Page 1

arXiv:1003.2179v1 [math.RT] 10 Mar 2010

REPRESENTATION THEORY OF RECTANGULAR FINITE

W-ALGEBRAS

JONATHAN BROWN

Abstract. We classify the finite dimensional irreducible representations of rectan-

gular finite W-algebras, i.e., the finite W-algebras U(g,e) where g is a symplectic or

orthogonal Lie algebra and e ∈ g is a nilpotent element with Jordan blocks all the

same size.

1. Introduction

This paper concerns the representation theory of the finite W-algebra U(g,e) associ-

ated to a nilpotent element e in a reductive Lie algebra g. The main focus of this paper

is the representation theory of the finite W-algebras associated to nilpotent elements

in the symplectic or orthogonal Lie algebras whose Jordan blocks are all the same size.

We refer to these simply as rectangular finite W-algebras.

The general definition of finite W-algebras is due to Premet in [P1], though in some

cases they had been introduced much earlier by Lynch in [Ly] following Kostant’s

celebrated work on Whittaker modules in [K]. The terminology “finite W-algebra”

comes from the mathematical physics literature, where finite W-algebras are the finite

type analogs of the vertex W-algebras defined and studied for example by Kac, Roan,

and Wakimoto in [KRW]. The precise identification between the definitions in [P1]

and [KRW] was made only recently by D’Andrea, De Concini, De Sole, Heluani, and

Kac in [D3HK].

There are many remarkable connections between finite W-algebras and other ar-

eas of mathematics. The finite W-algebra U(g,e) possesses two natural filtrations,

the Kazhdan and loop filtrations. The main structure theorem for finite W-algebras,

proved in [P1] and reproved in [GG], is that the associated graded algebra to U(g,e)

with respect to the Kazhdan filtration is isomorphic to the coordinate algebra of the

Slodowy slice, i.e. U(g,e) is a quantization of the Slodowy slice through the nilpotent

orbit containing e. On the other hand, by [P2] the associated graded algebra with

respect to the loop filtration is isomorphic to U(ge), the universal enveloping algebra

of the centralizer of e in g. Because of this, the structure of U(g,e) is intimately related

to the invariant theory of the centralizer ge. In [BB] this connection was used to con-

struct a system of algebraically independent generators for the center of the universal

enveloping algebra U(ge) in the case g = gln(C), giving a constructive proof of the

freeness of this center (which had been established earlier by Panyushev, Premet and

Yakimova in [PPY] by a different method) and also verifying [PPY, Conjecture 4.1].

The work of Premet in [P2, P3], Losev in [Lo1, Lo2], and Ginzburg in [Gi] has

highlighted the importance of the study of finite dimensional representations of U(g,e),

revealing an intimate relationship with the theory of primitive ideals of the universal

enveloping algebra U(g) itself. At the heart of this connection is an equivalence of

categories due to Skryabin in [Sk] between the category of U(g,e)-modules and a

1

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2JONATHAN BROWN

certain category of generalized Whittaker modules for g.

about the representation theory of finite W-algebras see e.g. [Lo3], [Lo4], [Go], [GRU].

For other recent results

1.1. Statement of the main results. Throughout this paper we denote the general

linear, symplectic, and orthogonal Lie algebras gln(C), spn(C), and son(C) as gn, g−

and g−

nfor short, assuming that n is even if g = spn(C). We will also need the following

index set defined in terms of a positive integer n:

n,

In= {1 − n,3 − n,...,n − 1}.

n denote the twisted Yangians associated to g+

These are certain associative algebras with generators {S(r)

[MNO] for the full relations. Fix positive integers n and l, and a sign ǫ ∈ {±}, now let

g = gǫ

such a nilpotent exists one must further assume that if ǫ = + and l is even then n is

even, and that if ǫ = − and l is odd then n is even. Let U(g,e) be the finite W-algebra

attached to g and the nilpotent element e; see §2.1 below for the general definition.

We will also need another sign φ defined to be ǫ if l is odd, and −ǫ if l is even. Set

Y = Yφ

n. The main result of [B1] is the following theorem:

Let Y+

n and Y−

nand g−

i,j| i,j ∈ In,r ∈ Z>0}; see

n, respectively.

nl. Let e be a nilpotent element of Jordan type (ln) in g. In order to ensure that

Theorem 1.1. There exists a surjective algebra homomorphism Y ։ U(g,e) with

kernel generated by the elements

?

?

Results along these lines were first noticed by Ragoucy in [R], where he observed

that a similar homomorphism exists in the case that l is odd for certain commutative

analogs of these algebras.

The main aim of the present article is to combine this theorem with Molev’s classi-

fication of the finite dimensional irreducible representations of twisted Yangians from

[M] to deduce a classification of finite dimensional irreducible representations of the

rectangular finite W-algebras. The main combinatorial objects in this classification

are skew-symmetric n × l tableaux. A skew-symmetric n × l tableaux is an n × l ma-

trix of complex numbers, with rows labeled in order from top to bottom by the set

Inand columns labeled in order from left to right by the set Il, and which is skew-

symmetric with respect to the center of the matrix, that is, if A = (ai,j)i∈In,j∈Ilis

a skew-symmetric n × l tableaux then ai,j = −a−i,−j. Let Tabn,ldenote the set of

skew-symmetric n × l tableaux. We say that two skew-symmetric n × l tableaux are

row equivalent if one can be obtained from the other by permuting entries within rows.

Let Rown,ldenote the set of row equivalence classes of skew-symmetric n×l tableaux.

In the following definition (and from here on) we use the partial order ≥ on C

defined by a ≥ b if a − b ∈ Z≥0. A skew-symmetric n × l tableaux A = (ai,j)i∈In,j∈Il

is ǫ-column strict if

- the entries in every column except for the middle column (which exists only

when l is odd) are strictly decreasing from top to bottom, i.e., a1−n,j >

a3−n,j> ··· > an−1,jfor all 0 ?= j ∈ Il;

S(r)

i,j

??? i,j ∈ In,r > l

?

?

if l is even;

S(r)

i,j+φ

2S(r−1)

i,j

??? i,j ∈ In,r > lif l is odd.

(1.1)

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REPRESENTATION THEORY OF RECTANGULAR FINITE W-ALGEBRAS3

- if l is odd and n is even then the entries in the middle column satisfy a1−n,0>

a3−n,0> ··· > a−1,0, and they also satisfy a−1,0> 0 if ǫ = −, and they satisfy

a−3,0+ a−1,0> 0 if ǫ = + and n ≥ 4;

- if l is odd and n is odd then the entries in the middle column satisfy a1−n,0>

a3−n,0> ··· > a−2,0, and they also satisfy 2a−2,0> 0.

Let Colǫ

n,ldenote the set of all ǫ-column strict skew-symmetric n×l tableaux, and let

Stdǫ

We relate these sets to certain representations of the twisted Yangian Y .

convenient to use the power series

?

where S(0)

i,j= δi,j. A Y -module V is called a highest weight module if it generated by a

vector v such that Si,j(u)v = 0 for all i < j and if for all i we have that Si,i(u)v = µi(u)v

for some power series µi(u) ∈ 1 + u−1C[[u−1]]. To a skew-symmetric n × l tableaux

A = (ai,j)i∈In,j∈Ilwe associate a (unique up to isomorphism) irreducible highest weight

Y -module generated by a highest weight vector v for which

(u −i

2)v = (u + ai,1−l)(u + ai,3−l)...(u + ai,l−1)v

if l is even and i ≥ 0, or

(u −i

2

= (u + ai,1−l)(u + ai,3−l)...(u + ai,−2)(u + ai,0+ δi,0/2)(u + ai,2)...(u + ai,l−1)v

n,ldenote the set of elements of Rown,lwhich have a representative in Colǫ

n,l.

It is

Si,j(u) =

r≥0

S(r)

i,ju−r∈ Y [[u−1]],(1.2)

2)lSi,i(u −i

2)l−1(u +φ − i

)Si,i(u −i

2)v

if l is odd and i ≥ 0. This Y -module factors through the surjection Y ։ U(g,e) from

Theorem 1.1 to yield a (not necessarily finite dimensional) irreducible U(g,e)-module

denoted L(A) for each A ∈ Rown,l. Moreover these are the only highest weight Y -

modules which descend to U(g,e), so the problem of classifying the finite dimensional

irreducible representations is reduced to determining exactly which L(A)’s are finite

dimensional, which can be deduced from Molev’s results in [M]. The following is the

main theorem of this paper:

Theorem 1.2. Suppose A ∈ Rown,l.

(i) If l is odd or if l is even and ǫ = + then L(A) is finite dimensional if and only

if A has a representative in Colǫ

n,l. Hence

{L(A) | A ∈ Stdǫ

n,l}

is a complete set of isomorphism classes of the finite dimensional irreducible

representations of U(g,e).

(ii) If l is even and ǫ = − then L(A) is finite dimensional if and only if A+has a

representative in Col+

n,l+1. Hence

{L(A) | A ∈ Rown,l,A+∈ Std+

n,l+1}

is a complete set of isomorphism classes of the finite dimensional irreducible

representations of U(g,e).

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4JONATHAN BROWN

In the theorem A+denotes the skew-symmetric n × (l + 1) tableaux obtained by

inserting a middle column into A with entries

n

2− 1,n

if n is even and

n

2− 1,n

if n is odd down the middle column.

The classification in Theorem 1.2 meshes well with the general framework of highest

weight theory for finite W-algebras developed in [BGK]. Under this framework for

each A ∈ Rown,lone can associate an irreducible U(g,e)-module. In §5 we show that

this module is isomorphic to L(A) for each A ∈ Rown,l.

The theorem also helps illuminate the connection between U(g)-modules and U(g,e)-

modules via primitive ideals. For an algebra A let Prim A denote the set of primitive

ideals in A. In [Lo2] Losev showed that there exists a surjective map

2− 2,...,1,0,0,−1,−2,... ,1 −n

2

2− 2,...,1

2,0,−1

2,−3

2,...,1 −n

2

† : PrimfinU(g,e) → PrimG.eU(g).

Here G is the adjoint group of g, PrimfinU(g,e) denotes the primitive ideals of U(g,e)

of finite co-dimension, and

PrimG.eU(g) = {I ∈ Prim U(g) | VA(I) = G.e},

where VA(I) denotes the associated variety of an ideal I in U(g). Moreover, Lo-

sev showed that the fibers of the map † are C-orbits, where C = CG(e)/CG(e)◦is

the component group associated to the nilpotent element e, which acts naturally as

automorphisms on U(g,e) (induced ultimately by its adjoint action on U(g)).

In our special cases we can calculate explicitly the action of C on the set of finite di-

mensional irreducible U(g,e)-modules, and therefore on PrimfinU(g,e). By [C, Chapter

13] the only rectangular finite W-algebras for which C is not trivial are the ones where

ǫ = −, and n and l are both even, in which case C∼= Z2. To explicitly state the C-

action we need to define the notion of a ♯-special element of a list of complex numbers.

Given a list (a1,...,a2k+1) of complex numbers let {(a(i)

set of all permutations of this list which satisfy a(i)

Assuming such rearrangements exist, we define the ♯-special element of (a1,...,a2k+1)

to be the unique maximal element of the set {a(i)

such rearrangements exist, we say that the ♯-special element of (a1,...,a2k+1) is un-

defined. For example, the ♯-special element of (−3,−1,2) is −3, whereas the ♯-special

element of (−3,−2,1) is undefined.

We define an action of Z2on Rown,las follows. Let A = (ai,j)i∈In,j∈Il∈ Rown,l,

let a be the ♯-special element of (0,a−1,l−1,a−1,l−3,...,a−1,l−1), and let c denote the

generator of Z2. If a is undefined or a = 0 then we declare that c · A = A. Otherwise

we declare that c · A = B where B ∈ Rown,lhas the same rows as A, except with one

occurrence of a replaced with −a in row −1, and one occurrence of −a replaced with

a in row 1. It is an immediate corollary of Lemma 3.19 below that this action is well

defined. For example,

1,...,a(i)

2j> 0 for each j = 1,...,k.

2k+1) | i ∈ I} be the

2j−1+ a(i)

2k+1| i ∈ I}. On the other hand, if no

c ·-3 1 2 4

-4 -2 -1 3

=-3 -2 1 4

-4 -1 2 3.

since the ♯-special element of (0,−3,1,2,4) is 2.

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REPRESENTATION THEORY OF RECTANGULAR FINITE W-ALGEBRAS5

In §6 we prove the following theorem:

Theorem 1.3. Suppose that n and l are even positive integers and ǫ = −. Let A =

(ai,j)i∈In,j∈Il∈ Rown,lbe such that A+∈ Std+

finite dimensional irreducible representation of U(g,e). Then the ♯-special element of

(0,a−1,l−1,a−1,l−3,...,a−1,l−1) is defined, and c · L(A) = L(c · A).

n,land let L(A) denote the corresponding

Understanding the C-action for the rectangular finite W-algebras turns out to be

key to understanding the C-action for more complicated finite W-algebras. In the

forthcoming paper [BroG] we use these results as well as the results in [BGK] to

classify the finite dimensional irreducible representations of U(g,e) for a large class of

nilpotent elements in the symplectic and orthogonal Lie algebras.

Acknowledgements.

enlightening conversations, and he would like to thank Simon Goodwin for pointing

out and correcting an error in §6.

The author would like to thank Jonathan Brundan for many

2. Rectangular finite W-algebras

2.1. Overview of finite W-algebras. Throughout this subsection g denotes a reduc-

tive Lie algebra and e denotes a nilpotent element of g. To define the finite W-algebra

U(g,e), one first applies the Jacobson-Morozov Theorem to embed e into an sl2-triple

(e,h,f). Now the ad h eigenspace decomposition gives a grading on g:

?

where g(i) = {x ∈ g | [h,x] = ix}. Finite W-algebras are defined for any grading,

however to simplify the definition of U(g,e), we assume that this grading is an even

grading, i.e., g(i) = 0 if i is odd. Define a character χ : g → C by χ(x) = (x,e),

where (.,.) is a fixed non-degenerate symmetric invariant bilinear form on g. Let

m =?

U(g) = U(p) ⊕ I.

g =

i∈Z

g(i),(2.1)

i<0g(i), and let p =?

i≥0g(i). Let I be the left ideal of U(g) generated by

{m − χ(m) | m ∈ m}. By the PBW Theorem,

(2.2)

Define pr : U(g) → U(p) to be the projection along this direct sum decomposition.

Now we define

U(g,e) = {u ∈ U(p) | pr([m,u]) = 0 for all m ∈ m},

so U(g,e) is a subalgebra of U(p) in these even grading cases.

The finite W-algebra U(g,e) possesses two natural filtrations. The first of these,

the Kazhdan filtration, is the filtration on U(g,e) induced by the filtration on U(g)

generated by declaring that each element x ∈ g(i) in the grading (2.1) is of degree

i+2. The fundamental PBW theorem for finite W-algebras asserts that the associated

graded algebra to U(g,e) under the Kazhdan filtration is canonically isomorphic to the

coordinate algebra of the Slodowy slice at e; see e.g. [GG, Theorem 4.1].

The second important filtration is called the good filtration. The good filtration is the

filtration induced on U(g,e) by the grading (2.1) on U(p). According to this definition,

the associated graded algebra grU(g,e) is identified with a graded subalgebra of U(p).

Page 6

6JONATHAN BROWN

The fundamental result about the good filtration, which is a consequence of the PBW

theorem and [P2, (2.1.2)], is that

grU(g,e) = U(ge)

as graded subalgebras of U(p), where gedenotes the centralizer of e in g; see also

[BGK, Theorem 3.5].

(2.3)

2.2. Rectangular finite W-algebras and twisted Yangians. Recall that a rectan-

gular finite W-algebra is a finite W-algebra U(g,e) for which g is son(C) or spn(C) and e

has Jordan blocks all the same size. We need to recall the many of the results from [B1]

about the relationship between twisted Yangians and rectangular finite W-algebras.

We begin by fixing explicit matrix realizations for the classical Lie algebras. Recall

that for any integer n ≥ 1, we have defined the index set In= {1−n,3−n,...,n−1}.

Let gn= gln(C) with standard basis given by the matrix units {ei,j| i,j ∈ In}. Let

J+

nbe the n × n matrix with (i,j) entry equal to δi,−j, and set

g+

where xTdenotes the usual transpose of an n × n matrix. Assuming in addition that

n is even, let J−

nbe the n×n matrix with (i,j) entry equal to δi,−jif j > 0 and −δi,−j

if j < 0, and set

g−

We adopt the following convention regarding signs. For i ∈ In, define ˆ ı ∈ Z/2Z by

?

1

We will often identify a sign ± with the integer ±1 when writing formulae. For example,

ǫˆ ıdenotes 1 if ǫ = + or ˆ ı = 0, and it denotes −1 if ǫ = − and ˆ ı = 1. With this notation,

gǫ

nis spanned by the matrices {fi,j| i,j ∈ In}, where

fi,j= ei,j− ǫˆ ı+ˆ e−j,−i.

n= son(C) = {x ∈ gn| xTJ+

n+ J+

nx = 0},

n= spn(C) = {x ∈ gn| xTJ−

n+ J−

nx = 0}. (2.4)

ˆ ı =

0if i ≥ 0;

if i < 0.

(2.5)

Next we fix integers n,l ≥ 1 and signs ǫ,φ ∈ {±}, assuming that φ = ǫ if l is odd,

φ = −ǫ if l is even, and φ = + if n is odd; now let g = gǫ

e ∈ g of Jordan type (ln) we introduce an n×l rectangular array of boxes, labeling rows

in order from top to bottom by the index set Inand columns in order from left to right

by the index set Il. Also label the individual boxes in the array with the elements of

the set Inl. For a ∈ Inlwe let row(a) and col(a) denote the row and column numbers of

the box in which a appears. We require that the boxes are labeled skew-symmetrically

in the sense that row(−a) = −row(a) and col(−a) = −col(a). If ǫ = − we require in

addition that a > 0 either if col(a) > 0 or if col(a) = 0 and row(a) > 0; this additional

restriction streamlines some of the signs appearing in formulae below. For example, if

n = 3,l = 2 and ǫ = −,φ = +, one could pick the labeling

nl. To define a nilpotent element

-5 1

-3 3

-1 5

and get that row(1) = −2 and col(1) = 1. We remark that the above arrays are a

special case of the pyramids introduced by Elashvili and Kac in [EK]; see also [BruG].

Page 30

30JONATHAN BROWN

So in all cases we have that

S−η(S−γ(si,i(ep,p+ ρp))) =

si,i(ep,p) +i

si,i(ep,p) +i

si,i(ep,p)

2

2−ǫ

if p ?= 0;

if p = 0, i ?= 0;

if p,i = 0.

2

?

Let E(r)

i

denote the rth elementary symmetric function in

{fa,a+row(a)

2

| a ∈ Inl,row(a) = i,col(a) ∈ Il}.

Recall the definition of ωrfrom (2.27)

Lemma 5.9. Let i ∈ InIf i > 0, and l is even then

S−η◦ π−γ(si,i(ωr)) = E(r)

i.

If i ≥ 0 and l is odd then

S−η◦ π−γ(si,i(ωr)) =

r−1

?

i=0

(−2ǫ)iE(r−i)

i

.

Proof. If l is even then by Lemma 5.7, (5.6), and Lemma 5.8 we have that

S−η◦ π−γ(si,i(ω(u)) = si,i(u + e1−l,1−l+ i/2)...si,i(u + el−1,l−1+ i/2)(5.8)

so the lemma holds in this case.

Now we consider the l odd case. Let

Pi(u) = si,i(u + e1−l,1−l+ i/2)...si,i(u + e−2,−2+ i/2)

× si,i(u + e0,0+ i/2 − ǫ/2)si,i(u + e2,2+ i/2) + ...si,i(u + el−1,l−1+ i/2),

and

Qi(u) = si,i(u + e1−l,1−l+ i/2)...si,i(u + e−2,−2+ i/2)

× si,i(e0,0+ i/2 − ǫ/2)si,i(u + e2,2+ i/2) + ...si,i(u + el−1,l−1+ i/2).

So

S−η◦ π−γ(si,i(ω(u)) = Pi(u) +

∞

?

r=1

(−2ǫu)−rQi(u).

Observe that Pi(u) = P′

P′

i(u) = si,i(u + e1−l,1−l+ i/2)...si,i(u + e−2,−2+ i/2)

× si,i(u + e0,0+ i/2)si,i(u + e2,2+ i/2)...si,i(u + el−1,l−1+ i/2),

i(u) −ǫ

2P′′

i(u) and Qi(u) = Q′

i(u) −ǫ

2P′′

i(u) where

(5.9)

P′′

i(u) = si,i(u + e1−l,1−l+ i/2)...si,i(u + e−2,−2+ i/2)

× si,i(u + e2,2+ i/2)...si,i(u + el−1,l−1+ i/2),

and

Q′

i(u) = si,i(u + e1−l,1−l+ i/2)...si,i(u + e−2,−2+ i/2)

× si,i(e0,0+ i/2)si,i(u + e2,2+ i/2)...si,i(u + el−1,l−1+ i/2).

Also observe that

P′′

i(u) +1

uQ′

i(u) =1

uP′

i(u).

Page 31

REPRESENTATION THEORY OF RECTANGULAR FINITE W-ALGEBRAS 31

So

Pi(u) −

ǫ

2uQi(u) = P′

i(u) −ǫ

2P′′

ǫ

2uP′

ǫ

2uPi(u).

i(u) −

ǫ

2uQ′

1

4uP′′

i(u) +

1

4uP′′

i(u)

= P′

i(u) −

i(u) +

i(u)

= P′

i(u) −

Thus

S−η◦ π−γ(si,i(ω(u)) = Pi(u) +

∞

?

(−2ǫu)−rQi(u)

r=1

(−2ǫu)−rQi(u)

= Pi(u) −

ǫ

2uQi(u) +

∞

?

ǫ

2u

r=2

= P′

i(u) −

ǫ

2uPi(u) −

?

∞

?

∞

?

?

r=1

(−2ǫu)−rQi(u)

= P′

i(u) −

ǫ

2u

Pi(u) +

r=1

(−2ǫu)−rQi(u)

?

.

So we have that

Pi(u) +

∞

?

r=1

(−2ǫu)−rQi(u) = P′

i(u) −

ǫ

2u

Pi(u) +

∞

?

r=1

(−2ǫu)−rQi(u)

?

,

and solving this equation for

Pi(u) +

∞

?

r=1

(−2ǫu)−rQi(u)

gives that

S−η◦ π−γ(si,i(ω(u)) =

∞

?

r=0

(−2ǫu)−rP′

i(u),(5.10)

which implies the lemma.

?

Now we explain how irreducible highest weight U(g,e)-modules under the BGK

highest weight theory are related to the irreducible highest weight U(g,e)-modules

from Theorem 1.2. To each skew-symmetric n×l tableaux we associate an element of

t∗in the following way. For each A = (ai,j)i∈In,j∈Il∈ Tabn,lwe define the weight

?

Under this association, t∗= Tabn,l, and t∗/W0 = Rown,l. Let ΛA denote the one-

dimensional U(g0,e)-module obtained by lifting the one-dimensional S(t)W0-module

corresponding to λAthrough ξ−η.

Theorem 5.10. Let A ∈ Rown,l. Then L(A)∼= L(ΛA,q).

λA=

b∈Inl∩Z>0

arow(b),row(b)ǫb∈ t∗.

Page 32

32JONATHAN BROWN

Proof. First note that L(A) is a BGK-highest weight module, since if v is a highest

weight vector for L(A) then si,j(ωr)v = 0 when i < j and si,i(u−lω(u))v = µi(u) for

some µi(u) ∈ 1 + u−1C[[u−1]]. Next by conferring with the definition of L(A) in §1,

(2.28), (5.8), (5.9), and (5.10) we see that the action of si,i(u−lω(u)) on the highest

weight vector for L(A) and on the the highest weight vector for L(ΛA,q) are the same.

Thus the theorem follows from [BGK, Theorem 4.5].

?

6. Action of the Component Group C

In this section we show how to explicitly calculate the action of the component

group C = CG(e)/Cg(e)◦= CG(e,h,f)/CG(e,h,f)◦on the set of finite dimensional

irreducible U(g,e)-modules. Here CG(e,h,f) denotes the centralizer of the sl2-triple

(e,h,f) in the adjoint group G of g. Recall Losev’s near classification of finite di-

mensional irreducible representations of U(g,e) from the introduction: there exists a

surjective map

† : PrimfinU(g,e) ։ PrimG.eU(g),

and the fibers of this map are precisely C-orbits.

In our special cases we can calculate explicitly the action of C on the set of fi-

nite dimensional irreducible U(g,e)-modules, and therefore on PrimfinU(g,e). By [C,

Chapter 13] the only cases where C is not trivial are the cases when ǫ = −, and n

and l are both even; so unless otherwise indicated we assume this for the rest of this

section.

Recall that in §2.2 we introduced an n × l rectangular array to specify coordinates.

Now we claim that

?

row(a)/ ∈{±1}

c =

a∈Inl

ea,a+

?

a,b∈Inl

col(a)=col(b)

row(a)=1

row(b)=−1

ea,b+ eb,a

generates C. To see this note that conjugating with c simply transposes each pair

of indices a,b ∈ Inl where col(a) = col(b),row(a) = 1,row(b) = −1.

is an even number of transpositions, we have that detc = 1. It is also clear that

c.J−.c = J−(recall that g is defined with J−in (2.4)) since for a ∈ Inl∩ Z>0 we

have that c.e−a,a.c = e−b,band c.ea,−a.c = eb,−bfor some b ∈ Inl∩ Z>0. Thus we have

that c ∈ G. Furthermore, c.h.c = h (see (2.8) for the definition of h) since for a ∈ Inl

c.ea,a.c = eb,bfor some b such that col(b) = col(a). Next note c.e.c = e (see (2.6) for

the definition of e) since for a,b ∈ Inlsuch that row(a) = row(b),col(a) + 2 = col(b),

c.fa,b.c = fa,bif row(a) / ∈ {±1}, and if row(a) = 1 and col(b) ≥ 1 then c.fa,b.c = fa′,b′

where col(a′) = col(a),row(a′) = −1 and col(b′) = col(b),row(b′) = −1. So we have

that c ∈ CG(e,h,f). Next we show that c / ∈ CG(e,h,f)◦. By [J, §3.8] we have that

CG(e,h,f)∼= On(C). Next observe that CG(h)∼= GLn(C)×l/2(confer (2.10)), and

that the projection of c into any of these copies of GLn(C) has determinant -1, thus

c / ∈ CG(e,h,f)◦. Therefore C = ?c?.

To understand the action of C on the set of finite dimensional irreducible U(g,e)-

modules, we calculate the action of C on {prsi,j(ω(u))|i,j ∈ In}. Recall the definition

of si,jfrom (2.21). Note that c.si,j(ep,p) = si′,j′(ep,p) where i′= i if i / ∈ {±1}, i′= −i

otherwise. Thus c.prs1,1(ω(u)) = prs−1,−1(ω(u)), and c.prsi,i(ω(u)) = prsi,i(ω(u))

Since this

Page 33

REPRESENTATION THEORY OF RECTANGULAR FINITE W-ALGEBRAS33

for i / ∈ {±1}. Since by Theorem 2.3 κl(Si,j(u) = µ(si,j(ω(u))), we see that the action

of c on the U(g,e)-module L(A) is the same as the action of ψ.

We can now prove Theorem 1.3:

Proof. We have that L(A) = L(A+) = L(¯ µ(u)) as Y+

1

2u−1)−1(1 + ci,−lu−1)(1 + ci,2−lu−1)...(1 + ci,lu−1) are given from (4.1). Since µi(u)

must be a polynomial of degree at most k, we must also have for each i ∈ {1,...,n−1}

that ci,k=1

ement of (c1,−l,...,c1,l). By Theorem 3.18 L(¯ µ(u))♯= L((µ♯

where µ♯

skew symmetric n × l tableaux B = (bi,j) associated to L((µ♯

by (4.1) satisfies b1,0= −1/2 + (1 − c1,0) = −1/2 + (1 − (a1,0+ 1/2)) = −a1,0, and

bi,j= ai,jfor all (i,j) ?= (±1,0).

n-modules where µi(u) = (1 +

2for some k. After re-indexing we may assume that c1,0is the ♯′-special el-

1(u),µ2(u),...,µn−1(u))),

1(u) = (1+1

2u−1)−1(1+c1,−lu−1)...(1+(1−c1,0)u−1)...(1+c1,lu−1). So the

1(u),µ2(u),...,µn−1(u)))

?

Throughout this paper G denotes the adjoint group associated to g. It will be useful

in future work to consider the action of the group C′= COnl(C)(e,h,f)/COnl(C)(e,h,f)◦

on the set of finite dimensional irreducible U(g,e)-modules in the case when ǫ = +

and n is even and l is odd. In these cases, C′ ∼= Z2and is generated by c where

?

row(a)/ ∈{±1}

c =

a∈Inl

ea,a+

?

a,b∈Inl

col(a)=col(b)

row(a)=1

row(b)=−1

ea,b+ eb,a.

As before, the action of c on a finite dimensional U(g,e)-module L(A) is the same as

the action of the Y+

n-automorphism ψ, and so we obtain the following theorem, whose

proof is essentially the same as the proof for Theorem 1.3.

Theorem 6.1. Suppose that n is even l is odd integers and ǫ = +. Let A = Std+

and let L(A) denote the corresponding finite dimensional irreducible representation

of U(g,e). Then the ♯-special element of (a−1,l−1,a−1,l−3,...,a−1,l−1) is defined; let

a denote the ♯-special element of this list. Let c denote the generator of C′. Then

c.L(A) = L(B) where B ∈ Std+

n,lhas the same rows as A, except with one occurrence

of a replaced with −a in row −1, and one occurrence of −a replaced with a in row 1.

n,l,

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[C]

[DK]

School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT,

UK

E-mail address: brownjs@maths.bham.ac.uk