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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., revised exposition

This paper gives explicit constructions of all irreducible representations of unipotent radicals $$N_{n_1,n_2,n_3}({\mathbb {F}}_q)$$ of the standard parabolic subgroups $$P_{n_1,n_2,n_3}({\mathbb {F}}_q)$$ of $${\mathrm {GL}}_n({\mathbb {F}}_q),$$ corresponding to the ordered partition $$ (n_1,\ n_2,\ n_3)$$ of n. The construction gives a bijection between coadjoint orbits of $$N_{n_1,n_2,n_3}({\mathbb {F}}_q)$$ and irreducible representations inducing from degree 1 characters in the sense of Boyarchenko’s construction. The result shows that the number of irreducible characters of $$N_{n_1,n_2,n_3}({\mathbb {F}}_q)$$ with a fixed degree is a polynomial in $$q-1$$ with nonnegative integer coefficients and verifies analogue conjectures of Higman, Lehrer, and Isaacs.

We discuss some of the contributions of T.A. Springer (1926–2011) to the theory of algebraic groups, with emphasis on his work on unipotent classes and representations of Weyl groups.

In this paper we describe $${\mathcal {R}}$$ -unipotent semigroups being regular extensions of a left regular band by an $$\mathcal {R}$$ -unipotent semigroup T as certain subsemigroups of a wreath product of a left regular band by T. We obtain Szendrei’s result that each E-unitary $${\mathcal {R}}$$ -unipotent semigroup is embeddable into a semidirect product of a left regular band by a group. Further, specialising the first author’s notion of $$\lambda $$ -semidirect product of a semigroup by a locally $${\mathcal {R}}$$ -unipotent semigroup, we provide an answer to an open question raised by the authors in [Extensions and covers for semigroups whose idempotents form a left regular band, Semigroup Forum 81 (2010), 51-70].

The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of $Rep_R(G)$, the category of smooth representations of a $p$-adic group $G$ with coefficients in $R$. In particular, they were used to construct level 0 decompositions when $R=\overline{\mathbb{Z}}_{\ell}$, $\ell \neq p$, by Dat for $GL_n$ and the author for a more general group. Wang proved in the case of $GL_n$ that the subcategory associated with a system of idempotents is equivalent to a category of coefficient systems on the Bruhat-Tits building. This result was used by Dat to prove an equivalence between an arbitrary level zero block of $GL_n$ and a unipotent block of another group. In this paper, we generalize Wang's equivalence of category to a connected reductive group on a non-archimedean local field., 17 pages, in English

For a general linear supergroup $G=GL(m|n)$, we consider a natural isomorphism $\phi: G \to U^-\times G_{ev} \times U^+$, where $G_{ev}$ is the even subsupergroup of $G$, and $U^-$, $U^+$ are appropriate odd unipotent subsupergroups of $G$. We compute the action of odd superderivations on the images $\phi^*(x_{ij})$ of the generators of $K[G]$. We describe a specific ordering of the dominant weights $X(T)^+$ of $GL(m|n)$ for which there exists a Donkin-Koppinen filtration of the coordinate algebra $K[G]$. Let $\Gamma$ be a finitely generated ideal $\Gamma$ of $X(T)^+$ and $O_{\Gamma}(K[G])$ be the largest $\Gamma$-subsupermodule of $K[G]$ having simple composition factors of highest weights $\lambda\in \Gamma$. We apply combinatorial techniques, using generalized bideterminants, to determine a basis of $G$-superbimodules appearing in Donkin-Koppinen filtration of $O_{\Gamma}(K[G])$.

We examine the cohomology and representation theory of a family of finite supergroup schemes of the form $(\mathbb G_a^-\times \mathbb G_a^-)\rtimes (\mathbb G_{a(r)}\times (\mathbb Z/p)^s)$. In particular, we show that a certain relation holds in the cohomology ring, and deduce that for finite supergroup schemes having this as a quotient, both cohomology mod nilpotents and projectivity of modules is detected on proper sub-super\-group schemes. This special case feeds into the proof of a more general detection theorem for unipotent finite supergroup schemes, in a separate work of the authors joint with Iyengar and Krause. We also completely determine the cohomology ring in the smallest cases, namely $(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb G_{a(1)}$ and $(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb Z/p$. The computation uses the local cohomology spectral sequence for group cohomology, which we describe in the context of finite supergroup schemes., 19 pages

Generalizing Lusztig's work, Malle as associated to any imprimitive complex reflection $W$ group a set of "unipotent characters", which are in bijection of the usual unipotent characters of the associated finite reductive group if $W$ is a Weyl group. He also obtained a partition of these characters into families and associated to each family a $\mathbb{Z}$-modular datum. We construct a categorification of some of these data, by studying the category of tilting modules of the Drinfeld double of the quantum enveloping algebra of the Borel of a simple complex Lie algebra.

We give a short combinatorial proof of Asai's decomposition formula for Lusztig induction of unipotent characters in groups of classical type, relying solely on the Mackey formula.

The Malle–Navarro conjecture relates central block theoretic invariants in two inequalities. In this article, we prove the conjecture for the 2-blocks and the unipotent 3-blocks of the general line...

S.-Y. Pan decomposes the uniform projection of the Weil representation of a finite symplectic-odd orthogonal dual pair, in terms of Deligne-Lusztig virtual characters, assuming that the order of the finite field is large enough. In this paper we use Pan's decomposition to study the theta correspondence for this kind of dual pairs, following the approach of Adams-Moy and Aubert-Michel-Rouquier. Our results give the theta correspondence between unipotent representations and certain quadratic unipotent representations., Comment: Revised version, accepted by Pacific J. Math

In this paper we study the Gan–Gross–Prasad problem for unitary groups over finite fields. Our results provide complete answers for unipotent representations, and we obtain the explicit branching of these representations.

Let G(F_q) be the group of rational points of a simple algebraic group defined and split over a finite field F_q. In this paper we define a new basis for the Grothendieck group of unipotent representations of G(F_q)., 36 pages. arXiv admin note: text overlap with arXiv:1805.03770

We study the local zeta integrals attached to a pair of generic representations ( π , τ ) of GL n × GL m , n > m , over a p-adic field. Through a process of unipotent averaging we produce a pair of corresponding Whittaker functions whose zeta integral is non-zero, and we express this integral in terms of the Langlands parameters of π and τ. In many cases, these Whittaker functions also serve as a test vector for the associated Rankin–Selberg (local) L-function.

This article is part of a project which consists of investigating Arthur packets for real classical groups. Our goal is to give an explicit description of these packets and to establish the multiplicity one property (which is known to hold for $p$-adic and complex groups). The main result in this paper is a construction of packets from unipotent packets on $c$-Levi factors using cohomological induction. An important tool used in the argument is a statement of commutativity between cohomological induction and spectral endoscopic transfer.

This article deals with the tamely ramified geometric Langlands correspondence for GL_2 on $\mathbf{P}_{\mathbf{F}_q}^1$, where $q$ is a prime power, with tame ramification at four distinct points $D = \{\infty, 0,1, t\} \subset \mathbf{P}^1(\mathbf{F}_q)$. We describe in an explicit way (1) the action of the Hecke operators on a basis of the cusp forms, which consists of $q$ elements; and (2) the correspondence that assigns to a pure irreducible rank 2 local system $E$ on $\mathbf{P}^1 \setminus D$ with unipotent monodromy its Hecke eigensheaf on the moduli space of rank 2 parabolic vector bundles. We define a canonical embedding $\mathbf{P}^1$ into this module space and show with a new proof that the restriction of the eigensheaf to the degree 1 part of this moduli space is the intermediate extension of $E$., 34 pages

Let G be a special orthogonal group SO(2n+1) defined over a p-adic field F. Let $\pi$ be an admissible irreducible representation of G(F) which is tempered and of unipotent reduction. We prove that $\pi$ has a wave front set. In some particular cases, for instance if $\pi$ is of the discrete series, we give a method to compute this wave front set., Comment: in French

We construct two supercharacter theories (in the sense of P. Diaconis and I.M. Isaacs) for the parabolic subgroups in orthogonal and symplectic groups. For each supercharacter theory, we obtain a supercharacter analog of the A.A. Kirillov formula for irreducible characters of finite unipotent groups.

In this paper we start the study of configurations of flags in closed orbits of real forms using mainly tools of GIT. As an application, using cross ratio coordinates for generic configurations, we identify boundary unipotent representations of the fundamental group of the figure eight knot complement into real forms of $${{\,\mathrm{PGL}\,}}(4,{\mathbb {C}})$$ .

Let $G$ be a finite Chevalley group. We are concerned with computing the values of the unipotent characters of $G$ by making use of Lusztig's theory of character sheaves. In this framework, one has to find the transformation between several bases for the class functions on $G$. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. We shall determine these scalars in the specific case where $G$ is the (twisted or non-twisted) group of type $E_6$ over the finite field with $q$ elements, for $q$ a power of the bad prime $p=3$, by exploiting known facts about the representation theory of the Hecke algebra associated with $G$., Comment: 12 pages

This paper has two main parts. Firstly, we give a classification of the $\ell$-blocks of finite special linear and unitary groups $SL_n(\epsilon q)$ in the non-defining characteristic $\ell\ge 3$. Secondly, we describe how the $\ell$-weights of $SL_n(\epsilon q)$ can be obtained from the $\ell$-weights of $GL_n(\epsilon q)$ when $\ell\nmid\mathrm{gcd}(n,q-\epsilon)$, and verify the Alperin weight conjecture for $SL_n(\epsilon q)$ under the condition $\ell\nmid\mathrm{gcd}(n,q-\epsilon)$. As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for any unipotent $\ell$-block of $SL_n(\epsilon q)$ if $\ell\nmid\mathrm{gcd}(n,q-\epsilon)$., Comment: revised version, to appear in Journal of Algebra

Hiraga, Ichino and Ikeda have conjectured an explicit expression for the Plancherel density of the group of points of a reductive group defined over a local field F, in terms of local Langlands parameters. In these lectures we shall present a proof of these conjectures for Lusztig’s class of representations of unipotent reduction if F is p-adic and G is of adjoint type and splits over an unramified extension of F. This is based on the author’s paper [Spectral] transfer morphisms for unipotent affine Hecke algebras, Selecta Math. (N.S.) 22 (2016), no. 4, 2143–2207]. More generally for G connected reductive (still assumed to be split over an unramified extension of F), we shall show that the requirement of compatibility with the conjectures of Hiraga, Ichino and Ikeda essentially determines the Langlands parameterisation for tempered representations of unipotent reduction. We shall show that there exist parameterisations for which the conjectures of Hiraga, Ichino and Ikeda hold up to rational constant factors. The main technical tool is that of spectral transfer maps between normalised affine Hecke algebras used in op. cit

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (resp. subalgebra), or a natural generalisation if $G$ is of type $D_n$. From these representatives, it is straightforward to generate representatives for any given class. Along the way we also prove recognition theorems for identifying both the unipotent classes and nilpotent orbits of exceptional algebraic groups., 26 pages

This work concerns the representation theory and cohomology of a finite unipotent supergroup scheme $G$ over a perfect field $k$ of positive characteristic $p\ge 3$. It is proved that an element $x$ in the cohomology of $G$ is nilpotent if and only if for every extension field $K$ of $k$ and every elementary sub-supergroup scheme $E\subseteq G_K$, the restriction of $x_K$ to $E$ is nilpotent. It is also shown that a $kG$-module $M$ is projective if and only if for every extension field $K$ of $k$ and every elementary sub-supergroup scheme $E\subseteq G_K$, the restriction of $M_K$ to $E$ is projective. The statements are motivated by, and are analogues of, similar results for finite groups and finite group schemes, but the structure of elementary supergroups schemes necessary for detection is more complicated than in either of these cases. One application is a detection theorem for the nilpotence of cohomology, and projectivity of modules, over finite dimensional Hopf subalgebras of the Steenrod algebra., 46 pages; Sections 12 on Z-graded group schemes and the Steenrod algebra is revised compared to the previous version

We give a new realization of the prefundamental representations $L^\pm_{r,a}$ introduced by Hernandez and Jimbo, when the quantum loop algebra $U_q(\mathfrak{g})$ is of types $A_n^{(1)}$ and $D_n^{(1)}$, and the $r$-th fundamental weight $\varpi_r$ for types $A_n$ and $D_n$ is minuscule. We define an action of the Borel subalgebra $U_q(\mathfrak{b})$ of $U_q(\mathfrak{g})$ on the unipotent quantum coordinate ring associated to the translation by $-\varpi_r$, and show that it is isomorphic to $L^\pm_{r,a}$. We then give a combinatorial realization of $L^+_{r,a}$ in terms of the Lusztig data of the dual PBW vectors., Comment: 41 pages, introduction is revised, Lemma 3.2 is revised, the proofs of Lemma 3.8 and Corollary 4.8 are revised, added Example 3.9, Example 4.4 and Example 4.19, comments added on Remark 3.11, Remark 4.9, Remark 4.21 and Remark 4.30, typos corrected, minor corrections

Let u be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at u with the totally positive part of the flag manifold. We show that this intersection has a natural cell decomposition which is part of the cell decomposition (Rietsch) of the totally positive flag manifold., Comment: 20 pages. A new section has been added

Using the geometric Satake correspondence, the Mirkovic-Vilonen cycles in the affine Grasssmannian give bases for representations of a semisimple group G . We prove that these bases are "perfect", i.e. compatible with the action of the Chevelley generators of the positive half of the Lie algebra g. We compute this action in terms of intersection multiplicities in the affine Grassmannian. We prove that these bases stitch together to a basis for the algebra C[N] of regular functions on the unipotent subgroup. We compute the multiplication in this MV basis using intersection multiplicities in the Beilinson-Drinfeld Grassmannian, thus proving a conjecture of Anderson. In the third part of the paper, we define a map from C[N] to a convolution algebra of measures on the dual of the Cartan subalgebra of g. We characterize this map using the universal centralizer space of G. We prove that the measure associated to an MV basis element equals the Duistermaat-Heckman measure of the corresponding MV cycle. This leads to a proof of a conjecture of Muthiah. Finally, we use the map to measures to compare the MV basis and Lusztig's dual semicanonical basis. We formulate conjectures relating the algebraic invariants of preprojective algebra modules (which underlie the dual semicanonical basis) and geometric invariants of MV cycles. In the appendix, we use these ideas to prove that the MV basis and the dual semicanonical basis do not coincide in SL_6., 82 pages, with an appendix by Calder Morton-Ferguson and Anne Dranowski

We study unipotent fundamental groups for open varieties over [math] -adic fields with base point degenerating to the boundary. In particular, we show that the Galois representations associated to the étale unipotent fundamental group are semistable.

The formal degree of a unipotent discrete series character of a simple linear algebraic group over a non-archimedean local field (in the sense of Lusztig [17] ), is a rational function of q evaluated at q = q , the cardinality of the residue field. The irreducible factors of this rational function are q and cyclotomic polynomials. We prove that the formal degree of a supercuspidal unipotent representation determines its Lusztig-Langlands parameter, up to twisting by weakly unramified characters. For split exceptional groups this result follows from the work of M. Reeder [28] , and for the remaining exceptional cases this is verified in [7] . In the present paper we treat the classical families. The main result of this article characterizes unramified Lusztig-Langlands parameters which support a cuspidal local system in terms of formal degrees. The result implies the uniqueness of so-called cuspidal spectral transfer morphisms (as introduced in [22] ) between unipotent affine Hecke algebras (up to twisting by unramified characters). In [23] the essential uniqueness of arbitrary unipotent spectral transfer morphisms was reduced to the cuspidal case.

In his theory of unipotent characters of finite groups of Lie type, Lusztig constructed modular categories from two-sided cells in Weyl groups. Brou\'e,Malle and Michel have extended parts of Lusztig's theory to complex reflection groups. This includes generalizations of the corresponding fusion algebras, although the presence of negative structure constants prevents them from arising from modular categories. We give here the first construction of braided pivotal monoidal categories associated with non-real reflection groups (later reinterpreted by Lacabanne as super modular categories). They are associated with cyclic groups, and their fusion algebras are those constructed by Malle., Comment: 26 pages

In this paper we study the Gan-Gross-Prasad problem for finite classical groups. Our results provide complete answers for unipotent representations, and we obtain the explicit branching laws for these representations. Moreover, for arbitrary representations, we give a formula to reduce the Gan-Gross-Prasad problem to the restriction problem of Deligne-Lusztig characters.

Let $G$ be a semisimple real algebraic Lie group of real rank at least two and $U$ be the unipotent radical of a non-trivial parabolic subgroup. We prove that a discrete Zariski dense subgroup of $G$ that contains an irreducible lattice of $U$ is an arithmetic lattice of $G$. This solves a conjecture of Margulis and extends previous work of Hee Oh., Comment: 54 pages

Suppose that $G$ is a simple adjoint reductive group over $\mathbf{Q}$, with an exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of Gross-Wallach). Then there is a notion of modular forms for $G$, anchored on the so-called quaternionic discrete series representations of $G(\mathbf{R})$. The purpose of this paper is to give an explicit form of the Fourier expansion of modular forms on $G$, along the unipotent radical $N$ of the Heisenberg parabolic $P = MN$ of $G$., Comment: changed title; broadened definition of modular form; added discussion of constant term and Klingen Eisenstein series

We provide a “soft” proof for nontrivial bounds on spherical, hyperbolic, and unipotent Fourier coefficients of a fixed Maass form for a general cofinite lattice Γ \Gamma in PGL 2 ⁡ ( R ) {\operatorname {PGL}_2(\mathbb {R})} . We use the amplification method based on the Airy type phenomenon for corresponding matrix coefficients and an effective Selberg type pointwise asymptotic for the lattice points counting in various homogeneous spaces for the group PGL 2 ⁡ ( R ) {\operatorname {PGL}_2(\mathbb {R})} . This requires only L 2 L^2 -theory. We also show how to use the uniform bound for the L 4 L^4 -norm of K K -types in a fixed automorphic representation of PGL 2 ⁡ ( R ) {\operatorname {PGL}_2(\mathbb {R})} in order to slightly improve these bounds.

We introduce $q$-analogues of uniform matroids, which we call $q$-niform matroids. While uniform matroids admit actions of symmetric groups, $q$-niform matroids admit actions of finite general linear groups. We show that the equivariant Kazhdan-Lusztig polynomial of a $q$-niform matroid is the unipotent $q$-analogue of the equivariant Kazhdan-Lusztig polynomial of the corresponding uniform matroid, thus providing evidence for the positivity conjecture for equivariant Kazhdan-Lusztig polynomials.

Introduced by Kawanaka in order to find the unipotent representations of finite groups of Lie type, gener- alized Gelfand–Graev characters have remained somewhat mysterious. Even in the case of the finite general linear groups, the combinatorics of their decompositions has not been worked out. This paper re-interprets Kawanaka's def- inition in type A in a way that gives far more flexibility in computations. We use these alternate constructions to show how to obtain generalized Gelfand–Graev representations directly from the maximal unipotent subgroups. We also explicitly decompose the corresponding generalized Gelfand–Graev characters in terms of unipotent representations, thereby recovering the Kostka–Foulkes polynomials as multiplicities.

Given a toric affine algebraic variety $X$ and a collection of one-parameter unipotent subgroups $U_1,\ldots,U_s$ of $\mathop{\rm Aut}(X)$ which are normalized by the torus acting on $X$, we show that the group $G$ generated by $U_1,\ldots,U_s$ verifies the following alternative of Tits' type: either $G$ is a unipotent algebraic group, or it contains a non-abelian free subgroup. We deduce that if $G$ is $2$-transitive on a $G$-orbit in $X$, then $G$ contains a non-abelian free subgroup, and so, is of exponential growth., 24 pages. The main result strengthened, the proof of Proposition 4.8 written in more detail; some references added; the referee remarks taken into account; the title changed

Let $G(q)$ be a Chevalley group over a finite field $F_q$. By Lusztig's and Shoji's work, the problem of computing the values of the unipotent characters of $G(q)$ is solved, in principle, by the theory of character sheaves; one issue in this solution is the determination of certain scalars relating two types of class functions on $G(q)$. We show that this issue can be reduced to the case where $q$ is a prime, which opens the way to use computer algebra methods. Here, and in a sequel to this article, we use this approach to solve a number of cases in groups of exceptional type which seemed hitherto out of reach., 20 pages; minor corrections and additions

The notion of the supercharacter theory was introduced by P. Diaconis and I. M. Isaaks in 2008. In this paper, we present a review of the main notions and facts of the general theory and discuss the construction of the supercharacter theory for algebra groups and the theory of basic characters for unitriangular groups over a finite field. Based on his earlier papers, the author constructs the supercharacter theory for finite groups of triangular type. The structure of the Hopf algebra of supercharacters for triangular groups over finite fields is also characterized.

We use the progenerator constructed in our previous paper to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees., Comment: arXiv admin note: text overlap with arXiv:1611.07373

Let $$\mathrm{F}$$ (resp. $$\mathbb F$$ ) be a nonarchimedean locally compact field with residue characteristic p (resp. a finite field with characteristic p). For $$k=\mathrm{F}$$ or $$k=\mathbb F$$ , let $$\mathbf {G}$$ be a connected reductive group over k and R be a commutative ring. We denote by $$\mathrm{Rep}( \mathbf G(k)) $$ the category of smooth R-representations of $$ \mathbf G(k) $$ . To a parabolic k-subgroup $${\mathbf P}=\mathbf {MN}$$ of $$\mathbf G$$ corresponds the parabolic induction functor $$\mathrm{Ind}_{\mathbf P(k)}^{\mathbf G(k)}:\mathrm{Rep}( \mathbf M(k)) \rightarrow \mathrm{Rep}( \mathbf G(k))$$ . This functor has a left and a right adjoint. Let $${{\mathcal {U}}}$$ (resp. $${\mathbb {U}}$$ ) be a pro-p Iwahori (resp. a p-Sylow) subgroup of $$ \mathbf G(k) $$ compatible with $${\mathbf P}(k)$$ when $$k=\mathrm{F}$$ (resp. $$\mathbb F$$ ). Let $${H_{ \mathbf G(k)}}$$ denote the pro-p Iwahori (resp. unipotent) Hecke algebra of $$ \mathbf G(k) $$ over R and $$\mathrm{Mod}({H_{ \mathbf G(k)}})$$ the category of right modules over $${H_{ \mathbf G(k)}}$$ . There is a functor $$\mathrm{Ind}_{{H_{ \mathbf M(k)}}}^{{H_{ \mathbf G(k)}}}: \mathrm{Mod}({H_{ \mathbf M(k)}}) \rightarrow \mathrm{Mod}({H_{ \mathbf G(k) }})$$ called parabolic induction for Hecke modules; it has a left and a right adjoint. We prove that the pro-p Iwahori (resp. unipotent) invariant functors commute with the parabolic induction functors, namely that $$\mathrm{Ind}_{\mathbf P(k)}^{\mathbf G(k)}$$ and $$\mathrm{Ind}_{{H_{ \mathbf M(k)}}}^{{H_{ \mathbf G(k)}}}$$ form a commutative diagram with the $${{\mathcal {U}}}$$ and $${{\mathcal {U}}}\cap \mathbf M(\mathrm{F})$$ (resp. $${\mathbb {U}}$$ and $${\mathbb {U}}\cap \mathbf M(\mathbb F) $$ ) invariant functors. We prove that the pro-p Iwahori (resp. unipotent) invariant functors also commute with the right adjoints of the parabolic induction functors. However, they do not commute with the left adjoints of the parabolic induction functors in general; they do if p is invertible in R. When R is an algebraically closed field of characteristic p, we show that an irreducible admissible R-representation of $$ \mathbf G(\mathrm{F}) $$ is supercuspidal (or equivalently supersingular) if and only if the $${H_{ \mathbf G(\mathrm{F})}}$$ -module $${\mathfrak {m}}$$ of its $${{\mathcal {U}}}$$ -invariants admits a supersingular subquotient, if and only if $${\mathfrak {m}}$$ is supersingular.

We give a combinatorial formula to calculate the Euler characteristic of an analogue of a Deligne–Lusztig variety, denoted Y w , g , which is attached to an element w in the Weyl group of G L n and g ∈ G L n . The main theorem of this paper states that the Euler characteristic of Y w , g only depends on the unipotent part of the Jordan decomposition of g and the conjugacy class of w. It generalizes the formula of the Euler characteristic of Springer fibers for type A.

Let $\pi$ be an irreducible cuspidal representation of $\mathrm{GL}_{kn}\left(\mathbb{F}_q\right)$. Assume that $\pi = \pi_{\theta}$, corresponds to a regular character $\theta$ of $\mathbb{F}_{q^{kn}}^{*}$. We consider the twisted Jacquet module of $\pi$ with respect to a non-degenerate character of the unipotent radical corresponding to the partition $(n^k)$ of $kn$. We show that, as a $\mathrm{GL}_{n}\left(\mathbb{F}_q\right)$-representation, this Jacquet module is isomorphic to $\pi_{\theta \upharpoonright_{\mathbb{F}_n^*}} \otimes \mathrm{St}^{k-1}$, where $\mathrm{St}$ is the Steinberg representation of $\mathrm{GL}_{n}\left(\mathbb{F}_q\right)$. This generalizes a theorem of D. Prasad, who considered the case $k=2$. We prove and rely heavily on a formidable identity involving $q$-hypergeometric series and linear algebra., Comment: 27 pages. Comments are welcome

Horospheres for an action of a semisimple algebraic group G on an affine variety X are the generic orbits of a maximal unipotent subgroup U ⊂ G or, equivalently, the generic fibers of the categorical quotient of the variety X by the action of U, which is defined by the values of the highest weight functions. The remaining fibers of this quotient (which we call degenerate horospheres) for a certain class of spherical G-varieties containing all simply connected symmetric spaces are studied.

We study the decomposition of certain reducible characters of classical groups as the sum of irreducible ones. Let G be an algebraic group of classical type with defining characteristic p > 0 , μ a dominant weight and W the Weyl group of G. Let G = G ( q ) be a finite classical group, where q is a p-power. For a weight μ of G the sum s μ of distinct weights w ( μ ) with w ∈ W viewed as a function on the semisimple elements of G is known to be a generalized Brauer character of G called an orbit character of G. We compute, for certain orbit characters and every maximal torus T of G, the multiplicity of the trivial character 1 T of T in s μ . The main case is where μ = ( q − 1 ) ω and ω is a fundamental weight of G. Let St denote the Steinberg character of G. Then we determine the unipotent characters occurring as constituents of s μ ⋅ S t defined to be 0 at the p-singular elements of G. Let β μ denote the Brauer character of a representation of S L n ( q ) arising from an irreducible representation of G with highest weight μ. Then we determine the unipotent constituents of the characters β μ ⋅ S t for μ = ( q − 1 ) ω , and also for some other μ (called strongly q-restricted). In addition, for strongly restricted weights μ, we compute the multiplicity of 1 T in the restriction β μ | T for every maximal torus T of G.

We construct a monoidal category C w , v which categorifies the doubly-invariant algebra C N ′ ( w ) [ N ] N ( v ) associated with Weyl group elements w and v. It gives, after a localization, the coordinate algebra C [ R w , v ] of the open Richardson variety associated with w and v. The category C w , v is realized as a subcategory of the graded module category of a quiver Hecke algebra R. When v = id , C w , v is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent coordinate algebra A q ( n ( w ) ) Z [ q , q − 1 ] given by Kang–Kashiwara–Kim–Oh. We show that the category C w , v contains special determinantial modules M ( w ≤ k Λ , v ≤ k Λ ) for k = 1 , … , l ( w ) , which commute with each other. When the quiver Hecke algebra R is symmetric, we find a formula of the degree of R-matrices between the determinantial modules M ( w ≤ k Λ , v ≤ k Λ ) . When it is of finite ADE type, we further prove that there is an equivalence of categories between C w , v and C u for w , u , v ∈ W with w = v u and l ( w ) = l ( v ) + l ( u ) .

In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive p-adic groups, analogous to Deligne–Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig’s program. Precisely, let X be the Deligne–Lusztig (ind-pro-)scheme associated to a division algebra D over a non-Archimedean local field K of positive characteristic. We study the $$D^\times $$ -representations $$H_\bullet (X)$$ by establishing a Deligne–Lusztig theory for families of finite unipotent groups that arise as subquotients of $$D^\times $$ . There is a natural correspondence between quasi-characters of the (multiplicative group of the) unramified degree-n extension of K and representations of $$D^{\times }$$ given by $$\theta \mapsto H_\bullet (X)[\theta ]$$ . For a broad class of characters $$\theta ,$$ we show that the representation $$H_\bullet (X)[\theta ]$$ is irreducible and concentrated in a single degree. After explicitly constructing a Weil representation from $$\theta $$ using $$\chi $$ -data, we show that the resulting correspondence matches the bijection given by local Langlands and therefore gives a geometric realization of the Jacquet–Langlands transfer between representations of division algebras.

Let p and $$\ell $$ be two distinct primes, F a p-adic field and n an integer. We show that any level 0 block of the category of smooth $$\overline{{\mathbb {Z}}}_{\ell }$$ -valued representations of $$ \mathrm{GL}_{n}(F)$$ is equivalent to the unipotent block of an appropriate product of $$\mathrm{GL}_{n_{i}}(F_{i})$$ . More precisely, we show that level 0 blocks of p-adic general linear groups obey the “functoriality principle for blocks” introduced in Dat (Contemp Math 691:103–131, 2017). The overall strategy is to use the “Jordan decomposition of blocks” of finite general linear groups, and the theory of coefficients systems on the relevant buildings as a bridge between the p-adic and finite worlds. To overcome the main technical difficulty, we need fine properties of Deligne–Lusztig cohomology proved in Bonnafe et al. (Ann Math 185(2):609–670, 2017).

We classify measures on a homogeneous space which are invariant under a certain solvable subgroup and ergodic under its unipotent radical. Our treatment is independent of characteristic. As a result we get the first measure classification for the action of semisimple subgroups without any characteristic restriction.

A ring $R$ is a UU ring if every unit is unipotent, or equivalently if every unit is a sum of a nilpotent and an idempotent that commute. These rings have been investigated in C\u{a}lug\u{a}reanu \cite{C} and in Danchev and Lam \cite{DL}. In this paper, two generalizations of UU rings are discussed. We study rings for which every unit is a sum of a nilpotent and an idempotent, and rings for which every unit is a sum of a nilpotent and two idempotents that commute with one another.