. Again the n = ∞ version of the theorem implies the finite n theorem. The following theorem is crucial.

Theorem 5.1. [CWZ] The map sending |λ > to |λ0

∗ > induces an isomorphism of Uq(sl(∞))-modules Λm(V ) ⊗ Λ∞(V ) and Λm(V ) ⊗ Λ∞(W ). Furthermore it com-patible with the respective bar involutions and partial orderings.

An immediate consequence of the theorem is that the canonical bases and the dual canonical bases correspond and hence the Kazhdan-Lusztig polynomials cor-respond. That is, if we denote the map by \, then we have

Corollary 5.2. In the n = ∞ case we have

Corollary 5.3. There exists an isomorphism between the Grothendieck groups of Om|∞₊ and Om+∞

+ , which sends K(λ), U(λ) and L(λ) to K(λ\), U(λ\) and L(λ\),

respectively.

The next corollary is an application of Corollary 5.2 together with the symmetric and skew-symmetric Howe dualities for the dual pair (GL, GL).

Corollary 5.4. [CWZ] \ is an isomorphism of Grothendieck rings. In particular

\ preserves the composition factors of the tensor products in Om+∞

+ and O+m|∞. We have mentioned earlier that the n = ∞ cases determine the finite n cases. This follows essentially from the stability properties of the Kazhdan-Lusztig poly-nomials on the combinatorial sides, while on the representation theory sides this follows from stability properties of certain truncation functors which we will not discuss here. From this and Corollary 5.2 we now derive Theorem 2.1: We have

chL(λ)= X

µ

lµλ(1)chK(µ).

Applying the involution ω+ we have

ω+(chL(λ)) = X µ lµλ(1)ω+(chK(µ)) = X µ lµλ(1)chK(µ\₎ = X µ `µ\_λ\(1)ch_K(µ\₎ = ch_L(λ\₎.

Now setting xn+1 = xn+2 = · · · = 0 and using the stability of `µ\_λ\ we get

X

µ

`_µ\_λ\(1)ch_Kn(µ)= ch_Ln(λ\₎.

References

[B] J. Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra gl(m|n), J. Amer. Math. Soc. 16 (2003), 185–231.

[BR] A. Berele and A. Regev, Hook Young Diagrams with Applications to Combinatorics and to

Representations of Lie Superalgebras, Adv. Math. 64 (1987), 118–175.

[CWZ] Cheng, S.-J.; Wang, W. and R. B. Zhang: Super Duality and Kazhdan-Lustig Polynomi-als, preprint.

[K] V. Kac, Lie Superalgebras, Adv. Math. 16 (1977), 8–96.

[KL] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184.

[KMS] M. Kashiwara, T. Miwa, and E. Stern, Decomposition of q-deformed Fock spaces, Selecta Math. (N.S.) 1 (1995), 787–805.

[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.

[S] V. Serganova, Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra gl(m|n), Selecta Math. (N.S.) 2 (1996), 607–651.

[Sv] A. Sergeev, The tensor algebra of the identity representation as a module over the Lie

su-peralgebras gl(n|m) and Q(n), Math. USSR Sbornik 51 (1985), 419–427.

Department of Mathematics, National Taiwan University, Taipei, Taiwan 106