Unipotent Conjugacy Classes Research Articles

On Lusztig's map for spherical unipotent conjugacy classes in disconnected groups

AbstractLet be a simple algebraic group over an algebraically closed field and let be a graph‐automorphism of . We classify the spherical unipotent conjugacy classes in the coset . As a by‐product, we show that J.‐H. Lu's characterization in characteristic zero of spherical conjugacy classes in by the dimension formula also holds for spherical unipotent conjugacy classes in in positive characteristic. If has order 2, we provide an alternative description of the restriction to spherical unipotent conjugacy classes in , of Lusztig's map from the set of unipotent conjugacy classes in to the set of twisted conjugacy classes of the Weyl group of . We also show that a twisted conjugacy class in the Weyl group has a unique maximal length element if and only if it has maximum in the Bruhat order (a result previously proved by X. He).

Covering Numbers of Unipotent Conjugacy Classes in Simple Algebraic Groups

Covering Numbers of Unipotent Conjugacy Classes in Simple Algebraic Groups

Partial orders on conjugacy classes in the Weyl group and on unipotent conjugacy classes

Let G be a reductive group over an algebraically closed field and let W be its Weyl group. In a series of papers, Lusztig introduced a map from the set [W] of conjugacy classes of W to the set [Gu] of unipotent classes of G. This map, when restricted to the set of elliptic conjugacy classes [We] of W, is injective. In this paper, we show that Lusztig's map [We]→[Gu] is order-reversing, with respect to the natural partial order on [We] arising from combinatorics and the natural partial order on [Gu] arising from geometry.

GENERALIZED GELFAND–GRAEV REPRESENTATIONS IN SMALL CHARACTERISTICS

Let$\mathbf{G}$be a connected reductive algebraic group over an algebraic closure$\overline{\mathbb{F}_{p}}$of the finite field of prime order$p$and let$F:\mathbf{G}\rightarrow \mathbf{G}$be a Frobenius endomorphism with$G=\mathbf{G}^{F}$the corresponding$\mathbb{F}_{q}$-rational structure. One of the strongest links we have between the representation theory of$G$and the geometry of the unipotent conjugacy classes of$\mathbf{G}$is a formula, due to Lusztig (Adv. Math.94(2) (1992), 139–179), which decomposes Kawanaka’s Generalized Gelfand–Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, the formula given in Lusztig (Adv. Math.94(2) (1992), 139–179) is only valid under the assumption that$p$is large enough. In this article, we show that Lusztig’s formula for GGGRs holds under the much milder assumption that$p$is an acceptable prime for$\mathbf{G}$($p$very good is sufficient but not necessary). As an application we show that every irreducible character of$G$, respectively, character sheaf of$\mathbf{G}$, has a unique wave front set, respectively, unipotent support, whenever$p$is good for $\mathbf{G}$.

On Lusztig's map for spherical unipotent conjugacy classes

We provide an alternative description of the restriction to spherical unipotent conjugacy classes, of Lusztig's map Ψ from the set of unipotent conjugacy classes in a connected reductive algebraic group to the set of conjugacy classes of its Weyl group. For irreducible root systems, we analyze the image of this restricted map and we prove that a conjugacy class in a finite Weyl group has a unique maximal length element if and only if it has a maximum.

CONSTRUCTIONS OF GLOBAL INTEGRALS IN THE EXCEPTIONAL GROUP F4

Motivated by known examples of global integrals which represent automorphic L-functions, this paper initiates the study of a certain two-dimensional array of global integrals attached to any reductive algebraic group, indexed by maximal parabolic subgroups in one direction and by unipotent conjugacy classes in the other. Fourier coefficients attached to unipotent classes, the Gelfand-Kirillov dimension of automorphic representations, and an identity which, empirically, appears to constrain the unfolding process are presented in detail with examples selected from the exceptional groups. Two new Eulerian integrals are included among these examples.

A classification of unipotent spherical conjugacy classes in bad characteristic

Let G be a simple algebraic group over an algebraically closed field k of bad characteristic. We classify the spherical unipotent conjugacy classes of G. We also show that if the characteristic of k is 2, then the fixed point subgroup of every involutorial automorphism (involution) of G is a spherical subgroup of G.

Complete reducibility and separability

Let G G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0 p > 0 . A subgroup of G G is said to be separable in G G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre’s concept of G G -complete reducibility for subgroups of G G . A separability hypothesis appears in many general theorems concerning G G -complete reducibility. We demonstrate that some of these results fail without this hypothesis. On the other hand, we prove that if G G is a connected reductive group and p p is very good for G G , then any subgroup of G G is separable; we deduce that under these hypotheses on G G , a subgroup H H of G G is G G -completely reducible provided Lie G G is semisimple as an H H -module. Recently, Guralnick has proved that if H H is a reductive subgroup of G G and C C is a conjugacy class of G G , then C ∩ H C\cap H is a finite union of H H -conjugacy classes. For generic p p — when certain extra hypotheses hold, including separability — this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztig’s deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick’s result is false if one considers conjugacy classes of n n -tuples of elements from H H for n > 1 n > 1 .

Conjugacy classes in parabolic subgroups of general linear groups

We prove a formula connecting the number of unipotent conjugacy classes in a maximal parabolic subgroup of a finite general linear group with the numbers of unipotent conjugacy classes in various parabolic subgroups in smaller dimensions. We generalise this formula and deduce a number of corollaries; in particular, we express the number of conjugacy classes of unitriangular matrices over a finite field in terms of the numbers of unipotent conjugacy classes in maximal parabolic subgroups over the same field. We show how the numbers of unipotent conjugacy classes in parabolic subgroups of small dimensions may be calculated.

Springer correspondence for disconnected groups

The Springer correspondence is a map from the set of unipotent conjugacy classes of a reductive algebraic group to the set of irreducible complex characters of the Weyl group. Here, we determine this map explicitly in the case of disconnected classical algebraic groups.

On the Numbers of 2-Weights, Unipotent Conjugacy Classes, and Irreducible Brauer 2-Characters of Finite Classical Groups

On the Numbers of 2-Weights, Unipotent Conjugacy Classes, and Irreducible Brauer 2-Characters of Finite Classical Groups

On the numbers of 2-weights, unipotent conjugacy classes, and irreducible Brauer 2-characters of finite classical groups

A closed expression for the number of unipotent conjugacy classes of a classical group G is given and Alperin’s weight conjecture is confirmed for G globally and for a symplectic or odd-dimensional orthogonal group block by block.

Unipotent Orbital Integrals of Hecke Functions for GL(n)

AbstractLet G = GL(n, F) where F is a p-adic field, and let 𝓗(G) denote the Hecke algebra of spherical functions on G. Let u1,..., up denote a complete set of representatives for the unipotent conjugacy classes in G. For each 1 ≤ i ≤ p, let μi be the linear functional on such that μi(f) is the orbital integral of f over the orbit of ui. Waldspurger proved that the μi, 1 ≤ i ≤ p, are linearly independent. In this paper we give an elementary proof of Waldspurger's theorem which provides concrete information about the Hecke functions needed to separate orbits. We also prove a twisted version of Waldspurger's theorem and discuss the consequences for SL(n, F).

Germs Associated to Regular Unipotent Classes in p-adic SL(n)

AbstractFor an elliptic torus in SL(n), explicit formulae are given for the germs which are associated to the regular unipotent conjugacy classes. Using them, a formula is found for the germ associated to the "subregular" class, the class whose Jordan canonical form contains a 1 x 1 and an (n - 1) x (n - 1) block.

On Springer's correspondence for simple groups of type En (n = 6, 7, 8)

Let G be a connected reductive algebraic group over complex numbers. To each unipotent element u ε G (up to conjugacy) and to the unit representation of the group of components of the centralizer of u, Springer (11), (12) associates an irreducible representation of the Weyl group W of G. The tensor product of that representation with the sign representation will be denoted ρu. (This agrees with the notation of (5).) This representation may be realized as a subspace of the cohomology in dimension 2β(u) of the variety of Borel subgroups containing u, where β(u) = dim . For example, when u = 1, ρu is the sign representation of W. The map u → ρu defines an injective map from the set of unipotent conjugacy classes in G to the set of irreducible representations of W (up to isomorphism). Our purpose is to describe this map in the case where G is simple of type Eu (n = 6, 7, 8). (When G is classical or of type F4, this map is described by Shoji (9), (10); the case where G is of type G2 is contained in (11).