We determine the admissible nilpotent coadjoint orbits of real and p-adic split exceptional groups of types G2, F4, E6 and E7. We find that all Lusztig-Spaltenstein special orbits are admissible. Moreover, there exist nonspecial admissible orbits, corresponding to “completely odd” orbits in Lusztig’s special pieces. In addition, we determine the number of, and representatives for, the non-even ...

Using Deodhar’s decomposition of a double Schubert cell, we study the regular representations of finite groups of Lie type arising in the cohomology of Deligne-Lusztig varieties associated to tori. We deduce that the Deligne-Lusztig restriction of a Gelfand-Graev module is a shifted Gelfand-Graev module. INTRODUCTION Let G be a connected reductive algebraic group defined over an algebraic closu...

Let Uq be the quantum group associated to a Lie algebra g of rank n. The negative part U− of U has a canonical basis B with favourable properties (see Kashiwara [6] and Lusztig [11, §14.4.6]). The approaches of Lusztig and Kashiwara lead to a set of alternative parametrizations of the canonical basis, one for each reduced expression for the longest word in the Weyl group of g. We show that if g...

In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by Lusztig led to the first general construction of PBW bases of these algebras. Also, they have important applications to the representation theory of these and related algebras. In the present paper the Drinfel’d double for a class of graded Hopf algebras is investigated. Various quantum algebras, in...

LET %’ be a highest weight category [CPSl] with finitely many simple objects and such that every object has finite length. Then Ce = mod S for a finite dimensional quasi-hereditary algebra S. In a series of papers, [CPS4,5,6,7], we have studied various Kazhdan-Lusztig theories for %. This concept arises naturally in, for example, the modular representation theory of semisimple algebraic groups....

Let (W,S) be a Coxeter system and let w → w∗ be an involution of W which preserves the set of simple generators S. Lusztig and Vogan have recently shown that the set of twisted involutions (i.e., elements w ∈ W with w−1 = w∗) naturally generates a module of the Hecke algebra of (W,S) with two distinguished bases. The transition matrix between these bases defines a family of polynomials Pσ y,w w...

Let K be a local non-Archimedean field of positive characteristic and let L be the degree-n unramified extension of K. Let θ be a smooth character of L× such that for each nontrivial γ ∈ Gal(L/K), θ and θ/θ have the same level. Via the local Langlands and Jacquet-Langlands correspondences, θ corresponds to an irreducible representation ρθ of D×, where D is the central division algebra over K wi...

Kazhdan and Lusztig have shown that the partition of the symmetric group Sn into left cells is given by the Robinson–Schensted correspondence. The aim of this paper is to provide a similar description of the left cells in type Bn for a special class of choices of unequal parameters. This is based on a generalization of the Robinson–Schensted correspondence in type Bn. We provide an explicit des...

In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by Lusztig led to the first general construction of PBW bases of these algebras. Also, they have important applications to the representation theory of these and related algebras. In the present paper the Drinfel’d double for a class of graded Hopf algebras is investigated. Various quantum algebras, in...

We determine the set of connected components of closed affine Deligne-Lusztig varieties for special maximal compact subgroups of split connected reductive groups. We show that there is a transitive group action on this set. Thus such an affine Deligne-Lusztig variety has isolated points if and only if its dimension is 0. We also obtain a description of the set of these varieties that are zero-d...