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J. London Math. Soc. (2) 69 (2004) 273­290

C

e 2004 London Mathematical Society
DOI: 10.1112/S0024610703004873

WEIGHT MULTIPLICITY FREE REPRESENTATIONS, g-ENDOMORPHISM ALGEBRAS, AND DYNKIN POLYNOMIALS
DMITRI I. PANYUSHEV

Introduction Throughout this paper, G is a connected semisimple algebraic group defined over an algebraically closed field k of characteristic zero, and g is its Lie algebra. Recently, Kirillov introduced an interesting class of associative algebras connected with the adjoint representation of G [16]. In our paper, such algebras are called gendomorphism algebras. Each g-endomorphism algebra is a module over the algebra of invariants k[g]G ; furthermore, it is a direct sum of modules of covariants. Hence it is a free graded finitely generated module over k[g]G . The aim of this paper is to show that commutative g-endomorphism algebras have intriguing connections with representation theory, combinatorics, commutative algebra, and equivariant cohomology. Let : G - GL(V ) be an irreducible representation, where stands for the highest weight of V . Following Kirillov, one can form an associative k-algebra by taking the G-invariant elements in the G-module End V k[g]. That is, we set C (g) = (End V k[g])G . This algebra will be referred to as the g-endomorphism algebra (of type ). We do not use Kirillov's term `classical family algebra' for C (g); nor we consider `quantum family algebras' in our paper. It is proved in [16] that C (g) is commutative if and only if all weight spaces in V are 1-dimensional. That paper also contains a description of g-endomorphism algebras for simplest representations of the classical Lie algebras. In our paper we do not attempt to dwell upon consideration of particular cases, but rather we try to investigate general properties of such algebras. As a tool for studying g-endomorphism algebras, we use t-endomorphism algebras. Let t be a Cartan subalgebra of g and let W be the corresponding Weyl group. Let End T (V ) denote the set of T -equivariant endomorphisms of V , where T is the maximal torus with Lie algebra t. Then C (t) = (End T (V ) k[t])W is called the t-endomorphism algebra of type . Both C (t)and C (g) are free graded k[g]G -modules of the same rank, and there exists an injective homomorphism of k[g]G -modules r : C (g) - C (t). Moreover, k[g]G is a subalgebra in both C (t) ^ ^ ^ and C (g), and r is a monomorphism of k[g]G -algebras. We show that r becomes an isomorphism after inverting the discriminant D k[g]G .
Received 11 June 2003. 2000 Mathematics Sub ject Classification 17B10, 14L30, 14F43, 16E65. Research supported in part by RFBI grant 01­01­00756.


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Most of our results concern the case in which C (g) is commutative, that is, V is weight multiplicity free (wmf). A considerable amount of wmf representations are minuscule ones. We show that if V is minuscule, then C (g) k[t]W . Here W W is the stabilizer of . The proof relies on a recent result of Broer concerning `small' G-modules [5]. If V is wmf but not minuscule, then both C (t) and C (g) have zero-divisors. We prove that Spec C (t) is a disjoint union of affine spaces of dimension rk g, while Spec C (g) is connected. However, both varieties have the same number of irreducible components, which is equal to the number of dominant ^ weights of V . As a by-product, we obtain the assertion that r is an isomorphism if and only if is minuscule. Since C (g) and C (t) are graded k-algebras, one may consider their Poincar´ series. We explicitly compute these series for any . e The principal result of the paper is that any commutative algebra C (g) is Gorenstein. The proof goes as follows. Any set f1 ,...,fl of algebraically independent homogeneous generators of k[g]G form a system of parameters for C (g). Therefore C (g) is Gorenstein if and only if R() := C (g)/C (g)f1 + ... + C (g)fl is. The finite-dimensional k-algebra R() is isomorphic with (End V )A , where A G is the connected centraliser of a regular nilpotent element. Using this fact, we prove that the socle of R() is one-dimensional, that is, R() is Gorenstein. It is also shown that the Poincar´ polynomial of R() is equal to the e Dynkin polynomial for V . The Dynkin polynomial is defined for any V . It can be regarded as a q -analogue of dim V that describes the distribution of weight spaces with respect to some level function. According to an old result of Dynkin [7], it is a symmetric unimodal polynomial with integral coefficients. Later on, Stanley observed that Dynkin polynomials have rich combinatorial applications and there is a multiplicative formula for them (see [20]). In our setting, the Dynkin e polynomial of a wmf G-module V appears as the numerator of the Poincar´ series of C (g). It is natural to suspect that any reasonable finite-dimensional Gorenstein k-algebra is the cohomology algebra of a `good' variety. Following this harmless idea, we construct for any wmf representation of a simple group G a certain variety R() and HGc (X ) C (g), where X P(V ). We conjecture that H (X ) Gc G is a maximal compact subgroup. If is a minuscule dominant weight, then X is nothing but G/P , a generalised flag variety. In this case, the conjecture follows from the equality C (g) = k[t]W (Theorem 2.6) and the well known description of HGc (G/P ). We also verify (a part of ) the conjecture for some nonminuscule weights. The paper is organised as follows. In Section 1, we collect necessary information on modules of covariants. Section 2 is devoted to basic properties of endomorphism algebras. We describe the structure of these algebras in the commutative case, in particular, for the minuscule weights. Dynkin polynomials and their applications are discussed in Section 3. In Section 4, we give explicit formulas for the Poincar´ series e of endomorphism algebras. The Gorenstein property is considered in Section 5. Finally, in Section 6, we construct varieties X P(V ) and discuss connections between g-endomorphism algebras and (equivariant) cohomology of X .

1. Generalities on modules of covariants Fix a Borel subgroup B G and a maximal torus T B . We will always work with roots, simple roots, positive roots, and dominant weights that are determined


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275

by this choice of the pair (B, T ). For instance, the roots of B are positive and a highest weight vector in some G-module is a B -eigenvector. More specifically, write P for the T -weight lattice, and P+ for the dominant weights in P . Next, (respectively + ) is the set of all (respectively positive) roots, is the set of simple roots, and Q is the root lattice. For a G-module M , let M µ denote the µ-weight space of M (µ P ). If P+ , then V stands for the simple G-module with highest weight . Set mµ = dim Vµ . The notation µ V means that mµ = 0. For instance, we have 0 V if and only if Q. Given P+ , the space (V k[g])G is called the module of covariants (of type ). We will write J (g) for it. Clearly, J (g) is a module over J0 (g) = k[g]G , and J (g) = 0 if and only if Q. The elements of J (g) can be identified with the Gvi fi J (g) equivariant morphisms from g to V . More precisely, an element fi (x)vi V . This interpretation of defines the morphism that takes x g to J (g) will freely be used in the sequel. Since k[g]G is a graded algebra, each J (g) is a graded module too: J (g) =
n0

J (g)n ,

the component of grade n being (V k[g]n )G . The Poincar´ series of J (g) is the e formal power series F (J (g); q ) =
n0

dim J (g)n q n Z[[q ]].

More generally, C being an arbitrary graded ob ject, we write F (C ; q ) for its Poincar´ series. e The following fundamental result is due to Kostant [17, Theorem 11]. Theorem 1.1. J (g) is a free graded J0 (g)-module of rank m0 . Let d1 ,...,dl be the degrees of basic invariants in k[g]G , where l = rk g. It follows from the theorem that F (J (g); q ) is a rational function of the form F (J (g); q ) =
j l i=1

q

ej ()

(1 - q di )

.

j m0 ), which are merely the degrees of a set of The numbers {ej ()} (1 free homogeneous generators of J (g), are called the generalised exponents for V . Another interpretation of generalised exponents is obtained as follows. Let f1 ,...,fl k[g]G be a set of basic invariants, with deg fi = di . It is a homogeneous system of parameters for J (g). Therefore J (g) = (f1 ,...,fl )J (g) H , where H is a graded finite-dimensional k-vector space such that dim H = m0 . Any homogeneous k-basis for H is also a basis for J (g) as k[g]G -module. Let N g denote the set of nilpotent elements of g (the nilpotent cone). Since k[g]G /(f1 ,...,fl ) k[N ], we see that
m

H

(V k[N ])

0

G

and F (H ; q ) =
i=1

q

ej ()

.

The polynomial F (H ; q ) has non-negative integral coefficients and F (H ; q )|q=1 = m0 . It is a q -analogue of m0 . A combinatorial formula for F (H ; q ) was found by Hesselink [13] and Peterson, independently.


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Now we describe another approach to computing F (H ; q ), which is due to Brylinski. Let e N be a regular nilpotent element. Then G·e is dense in N k[G]Ge [17]. Hence H (V )Ge , and in particular, dim(V )Ge = and k[N ] 0 m . Thus the space (V )Ge is equipped with a grading coming from the above isomorphism. A direct description of this grading can be obtained in terms of `jump polynomials'. Fix a principal sl2 -triple {e, h, f } such that Gh = T and e is a sum of root vectors corresponding to the simple roots. We have Ge B and Ge Z (G) â A, where Z (G) is the centre of G and A is a connected commutative unipotent group. As [h, e] = 2e, the space (V )A is ad h-stable. It follows from the sl2 -theory that ad h-eigenvalues on (V )A are nonnegative. Moreover, since Q, we see that Z (G) acts trivially on V , (V )Ge = (V )A , and these eigenvalues are ~ even. Therefore it is convenient to consider h = 1 h and its eigenvalues. Set 2 ~ (V )A = {x (V )A | [h, x] = ix} i and JV (q ) = V .
i

dim(V )A q i . This polynomial is called the jump polynomial for i For any Q P+ , we have

Theorem 1.2 [4, Theorem 2.4; 6, Theorem 3.4]. JV (q ) = F (H ; q ).

Let t denote the Lie algebra of T and let W be the Weyl group of T . For any Q, the space V0 is a W -module. Therefore, one can form the space J (t) = (V0 k[t])W . It is a module over J0 (t) = k[t]W . By Chevalley's theorem, the restriction homomorphism k[g] - k[t] induces an isomorphism of J0 (g)and J0 (t), so that this common algebra will be denoted by J . Since W is a finite reflection group in t, we have J (t) is a free graded J -module of rank m0 . Restricting a G-equivariant morphism g - V to t g yields a W -equivariant morphism t - V0 . In other words, we obtain a map res : J (g) - J (t), which, in view of Chevalley's theorem, is a homomorphism of J -modules. Since G·t is dense in g, the homomorphism res is injective. It is not, however, always surjective. The following elegant result is due to Broer [5, Theorem 1]. Theorem 1.3. Suppose that QP+ . Then the homomorphism res is onto m2µ = 0 for all µ . In other words, res is an isomorphism if and only if twice a root is not a weight for V . The G-modules satisfying the last condition are said to be small. Looking at the elements of J (g) as G-equivariant morphisms

g - V ,


one can consider the evaluation map
x

: J (g) - (V )G

x

for any x g. Namely, set x () = (x). The following is a particular case of a more general statement [18, Theorem 1], which applies to arbitrary G-actions. Theorem 1.4. Suppose that G·x is normal. Then
x

is onto.


g-endomorphism algebras
2. g-endomorphism and t-endomorphism algebras

277

Following Kirillov, define the g-endomorphism algebra of type by the formula C (g) = (End V k[g])G . It is immediate that C (g) is a J -module and an associative k-algebra. Since c V , one sees that C (g) is a direct sum of modules of End V V V = c J (g). It is important covariants (possibly with multiplicities). Hence C (g) c m0 . that all belong to Q. Therefore dim(End V )A = dim(End V )0 = Notice that C (g) is not only a J -module, but it also contains J as subalgebra, since idV End V . In particular, C (g) is a J -algebra. Clearly, C (g) is a graded k-algebra, the component of grade n being (End V k[g]n )G . The zero-weight space in the G-module End V is the set of T -equivariant endomorphisms of V ; that is, we have End T (V ) = (End V )0 . Define the tendomorphism algebra of type by the formula C (t) = (End T (V ) k[t])W . c J (t). It follows that, patching Using the above notation, one sees that C (t) = together the homomorphisms res , one obtains the monomorphism of J -algebras r : C (g) - C (t). Thus we have two associative J -algebras such that both are ^ free graded J -modules of the same rank, dim End T (V ). Lemma 2.1. Let D J be the discriminant. Then C (g)D and C (t)D are isomorphic as JD -modules. Proof. Both C (t) and C (g) are built of modules of covariants. Therefore the result stems from the analogous statement for the modules of covariants, which was proved in [5, Lemma 1(iii)]; see also [18, Proposition 4] for another proof in a more general context. We will primarily be interested in commutative g- and t-endomorphism algebras. The following proposition contains a criterion of commutativity. Part (i) has been proved in [16, Corollary 1]. However, we give a somewhat different proof for it, which has the potential to be applied in more general situations; cf. [18, Proposition 3]. Proposition 2.2. (i) C (g) is commutative mµ = 1 for all µ (ii) C (g) is commutative C (t) is commutative.

V .

µ Proof. (i) `': Because End T (V ) µ End (V ), the algebra C (t) is µ commutative whenever m = 1 for all µ V . Since r is injective, we are done. ^ `': Interpreting elements of C (g) as G-equivariant morphisms, consider the evaluation map x

: C (g) - (End V )G

x

(x g).


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Taking x t with Gx = T and applying Theorem 1.4, we obtain a surjective kµ algebra homomorphism C (g) - End T (V )= µ End (V ). Thus commutativity µ of C (g) forces that of End (V ) for all µ V , whence mµ = 1. (ii) This readily follows from part (i) and from the fact that C (g) is a subalgebra of C (t). Definition 2.3. A G-module V is called weight multiplicity free (wmf) if mµ = 1 for all µ V . Although we do not need this directly, it is worth mentioning that a classification of the wmf G-modules with G simple is contained in [14, 4.6]. Lemma 2.4. Suppose that V is wmf. Then End V is a multiplicity free Gc V , all nonzero coefficients module; that is, in the decomposition End V = c are equal to 1. Proof. An explicit formula for the multiplicities in tensor products an be found c in [19, Theorem 2.1]. In particular, that formula says that c m - ( 1). Here is the highest weight of the dual G-module V . We regard roots and weights as elements of the Q-vector space P Z Q sitting in t ; next, (, ) is a fixed W -invariant bilinear form on t that is positive-definite on P Z Q. Then, as usual, = 2/(, ) for all . Of all wmf G-modules, the minuscule ones occupy a distinguished position. Recall that V is called minuscule if is minuscule. Some properties of minuscule dominant weights are presented in the following proposition, where either of two items can be taken as a definition of a minuscule dominant weight. Proposition 2.5. For P+ , the following conditions are equivalent. (i) If µ V , then µ W. (ii) (, ) 1 for all + . It follows from Proposition 2.5(i) that the corresponding simple G-module is wmf, while Proposition 2.5(ii) implies that if g is simple, then a minuscule dominant weight is fundamental. Theorem 2.6. If is minuscule, then C (g) Proof. C (t)

k[t]W .

(1) Let us look again at the decomposition

V V =
I

c V = V+




<+

c V

.

By Lemma 2.4, all c = 1. (2) We claim that each V ( I ) is small in the sense of Broer. Indeed, assume that m2 = 0 for some dominant root . By a standard property of weights of V V , we have = + - n , where n 0. Then 4 = (2, ) ( + , ) 2, a contradiction!


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279

(3) It follows from the previous part and Theorem 1.3 that r = ^ I res : - C (g) - C (t) is an isomorphism of J -algebras. Since all weight spaces in V are one-dimensional and all weights are W -conjugate, End T (V ) =
µW

End V

µ

k[W/W ] k[t]W .

as W -modules. Thus C (t) = (k[W/W ] k[t])W

The last equality is a manifestation of the transfer principle in invariant theory; see [10, Chapter 2]. Remark 2.7. Another proof of this result, which does not appeal to Broer's theorem, follows from the description of Poincar´ series given in Theorem 4.1. e By a result of Steinberg, W is a reflection group in t. Hence C (g) is a polynomial algebra if V is minuscule. However, for the other wmf G-modules, the situation is not so good. Since C (g) and C (t) are commutative k-algebras in the wmf-case, one can consider the varieties M (t) := Spec C (t) and M (g) := Spec C (g). The chain of algebras J C (g) C (t) yields the following commutative diagram. ( Mt) M (g)

- g/ /G

t/W

All maps here are finite flat morphisms. Theorem 2.8. Let V be a wmf G-module. Then the following hold. (i) C (t) and C (g) are commutative Cohen­Macaulay k-algebras. (ii) C (t) and C (g) are reduced. (iii) M (t) is a disjoint union of affine spaces of dimension l. The connected (= irreducible) components of M (t) are parametrised by the dominant weights in V . (iv) The morphism yields a one-to-one correspondence between the irreducible components of M (t) and M (g). The variety M (g) is connected. Proof. (1) Cohen­Macaulayness follows, since both C (t) and C (g) are graded free modules over the polynomial ring J . ^ (2) Since C (g) is a subalgebra of C (t) via r , it is enough to prove that C (t) has no nilpotent elements. Let C (t). Regard as a T -equivariant morphism µ from t to µ End (V ). If = 0, then there is µ V such that (t)(vµ ) = cµ vµ µ for t t, 0 = vµ V , and some cµ k \{0}. Hence n (t)(vµ ) = (cµ )n vµ = 0. (3) (Cf. the proof of Theorem 2.6.) Let µ1 ,...,µk be all dominant weights in V . Then
k

End V
µ

µ

=
i=1 w W

End V

k

wµi

=
i=1

k W/W

µi

as W -modules. Hence
k W

C (t) =
i=1

k W/W

µi

k[t]

k

=
i=1

k W/W

µi

k[t]

k W

=
i=1

k[t]Wµ i .


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dmitri i. panyushev

(4) Because C (t) is a free J -module, none of the elements of J becomes a zerodivisor in C (t) (and hence in C (g)). Therefore, given any f J , the principal open subset M (t)f (respectively M (g)f ) has the same irreducible components as M (t) (respectively M (g)). Let us apply this to f = D J , the discriminant. In the commutative case, one can consider C (t)D not only as JD -module, which is a localisation of a J -module, but also as k-algebra in its own right. By Lemma 2.1, we then conclude that the k-algebras C (t)D and C (g)D are isomorphic. That M (g) is connected follows from the fact that C (g) is graded, and the component of grade 0 is just k. (Recall that C (g)n = (End V k[g]n )G .)
Wµ i We have shown that C (t) . Since J k[t]W and k[t]W k[t]Wµ i , i=1 k[t] it is not hard to realise that J embeds diagonally in the above sum. k

Corollary 2.9.

C (g)

C (t) if and only if is minuscule.

Proof. One implication is proved in Theorem 2.6. Conversely, if the algebras are isomorphic, then parts (iii) and (iv) show that k = 1, that is, is minuscule. In view of Theorem 1.3, the corollary states that if is not minuscule, then at least one simple g-module occurring in V V is not small. e The discrepancy between C (t) and C (g) can be seen on the level of Poincar´ series. We give below precise formulae for F (C (t); q ) and F (C (g); q ). 3. Dynkin polynomials In 1950, Dynkin showed that to any simple G-module V , one can attach a symmetric unimodal polynomial [7]. This polynomial represents the distribution of the weight spaces in V with respect to some level function. In Dynkin's paper, the properties of symmetricity and unimodality were also expressed in terms of the weight system of V being `spindle-like'. To prove this, Dynkin introduced what is now called `a principal sl2 -triple in g'. We say that the resulting polynomial is the Dynkin polynomial (of type ). In this section, we give some formulae for Dynkin polynomials and some applications of them. The lowest weight vector in V is - . Recall that is the set of simple roots. Given µ V , let us say that µ is on the nth floor if µ - (- ) = n with n = n. Thus the lowest weight is on the zero (ground) floor and the highest weight is on the highest floor.
µ Definition 3.1 (cf. [7, p. 222]). Letting (V )n = µ:flo or(µ)=n dim V and n an () = dim(V )n , define the Dynkin polynomial D (q ) to be n an ()q . (If we wish to indicate explicitly the dependence of this polynomial on g, we write D (g)(q ).

To obtain a more formal presentation, consider Since (, ) = 1 for any , we have (, ) . If µ Q, then (µ, ) Z. Since - (- see that ht(µ) := (µ, ) 1 Z for an arbitrary µ 2

the element = 1 + . 2 = ht , the height of , for any ) Q and (, ) = ( , ), we P . In view of these properties


g-endomorphism algebras
of , the following is an obvious reformulation of the previous definition: D (q ) = q
(, ) µV


281

mµ q

(µ, )

=
µV


mµ q

ht(+µ)

.

(3.1)

e Thus defines a grading of V , and D (q ) is nothing but the shifted Poincar´ polynomial of this grading. Theorem 3.2 [7, Theorem 4]. For any P+ , the polynomial D is symmetric (that is, ai () = am-i (), m = deg D ) and unimodal (that is, a0 () a1 () ... a[m/2] ()). Idea of proof. Having identified t and t , we see that 2 becomes h, the semisimple element of our fixed principal sl2 -triple. Therefore (V )n = {v V | h·v = 2(n - ht())v }. Hence, up to a shift of degree, D (q 2 ) gives the character of V as sl2 -module. From (3.1), one also sees that deg D =2( ,) = 2 ht(). The next proposition provides a multiplicative formula for the Dynkin polynomials. Apparently, this formula was first proved, in a slightly different form, in [20]. It was Stanley who realised that Dynkin's result has numerous combinatorial applications. Of course, the proof exploits Weyl's character formula. Proposition 3.3. D (q ) =

+

1 - q (+, ) , 1 - q (, )

where =

1 2



+

.

Remark 3.4. A similar formula also appears in [22, Lemma 2.5] (and probably in many other places) as a q -analogue of Weyl's dimension formula (or a specialization of Weyl's character formula). However, Stembridge did not make the degree shift q ( ,) and did not mention a connection of this q -analogue with Dynkin's results. Example 3.5. Suppose that g = sln+1 , and let i be the ith fundamental weight of it. Then a direct calculation based on Proposition 3.3 gives D
m1

(sln (sln

+1

)(q ) = )(q ) =

(1 - q m+1 ) ... (1 - q m (1 - q ) ... (1 - q n ) n +1 . m

+n

)

=:

m+n m+n = , n m

D



m

+1

As a consequence of this example, one can deduce the following (well known) assertions. Proposition 3.6. Suppose that dim V = 2. Then there are two isomorphisms of sl(V )-modules: S n (S m (V )) (S
m n+m-1

S m (S n (V ))
m n

(Hermite's reciprocity)

(V )) = S (S (V )).


282

dmitri i. panyushev

Proof. The first formula follows from the equality Dm1 (sln+1 ) = Dn1 (slm+1 ) and the fact that Vm1 (sln+1 )|sl2 = S m (S n (V )), where sl2 = sl(V ) is a principal sl2 in sln+1 . Similarly, the second formula follows from the equality Dm1 (sln+1 ) = Dm (slm+n ). Now we show that Dynkin polynomials arise in connection with g-endomorphism algebras for the minuscule dominant weights. For any P+ , we have W is a parabolic subgroup of W . Let di (W ), 1 i l, be the degrees of basic invariants in k[t]W . In particular, di = di (W0 ) = di (W ). Note that if = 0, then some di (W ) are equal to 1. Let n : W - N {0} - be the length function with respect to the set of simple reflections. Set t (q ) = n(w) . It is well known (see for example [15, 3.15]) that w W q
l

t (q ) =
i=1

1 - q di (W ) . 1-q

(3.2)

These polynomials will appear frequently in the following exposition. Proposition 3.7. Suppose that P+ is minuscule. Then D (q ) = Proof. obtain D (q ) =
+ \()
+

t0 (q ) = t (q )

l

i=1

1 - q di . 1 - q di (W )

Set () = { | (, ) = 0}. Using Propositions 3.3 and 2.5(ii), we 1 - q (, )+1 = 1 - q (, )




+

1 - q (, )+1 · 1 - q (, )



()

+

1 - q (, 1 - q (,



)

)+1

.

Note that (, ) = ht , the height of in the dual root system . By a result of Kostant (see [15, 3.20]), t0 (q ) =

+

1 - q ht()+1 . 1 - q ht()

Applying this formula to and () and substituting in the previous expression for D , we complete the proof. Let us look at the Poincar´ series of C (g), where is minuscule. Since e l W C (g) = k[t] (Theorem 2.6) and F (J ; q ) = i=1 (1/(1 - q di )), we deduce from Proposition 3.7 that F (C (g); q ) = D (q ) ·F (J ; q ). It is shown in Section 5 that this relation holds for all wmf G-modules. 4. The Poincar´ series of endomorphism algebras e In this section we find explicit formulas for the Poincar´ series F (C (g); q ) and e F (C (t); q ) with arbitrary P+ . Let mµ (q ) be Lusztig's q -analogue of weight multiplicity. It is a certain polynomial in q , with integral coefficients, such that mµ (1) = mµ . Defining these q -analogues (3.3)


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283

consists of two steps. First, one defines a q -analogue of Kostant's partition function P by 1 = Pq ( )e . 1 - e q +
Q

Then we set (q ) = wW det(w )Pq (w ( + ) - µ - ). Pro ofs of properties of polynomials mµ (q ) can be found in [4, 11]. (1) If both , µ are dominant, then the following hold. (i) The coefficients of mµ (q ) are nonnegative. (ii) mµ (q ) = 0 µ V . (iii) deg mµ (q ) = ( - µ, ) = ht( - µ). (2) If Q, then m0 (q ) is the numerator of the Poincar´ series for the e covariants J (g); that is, in the notation of Section 1, we have F (H ; q ) = m0 (q ). Theorem 4.1. Let be an arbitrary dominant weight. (i) F (C (g); q ) =
P+

mµ

the next

module of JV (q ) =

m (q ) m (q )
P+

2

t0 (q ) t (q )

l

(1 - q di )
i=1

= (ii) F (C (t); q ) =
V , P+

2

t0 (q ) ·F (J ; q ). t (q )

t0 (q ) t (q )

l

(1 - q di ) =
i=1 V , P+

t0 (q ) ·F (J ; q ). t (q )

Proof. (i) Recall that J = k[f1 ,...,fl ] and deg fi = di . Since f1 ,...,fl is a homogeneous system of parameters for C (g) as J -module, we have C (g)/C (g)f1 + ... + C (g)fl (End V k[N ])G
l

(End V )A

(cf. Section 1). Hence F (C (g); q ) = F ((End V )A ; q )/ i=1 (1 - q di ). Using the decomposition End V = I c V and Theorem 1.2, we see that the numerator is just the jump polynomial corresponding to the (reducible) G-module End V . c m0 (q ). It is remarkable, however, that, for the That is, F ((End V )A ; q ) = G-modules of the form V Vµ , there is a formula for the jump polynomial that does not appeal to the explicit decomposition of this tensor product. Namely, by [12, Corollary 2.4], we have t0 (q ) . (4.1) JV V (q ) = m (q )m (q ) µ µ t (q )
P+

Letting µ = , one obtains the required formula.
W (ii) Recall from Section 2 that C (t) , where ranges over all k[t] dominant weights of V . Making use of equation (3.2), we obtain

F (k[t]W ; q ) = 1 which completes the proof.

l

1-q
i=1

di (W )

=

t0 (q ) t (q )

l

(1 - q di) ,
i=1


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dmitri i. panyushev

In the case when is minuscule, Theorem 4.1 shows that F (C (g); q ) = F (C (t); q ) = t0 (q ) ·F (J ; q ), t (q )

since m (q ) = 1. Thus we recover in this way equation (3.3) and the fact that C (g) = C (t). From the last equality, we deduce the following claim. (This yields another proof for part of Broer's results.) Corollary 4.2. Suppose that is minuscule, and let V be any irreducible constituent of V V . Then the restriction homomorphism res : J (g) - J (t) is onto. Let F (q ) denote the right-hand side of equation (4.1) with = µ, that is, the jump polynomial for the G-module End V . Lemma 4.3. (i) deg F (q ) = 2 ht(). (ii) F (1) = µ V (mµ )2 . (iii) If V is wmf, then F (q ) =
µ V ,µP+

q

2ht(-µ)

·

t0 (q ) . tµ (q )

Proof. (1) F (q ) is a sum (with multiplicities) of the jump polynomials for the irreducible constituents of End V . The jump polynomial for V+ is of degree ht( + ) = 2ht(). For all other simple G-submodules in End V , the height of the highest weight is strictly less. µ (2) F (1) = dim(End V )A = dim(End V )T = µ dim End (V ). µ (3) In view of Theorem 4.1(i), it suffices to prove that m (q ) = q ht(-µ) . Because mµ = 1 and the coefficients of mµ (q ) are nonnegative integers, mµ (q ) = q a . Since deg mµ (q ) = ht( - µ), we are done. The last expression demonstrates an advantage of using equation (4.1) in the wmf case. We obtain a closed formula for the jump polynomial of a reducible representation that requires no bulky computations. 5. The Gorenstein property for the commutative g-endomorphism algebras The goal of this section is to prove that if V is wmf, then C (g) is a Gorenstein algebra. We also give another expression for the Poincar´ series of C (g), which e includes the Dynkin polynomial of type . To begin with, we recall some facts about graded Gorenstein algebras. A nice exposition of relevant material is found in [21]. Let C = n 0 Cn b e a graded Cohen­Macaulay k-algebra with C0 = k. Suppose that the Krull dimension of C is n and let f1 ,...,fn be a homogeneous system of parameters. Then C = C /(f1 ,...,fn ) d is a graded Artinian Cohen­Macaulay k-algebra. We have C = i=0 C i for some d, d and m = i=1 C i is the unique maximal ideal in C . The annihilator of m in C is called the socle of C : soc(C ) = {c C | c·m = 0}.


g-endomorphism algebras
Then the following are true. (1) C is Gorenstein if and only if C is. (2) C is Gorenstein if and only if soc(C ) is one-dimensional.

285

In the rest of the section, V is a wmf G-module, and hence C (g) is commutative. Consider the k-algebra (End V )A =: R() . Being a homomorphic image of C (g), it is commutative as well. We also have dim R() = dim V . Proposition 5.1. Let v-


be a lowest weight vector in V . Then R() (v
-

) = V .

Proof. Set N = R() (-v ), and consider N , the annihilator of N in the dual space V .Let v be a highest weight vector in V . By definition, N contains v- and therefore v N . Since A G End V and A is commutative, N is an A-module and hence N is an A-module too. Assume that N = 0. Since A is unipotent, N must contain a non-trivial A-fixed vector. By a result of Graham (see [9, 1.6]), dim(V )A = 1 in the wmf case. As A B , we conclude that (V )A = kv . Thus N (V )A = 0. This contradiction shows that N = 0. Proposition 5.2. F (q ) = D (q ) if and only if V is wmf. In particular, F (q ) is symmetric and unimodal in the wmf case. Proof. (1) `': Suppose that F (q ) = D (q ). Then m
µV


µ2

= F (1) = D (1) = dim V =
µV


mµ ,

whence V is wmf. (2) `': The polynomial F (q ) is determined via the 1 2 (End V )A , whereas D (q ) is determined via the shifted Identifying t and t , we obtain h =2 . Obviously, the R() - V , x - x(v- ), respects both gradings and has Ri
()

h-grading in R() = ` -grading' in V . bijective linear map degree zero. Hence

- (V )i .



Remark 5.3. If V is not wmf, then F (q ) can be neither symmetric nor unimodal. For instance, if g = sp6 and = 2 , the second fundamental weight, then F2 (q ) = 1 + q +2q 2 +2q 3 +3q 4 +2q 5 +3q 6 + q 7 + q 8 . By a theorem of Stanley [21, 12.7], if C is a Cohen­Macaulay domain, then the symmetricity of the Poincar´ polynomial of C /(f1 ,...,fn ) implies the Gorenstein e property for C . In our situation, C (g) is not a domain unless is minuscule; see Theorem 2.8. Therefore we still cannot conclude that C (g) is always Gorenstein. Proposition 5.4. The socle of R(
)

is one-dimensional.

Proof. Write R for R() in this proof. Recall from Section 3 that V is a disjoint union of `floors', the weight space V- being the zero floor and V 2ht() being the 2 ht()th floor. We know that R is commutative and R = Ri . i=0 By Propositions 5.1 and 5.2, we have Ri (v- ) (V )i , the ith floor in V , = µ and dim R2ht() = 1. Clearly, R2ht() takes V- to V , and R2ht() (V ) = 0


286

dmitri i. panyushev

for µ = - . We wish to show that soc(R) = R2ht() . Suppose x Ri , then xv- (V )i . Let , denote the natural pairing of V and V . Using the hinvariance, we see that (V )i , (V )j = 0 unless i + j = 2 ht(). Hence there exists (V )j , where j = 2 ht() - i, such that , xv- = 0. Applying Proposition 5.1 to V , we obtain = y (v- ) for some y Rj . Here v- is a lowest weight vector in V , and R is identified with (End V )A . Hence v- ,y x(v- ) = 0. Thus, for any x Ri , there exists y R2ht()-i such that xy = 0, which is exactly what we need. Combining all previous results of this section, we obtain the following. Theorem 5.5. Let V be a wmf G-module. Then the following hold. (i) The map (End V )A - V , x - x(v- ), is bijective. (ii) (End V )A is an Artinian Gorenstein k-algebra. (iii) D (q ) = F (q ). (iv) C (g) is a Gorenstein k-algebra. It is worth mentioning the following property of wmf G-modules whose proof also uses Graham's result; cf. Proposition 5.1. Proposition 5.6. For any n N, the vector en (v to all weight spaces in (V )n .
-

) has nonzero pro jections

Proof. Let E be the k-subalgebra of R() generated by e. Then M := (E (v- )) is an E -stable subspace of V , and, obviously, v M . Assume that E (v- ) has the zero pro jection to some weight space, say V- . Then M contains the weight space V . Let v be a nonzero vector in V and let k be the maximal integer such that ek (v ) = 0. By [6, 2.6], ek (v ) (V )A , and by [9, 1.6], (V )A = kv . As ek (v ) M , we obtain a contradiction. All wmf representations of simple algebraic groups were found by Howe in [14, 4.6]. This information is included in the first two columns of Table 1; the last column gives the corresponding Dynkin polynomial. We write i for the ith fundamental weight of G, according to the numbering of [23]. One can observe that each in Table 1 is a multiple of a fundamental weight. Using polynomials mµ (q ), one can give a conceptual proof of this fact. This will appear elsewhere. 6. A connection with (equivariant) cohomology Let C be an Artinian graded commutative associative k-algebra, C = i=0 C i . Suppose that dim C d = 1, and let be a nonzero linear form on C that annihilates e the space C 0 ··· C d-1 . Then C is called a Poincar´ duality algebra if the bilinear form (x, y ) - (xy ), x, y C , is nondegenerate. This name suggests that C looks very much as if it were the cohomology algebra of some `good' manifold. It is easily e seen that C is a Poincar´ duality algebra if and only if dim soc(C ) = 1, that is, C is Gorenstein. In the previous section, we proved that R() = (End V )A is an Artinian Gorenstein k-algebra if V is wmf. (Here d = 2 ht(), which is not necessarily
d


g-endomorphism algebras
Table 1. Dynkin polynomials for the weight multiplicity free representations. Type An i m1 ,mn (m 2) n 1 1 3 (n = 3) E E
6 1

287

Minuscule Yes No

D (q ) See Example 3.5 See Example 3.5

Bn

Yes No Yes No Yes Yes Yes Yes No

(1 + q )(1 + q 2 ) ... (1 + q n ) 1 + q + ... + q 2n 1 + q + ... + q 2n -1 1 + q + q +2(q 3 + ... + q 6 )+ q 7 + q 8 + q
2

C

n

9

Dn

n -1

,

n

(1 + q n -1 )(1 + q + ... + q n -1 ) (1 + q )(1 + q 2 ) ... (1 + q n -1 ) (1 + q 4 + q 8 )(1 + q + ... + q 8 ) (1 + q 5 )(1 + q 9 )(1 + q + ... + q 13 ) 1 + q + ... + q
6



1

7

1

G

2

1

even.) In this situation, one has more evidence in favour of the assertion that R() can be a cohomology algebra. Recall that e is a sum of root vectors corresponding () to the simple roots. Hence e(v- ) (V )1 , that is, e R1 . Proposition 6.1. The multiplication operator e : Ri for i [(2ht() - 1)/2] and surjective for i [ht()]. Proof.
()

- Ri

() +1

is injective

The proof follows from the sl2 -theory and the equality D (q ) = F (q ).

Hence, if R() = H (X ) for some algebraic variety X , then e R1 can be regarded as the class of a hyperplane section, and the hard Lefschetz theorem holds for X . If G is simple, then is necessarily fundamental. Therefore the subspace (V )1 V is one-dimensional. Indeed, if is the unique root such that () (, ) = 0, then (V )1 = V- . In terms of R() , this means that R1 = ke. That is, such X should satisfy the condition b2 (X ) = 1. In the rest of the paper, we assume that k = C, and consider cohomology with complex coefficients. Now we give a hypothetical description of such X in the case when G is simple. More precisely, we conjecture that there exists X P(V ) H (X ); moreover, the gsuch that odd cohomology of X vanishes and R() endomorphism algebra C (g) gives the equivariant cohomology of X . We refer to [3] for a nice introduction to equivariant cohomology. As usual, V is a wmf Gmodule. The variety we are seeking should satisfy the constraints (X ) = dim V
()


288

dmitri i. panyushev

and dim X = 2 ht(). Notice that the set of T -fixed points in P(V ) is finite:

P(V )T =
V


v ,

where v is the image of v V in the pro jective space. Since (Y ) = (Y T ) for any algebraic variety Y acted upon by a torus T [1], we see that our variety X must contain all v , V . This provides some explanation for the following description. Let µ V be the unique dominant minuscule weight. For instance, µ = 0 if and only if Q. (1) If (G, ) = (An ,m1 ) or (An ,mn ) with m > n + 1, then we define X to be the closure of the G-orbit of the line vµ in the pro jective space P(V ). 1, let X (2) Alternatively, for G = An and = m1 with any m be the pro jectivisation of the variety of decomposable forms of degree m in Vm1 = S m (V1 ), where a form of degree m is said to be decomposable if it is a product of m linear forms. (A similar definition applies to = mn .) As is easily seen, the (An ,m1 or mn ) with description of X in all example, for (An ,m1 ), two constructions coincide if they both apply, that is, for m n + 1. However, I do not see how to give a uniform cases. Direct calculations show that dim X = 2 ht(); for we have dim X = mn.

Conjecture 6.2. Let V be a wmf G-module. Then the following hold. (1) The variety X P(V ) is rationally smooth, odd cohomology of X vanishes, and H (X ) R() . In particular, the Poincar´ polynomial of H (X ) e 2 is equal to D (q ). (2) Let Gc G be a maximal compact subgroup of G. Then HGc (X ) C (g). Let J+ J be the augmentation ideal. Since C (g)/J+ C (g) R() , J H (X ) (see [3, Proposition 2]), the first part H c ({pt}), and HGc (X )/(J+ ) of the conjecture follows from the second one. Actually, the conjecture is true for most of items in Table 1, and in particular for all minuscule weights. We list below all the known results supporting the conjecture. (1) If is minuscule, then = µ, and therefore X = G/P , a generalised flag variety. Here the conjecture follows from Theorem 2.6 and the well known description of HGc (G/P ). Indeed, if P is an arbitrary parabolic subgroup (that is, is not necessarily minuscule), then HGc (G/P ) k[t]W . (2) For the simplest representations of Bn and G2 , we have µ = 0 and X = P(V ). On the other hand, the formulae for D (q ) in Table 1 shows that (End V )A is generated by e as k-algebra, and the equality H (X ) = R() follows. This also e shows that C (g) and HGc (X ) have the same Poincar´ series. (3) We have essentially four cases with nonminuscule : (Bn ,1 ), (C3 ,3 ), 2. For the first three cases and for the last one (G2 ,1 ), and (An ,m1 ), m with m = 2, X is a compact multiplicity-free Gc -space; in other words, X is a spherical G-variety. Therefore, making use of [3, Theorem 9], one obtains the description of HGc (X ). In these `spherical' cases, the number of dominant weights in V equals 2. Therefore the structure of C (g) is not too complicated, and this can be used for proving that C (g) = HGc (X ). (4) The pro jectivisation of the variety of decomposable forms of degree m in n + 1 variables is isomorphic with (Pn )m /m , where m is the symmetric group
G


g-endomorphism algebras

289

permuting factors in Pn â ... â Pn (m times); see [2, Theorem 1.3; 8, 4.2]. Notice that H ((Pn )m /m ) H ((Pn )m )
m

k[x1 ,...,xm ]/ xn+1 ,...,xn+1 m 1



m

,

the algebra of truncated symmetric polynomials. It is easily seen that its dimension is equal to m+n = dim Vm1 . m A somewhat more bulky but still elementary calculation shows that the Poincar´ polynomial of H ((Pn )m /m ) is equal to the Dynkin polynomial for e (An ,m1 ). Our proof is purely combinatorial. We establish a natural one-to-one correspondence between suitably chosen bases of H ((Pn )m /m ) and Vm1 such that H i ((Pn )m /m ) corresponds to (Vm1 )i . It is not, however, clear how to compare the multiplicative structure of H ((Pn )m /m ) and (End Vm1 )A . Acknowledgements. I would like to thank Michel Brion for several useful remarks and the suggestion to consider equivariant cohomology. References
1. A. Bialynicki-Birula, `On fixed points schemes of actions of multiplicative and additive groups', Topology 12 (1973) 99­103. 2. M. Brion, `Stable properties of plethysm: on two conjectures of Foulkes', Manuscripta Math. 80 (1993) 347­371. 3. M. Brion, `Equivariant cohomology and equivariant intersection theory', Representation theories and algebraic geometry (ed. A. Broer, Kluwer, 1998) 1­37. 4. A. Broer, `Line bundles on the cotangent bundle of the flag variety', Invent. Math. 113 (1993) 1­20. 5. A. Broer, `The sum of generalized exponents and Chevalley's restriction theorem', Indag. Math. 6 (1995) 385­396. 6. R. K. Brylinski, `Limits of weight spaces, Lusztig's q -analogs, and fiberings of adjoint orbits', J. Amer. Math. Soc. 2 (1989) 517­533. 7. E. B. Dynkin, `Some properties of the weight system of a linear representation of a semisimple Lie group', Doklady Akad. Nauk SSSR 71 (1950) 221­224 (Russian). 8. I. M. Gelfand, M. Kapranov and A. Zelevinsky, Discriminants, resultants, and multidimensional determinants (Birkh¨ auser, 1994). 9. W. Graham, `Functions on the universal cover of the principal nilpotent orbit', Invent. Math. 108 (1992) 15­27. 10. F. Grosshans, Algebraic homogeneous spaces and invariant theory, Lecture Notes in Mathematics 1673 (Springer, 1997). 11. R. K. Gupta, `Characters and q -analog of weight multiplicity', Bull. London Math. Soc. (2) 36 (1987) 68­76. 12. R. K. Gupta, `Generalized exponents via Hall­Littlewood symmetric functions', Bull. Amer. Math. Soc. 16 (1987) 287­291. 13. W. Hesselink, `Characters of the nullcone', Math. Ann. 252 (1980) 179­182. 14. R. Howe, `Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond', The Schur lectures (1992), Israel Mathematical Conference Proceedings 8 (BarIlan University, Ramat Gan, 1995) 1­182. 15. J. E. Humphreys, Reflection groups and Coxeter groups (Cambridge University Press, 1992). 16. A. A. Kirillov, `Introduction to family algebras', Moscow Math. J. 1 (2001) 49­64. 17. B. Kostant, `Lie group representations in polynomial rings', Amer. J. Math. 85 (1963) 327­ 404. 18. D. Panyushev, `On covariants of reductive algebraic groups', Indag. Math. 13 (2002) 125­129. 19. K. R. Parthasarathy, R. Ranga Rao and V. S. Varadarajan, `Representations of complex semisimple Lie groups and Lie algebras', Ann. of Math. 85 (1967) 383­429. 20. R. P. Stanley, `Unimodal sequences arising from Lie algebras', Combinatorics, representation theory and statistical methods in groups (ed. T. V. Narayana et al., Marcel Dekker, 1980) 127­136.


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21. R. P. Stanley, Combinatorics and commutative algebra, 2nd edn (Birkh¨ auser, 1996). 22. J. Stembridge, `On minuscule representations, plane partitions and involutions in complex Lie groups', Duke Math. J. 73 (1994) 469­490. 23. E. B. Vinberg and A. L. Onishchik (eds.), Seminar on Lie groups and algebraic groups (Nauka, Moscow, 1988) (Russian), Lie groups and algebraic groups (ed. A. L. Onishchik and E. B. Vinberg, Springer, 1990) (English).

Dmitri I. Panyushev Independent University of Moscow Bol'shoi Vlasevskii per. 11 121002 Moscow Russia panyush@mccme.ru