The algorithm of computing generalized Green functions of a finite reductive group contains some unkonwn scalars occuring from the F_q structure of irreducible local systems on unipotent classes on G. In this paper, we determine such scalars in the case where G = SL_n with Frobenius map F of split or non-split type. In the case where F is of non-split type, we use the theory of graded Hecke algebras and their representations on the equivariant homology group due to Lusztig.
We show the existence of a unitriangular basic set for unipotent blocks simple reductive groups of classical type in bad characteristic with some exceptions. Then,we introduce an algorithm to count irreducible unipotent Brauer characters and we verify that this algorithm is effective for exceptional adjoint groups.
Lusztig's algorithm of computing generalized Green functions of reductive groups involves an ambiguity of certain scalars. In this paper, for reductive groups of classical type with arbitrary characteristic, we determine those scalars explicitly, and eliminate the ambiguity. Our results imply that all the generalized Green functions of classical type are computable.
We compute special unipotent Arthur packets for real reductive groups in many cases. We list the cases that lead to incomplete answers, and in those cases, provide a suitable set of representations that could lead to a complete description of the special Arthur packet. In the process of achieving this goal we classify theta forms of a given even complex nilpotent orbit, and find methods to compute the associated varieties of irreducible group representations.
In a previous paper, "Generalized Green functions and unipotent classes for finite reductive groups, I", we have determined certain unknown scalars involved in the algorithm of computing generalized Green functions in the case of SL_n. In this paper, we consider a similar problem for the case of classical groups Sp_{2n}, SO_N with arbitrary characteristic. Our result enables us, in particular, to compute Green functions in bad characteristic case, which was not known before.
The wavefront set is a fundamental invariant of an admissible representation arising from the Harish-Chandra-Howe local character expansion. In this paper, we give a precise formula for the wavefront set of an irreducible representation of real infinitesimal character in Lusztig's category of unipotent representations in terms of the Deligne-Langlands-Lusztig correspondence. Our formula generalizes the main result of arXiv:2112.14354, where this formula was obtained in the Iwahori-spherical case. We deduce that for any irreducible unipotent representation with real infinitesimal character, the…
The set of strata of a reductive group can be viewed as an enlargement of the set of unipotent classes. In this paper the notion of distinguished unipotent class is extended to this larger set. The strata of a Weyl group are introduced and studied.
Let $k$ be a $p$-adic field and let $\mathbf{G}(k)$ be the $k$-points of a connected reductive group, inner to split. The set of Aubert-Zelevinsky duals of the constituents of a tempered L-packet form an Arthur packet for $\mathbf{G}(k)$. In this paper, we give an alternative characterization of such Arthur packets in terms of the wavefront set, proving in some instances a conjecture of Jiang-Liu and Shahidi. Pursuing an analogy with real and complex groups, we define some special unions of Arthur packets which we call \emph{weak} Arthur packets and describe their constituents in terms of thei…
The paper is devoted to the problem of finding free generators in the fields of invariants for actions of unipotent groups on affine varieties. We consider the case when the unipotent group is the unipotent radical in an arbitrary parabolic subgroup in the reductive group of classical type GL(n), SL(n), O(n) or Sp(2n). In the explicit form, we present a system of free generators in the field of invariants for the action of the unipotent radical on the reductive group by conjugation.
Let $G$ be a simple algebraic group defined over a finite field of good characteristic, with associated Frobenius endomorphism $F$. In this article we extend an observation of Lusztig, (which gives a numerical relationship between an ordinary character of $G^F$ and its unipotent support), to the case where $Z(G)$ is disconnected. We then use this observation in some applications to the ordinary character theory of $G^F$.
The wavefront set is a fundamental invariant arising from the Harish-Chandra-Howe local character expansion of an admissible representation. We prove a precise formula for the wavefront set of an irreducible Iwahori-spherical representation with `real infinitesimal character' and determine a lower bound for this invariant in terms of the Deligne-Langlands-Lusztig parameters. In particular, for the Iwahori-spherical representations with real infinitesimal character, we deduce that the algebraic wavefront set is a singleton, as conjectured by Moeglin and Waldspurger. As a corollary, we obtain an…
With a view to determining character values of finite reductive groups at unipotent elements, we prove a number of results concerning inner products of generalised Gelfand-Graev characters with characteristic functions of character sheaves, here called Lusztig functions. These are used to determine projections of generalised Gelfand-Graev characters to the space of unipo- tent characters, and to the space of characters with a given wave front set. Such projections are expressed largely in terms of Weyl group data. We show how the values of characters at their unipotent support or wave front se…
This paper contains an exposition of the theory of character sheaves for reductive groups and some attempts to extend it to other cases: unipotent groups, reductive groups modulo the unipotent radical of a parabolic.
Let $k'/k$ be a finite purely inseparable field extension and let $G'$ be a reductive $k'$-group. We denote by $G=\R_{k'/k}(G')$ the Weil restriction of $G'$ across $k'/k$, a pseudo-reductive group. This article gives bounds for the exponent of the geometric unipotent radical $\RR_{u}(G_{\bar{k}})$ in terms of invariants of the extension $k'/k$, starting with the case $G'=\GL_n$ and applying these results to the case where $G'$ is a simple group.
We show that various properties of unipotent elements in a reductive group over the complex numbers can be recovered purely in terms of the affine Weyl group of the dual group.
In 1979 Lusztig proposed a conjectural construction of supercuspidal representations of reductive p-adic groups, which is similar to the well known construction of Deligne and Lusztig in the setting of finite reductive groups. We present a general method for explicitly calculating the representations arising from Lusztig's construction and illustrate it with several examples. The techniques we develop also provide background for the author's joint work with Weinstein on a purely local and explicit proof of the local Langlands correspondence.
In this paper, we construct and classify the special unipotent representations of a real reductive group attached to the principal nilpotent orbit. We give formulas for the $\mathbf{K}$-types, associated varieties, and Langlands parameters of all such representations.
We define an involution on the space of elliptic unipotent Langlands parameters of a reductive $p$-adic group $G$ and verify that when $G$ is split adjoint exceptional, the composition of this involution with the hyperspecial parahoric restriction map agrees with Lusztig's nonabelian Fourier transform for unipotent representations of the finite reductive quotient. This is inspired by recent works of Lusztig on the almost unipotent characters of $p$-adic groups and of Moeglin and Waldspurger on the elliptic Fourier transform of odd orthogonal groups.
Let G be a connected reductive algebraic group over an algebraic closed field. We define a (surjective) map from the set of conjugacy classes in the Weyl group to the set of unipotent classes of G.
We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly when their connected component is a torus. This extends the earlier result of Testerman and Zalesski treating connected reductive subgroups.
Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the ``order formula'' of D. Testerman for unipotent elements in G; moreover, we show that the same formula determines the p-nilpotence degree of the corresponding nilpotent elements in the Lie algebra of G. Second, if G is semisimple and p is sufficiently large, we show that G always has a faithful representation (r,V) with the property that the exponential of dr(X) lies in r(G) for each p-nilpote…
In the monograph arXiv:2108.03453, we define the notion of a unipotent representation of a complex reductive group. The representations we define include, as a proper subset, all special unipotent representations in the sense of Barbasch-Vogan and form the (conjectural) building blocks of the unitary dual. In arXiv:2108.03453 we provide combinatorial formulas for the infinitesimal characters of all unipotent representations of linear classical groups. In this paper, we establish analogous formulas for spin and exceptional groups, thus completing the determination of the infinitesimal character…
Let G be a connected reductive group. We define a map from the set of unipotent classes in G to the set of conjugacy classes in the Weyl group (assuming that the characteristic is not bad). This map is a one sided inverse of a map in the opposite direction constructed in an earlier paper.
In this paper we discuss some of Springer's work on unipotent elements in a reductive groups and on representations of Weyl groups. Among the topics considered are Springer's bijection from the unipotent variety to the nilpotent variety, Springer's example of nonrationality of certain representations of the Hecke algebras associated to a Weyl group and the Springer representation of the Weyl group associated to a unipotent element.
We make a study of unipotent elements in a connected reductive group over an algebraically closed field with emphasis on the case where the characteristic is a bad prime. We try to see how much of the theory of Dynkin-Kostant extends to this case. We also study the variety of Borel subgroups containing a unipotent element and prove a purity property for its cohomology in certain cases in bad characteristic.
We consider the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. We propose a definition of a partition of this variety into smooth locally closed smooth subvarieties indexed by the unipotent classes in the corresponding group over complex numbers. We obtain explicit results in type A,C and D.
Let G be a reductive connected group over the algebraic closure of a finite field. In this paper we give the classification of character sheaves on G in categorical terms (as a categorical centre). Previously such a classification was known for unipotent character sheaves and also in the case where the ground field is replaced by the complex numbers.
We develop a theory of microlocalization for Harish-Chandra modules, adapting a construction of Losev (\cite{Losev2011}). We explore the applications of this theory to unipotent representations of real reductive groups. For complex groups, we deduce a formula for the $K$-multiplicities of unipotent representations attached to a nilpotent orbit $\OO$, proving an old conjecture of Vogan (\cite{Vogan1991}) in a large family of cases.
Let G be a connected reductive group over a finite field F_q. A map from irreducible representations of G(F_q) to semisimple classes in the dual group was defined by a cohomological method by Deligne and the author in 1976. Here we show that when q is large enough this map has an alternative (non-cohomological) definition.
Let G be a connected reductive group defined over a finite field F_q. We give a parametrization of the irreducible representations of G(F_q) in terms of (twisted) categorical centres of various monoidal categories associated to G. (Results of this type were known earlier for unipotent representations. Analogous results for the parametrization of character sheaves on G were also known earlier.)
Let \hat G be the semidirect product of a connected reductive group G over F_q with a finite cyclic group generated by a quasisemisimple automorphism of G defined over F_q. In this paper we prove a conjecture of G. Malle concerning the extensions of unipotent representations of G(F_q) to \hat G(F_q).
Let A be a character sheaf on a reductive connected group G over an algebraically closed field. Assuming that the characteristic is not bad, we show that for certain conjugacy classes D in G the restriction of A to D is a local system up to shift; we also give a parametrization of unipotent cuspidal character sheaves on G in terms of restrictions to conjugacy classes. Without restriction on characteristic we define canonical bijections from the set of unipotent character sheaves on G (and from the set of unipotent representations of the corresponding split reductive group over a finite field) …
An irreducible ordinary character of a finite reductive group is called quadratic unipotent if it corresponds under Jordan decomposition to a semisimple element $s$ in a dual group such that $s^2=1$. We prove that there is a bijection between, on the one hand the set of quadratic unipotent characters of $GL(n,q)$ or $U(n,q)$ for all $n \geq 0$ and on the other hand, the set of quadratic unipotent characters of $Sp(2n,q)$ for all $n \geq 0$. We then extend this correspondence to $\ell$-blocks for certain $\ell$ not dividing $q$.
In this article we extend independent results of Lusztig and Hézard concerning the existence of irreducible characters of finite reductive groups, (defined in good characteristic and arising from simple algebraic groups), satisfying a strong numerical relationship with their unipotent support. Along the way we obtain some results concerning quasi-isolated semisimple elements.
For a connected reductive group $G$ over ${\mathbb R}$, we study cohomological $A$-parameters, which are Arthur parameters with the infinitesimal character of a finite-dimensional representation of $G({\mathbb C})$. We prove a structure theorem for such $A$-parameters, and deduce from it that a morphism of $L$-groups which takes a regular unipotent element to a regular unipotent element respects cohomological $A$-parameters. This is used to give complete understanding of cohomological $A$-parameters for all classical groups. We review the parametrization of Adams-Johnson packets of cohomologic…
Let G be a reductive p-adic group which splits over an unramified extension of the ground field. Hiraga, Ichino and Ikeda conjectured that the formal degree of a square-integrable G-representation $π$ can be expressed in terms of the adjoint $γ$-factor of the enhanced L-parameter of $π$. A similar conjecture was posed for the Plancherel densities of tempered irreducible G-representations. We prove these conjectures for unipotent G-representations. We also derive explicit formulas for the involved adjoint $γ$-factors.
Let G be an affine algebraic group over an algebraically closed field such that the identity component G^0 of G is reductive. Let W be the Weyl group of G and let D be a connected component of G whose image in G/G^0 is a unipotent element. In this paper we define a map from the set of "twisted conjugay classes" in W to the set of unipotent G^0-conjugacy classes in D, generalizing an earlier construction which applied when G is connected.
This work was inspired by two natural questions. The first question is when Lie(G')=Lie(G)', where G is a connected algebraic supergroup defined over a field of characteristic zero. The second question is whether the unipotent radical of any normal supersubgroup H of G coincides with the intersection of H and G_u, where G_u is the unipotent radical of G. Both questions have affirmative answers in the category of algebraic groups (in the second case one has to assume additionally that G and H are reduced whenever char K >0). Surprisingly, using the technique of Harish-Chandra superpairs and a c…
Let G be a reductive algebraic group over the algebraic closure of a finite field F_q of good characteristic. In this paper, we demonstrate a remarkable compatibility between the Springer correspondence for G and the parametrization of unipotent characters of G(F_q). In particular, we show that in a suitable sense, "large" portions of these two assignments in fact coincide. This extends earlier work of Lusztig on Springer representations within special pieces of the unipotent variety.
The unipotent variety of a reductive algebraic group $G$ plays an important role in the representation theory. In this paper, we will consider the closure $\bar{\Cal U}$ of the unipotent variety in the De Concini-Procesi compactification $\bar{G}$ of a connected simple algebraic group $G$. We will prove that $\bar{\Cal U}-\Cal U$ is a union of some $G$-stable pieces introduced by Lusztig in \cite{L4}. This was first conjectured by Lusztig. We will also give an explicit description of $\bar{\Cal U}$. It turns out that similar results hold for the closure of any Steinberg fiber in $\bar{G}$.
Suppose $G$ is a real reductive group. The determination of the irreducible unitary representations of $G$ is one of the major unsolved problem in representation theory. There is evidence to suggest that every irreducible unitary representation of $G$ can be constructed through a sequence of well-understood operations from a finite set of building blocks, called the unipotent representations. These representations are `attached' (in a certain mysterious sense) to the nilpotent orbits of $G$ on the dual space of its Lie algebra. Inside this finite set is a still smaller set, consisting of the u…
Let G(F_q) be the group of rational points of a split connected reductive group G defined over the finite field F_q. In this paper we show that the category of representations of G(F_q) which are finite direct sums of unipotent representations in a fixed two-sided cell is equivalent to the centre of a certain monoidal category of sheaves on the product of two copies of the flag manifold of G. We also consider a version of this for nonsplit groups.
We show that the Morita Frobenius number of the blocks of the alternating groups, the finite groups of Lie type in describing characteristic, and the Ree and Suzuki groups is 1. We also show that the Morita Frobenius number of almost all of the unipotent blocks of the finite groups of Lie type in non-defining characteristic is 1, and that in the remaining cases it is at most 2.
We review the categorical representation of a Kac-Moody algebra on unipotent representations of finite unitary groups in non-defining characteristic given by the authors. Then, we extend this construction to finite reductive groups of types B or C, in non-defining characteristic. We show that the decategorified representation is isomorphic to a direct sum of level 2 Fock spaces. We deduce that the Harish-Chandra branching graph coincides with the crystal graph of these Fock spaces. We also obtain derived equivalences between blocks, yielding Broue's abelian defect group conjecture for unipoten…
Let $G$ be a connected reductive group over $F_q$, where $q$ is large enough and the center of $G$ is connected. We are concerned with Lusztig's theory of {\em character sheaves}, a geometric version of the classical character theory of the finite group $G(F_q)$. We show that under a certain technical condition, the restriction of a character sheaf to its {\em unipotent support} (as defined by Lusztig) is either zero or an irreducible local system. As an application, the generalized Gelfand-Graev characters are shown to form a $\Z$-basis of the $\Z$-module of unipotently supported virtual char…
