Eric Sommers

Exponents for B-stable ideals

Let G be a simple algebraic group over the complex numbers containing a Borel subgroup B. Given a B-stable ideal I in the nilradical of the Lie algebra of B, we define natural numbers m 1 , m 2 ,..., m k which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types A n , B n , C n and some other types. When / = 0, we recover the usual exponents of G by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincare polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.

Normality of Very Even Nilpotent Varieties in D2l

For the classical groups, Kraft and Procesi [4], [5] have resolved the question of which nilpotent orbits have closures which are normal and which are not, with the exception of the very even orbits in D2l which have partition of the form (a , b) for a, b distinct even natural numbers with ak + b = 2l. In this article, we show that these orbits do have normal closure. We use the technique of [8]. 1. SOME LEMMAS IN Al We retain the notation of [8]. Throughout,G is a connected simple algebraic group over C, B a Borel subgroup, T a maximal torus in B. The simple roots are denoted by Π, and they correspond to the Borel subgroup opposite to B. Let {ωi} be the fundamental weights of G corresponding to Π. If α ∈ Π, then Pα denotes the parabolic subgroup of semisimple rank one containing B and corresponding to α. If P is a parabolic subgroup of G, we denote by uP the Lie algebra of its unipotent radical. We recall Proposition 1.1. [3] Let V be a rational representation of B and assume that V extends to a representation of the parabolic subgroup Pα where α is a simple root. Let λ ∈ X (T ) be such that m = 〈λ, α∨〉 ≥ −1. Then there is a G-module isomorphism H (G/B, V ⊗ λ) = H (G/B, V ⊗ λ−(m+ 1)α) for all i ∈ Z. In particular, if m = −1, then all cohomology groups vanish. For the rest of this section and the next, let G = SLl+1(C). We index the simple roots Π = {αj} so that α1 is an extremal root and αj is next to αj+1 in the Dynkin diagram of type Al. The following lemma follows easily from several applications of the previous proposition. Lemma 1.2. [7] Let V be a rational representation of B which extends to a representation of Pαj for a ≤ j ≤ b. Let λ ∈ X(T ) be such that 〈λ, α∨j 〉 = 0 for a < j ≤ b. Set r = 〈λ, α ∨ a〉 and assume that a− b− 1 ≤ r ≤ −1. Then H(V ⊗ λ) = 0. A similar statement holds by applying the non-trivial automorphism to the Dynkin diagram of type Al. We use this lemma to prove Date: 7/23/03; 9/17/03. The author was supported in part by NSF grants DMS-0201826 and DMS-9729992. The author thanks Viktor Ostrik for directing him to this problem. 1

Local systems on nilpotent orbits and weighted Dynkin diagrams

We study the Lusztig-Vogan bijection for the case of a local system. We compute the bijection explicitly in type A for a local system and then show that the dominant weights obtained for different local systems on the same orbit are related in a manner made precise in the paper. We also give a conjecture (putatively valid for all groups) detailing how the weighted Dynkin diagram for a nilpotent orbit in the dual Lie algebra should arise under the bijection.

A family of affine Weyl group representations

In this paper we explicitly determine the virtual representations of the finite Weyl subgroups of the affine Weyl group on the cohomology of the space of affine flags containing a family of elementsnt in an affine Lie algebra. We also compute the Euler characteristic of the space of partial flags containingnt and give a connection with hyperplane arrangements.

Equivalence classes of ideals in the nilradical of a Borel subalgebra

An equivalence relation is defined and studied on the set of B-stable ideals in the nilradical of the Lie algebra of a Borel subgroup B. Techniques are developed to compute the equivalence relation and these are carried out in the exceptional groups. There is a natural partial order on equivalence classes coming from inclusion of one ideal in another. A main theorem is that this partial order is a refinement of the closure ordering on nilpotent orbits.

PIECES OF NILPOTENT CONES FOR CLASSICAL GROUPS

We compare orbits in the nilpotent cone of type Bn, that of type Cn, and Katos exotic nilpotent cone. We prove that the number of Fq-points in each nilpotent orbit of type Bn or Cn equals that in a corresponding union of orbits, called a type-B or type-C piece, in the exotic nilpotent cone. This is a ner version of Lusztigs result that corresponding special pieces in types Bn and Cn have the same number of Fq-points. The proof requires studying the case of characteristic 2, where more direct connections between the three nilpotent cones can be established. We also prove that the type-B and type-C pieces of the exotic nilpotent cone are smooth in any characteristic.