We give an overview of some of the main results in geometric representation theory that have been proved by means of the Steinberg variety. Steinberg's insight was to use such a variety of triples in order to prove a conjectured formula by Grothendieck.
The Steinberg variety was later used to give an alternative approach to Springer's representations and played a central role in the proof of the Deligne-Langlands conjecture for Hecke algebras by Kazhdan and Lusztig.
The aim of this paper is to give generalizationof the constructionof the Steinberg tempered character on a connected reductive p-adic group. We prove that this character is invariant by the weak restriction of the Jacquet module by analogy to finite reductive groups.
This is a preliminary version of the first chapter of a book project on the character theory of finite groups of Lie type. It provides the foundations from the general theory of reductive algebraic groups over a finite field.
Let G be a connected reductive group over an algebraic closure of a finite field Fq. In this paper it is proved that the infinite dimensional Steinberg module of kG defined by N. Xi in 2014 is irreducible when k is a field of positive characteristic and char k is not char Fq. For certain special linear groups, we show that the Steinberg modules of the groups are not quasi-finite with respect to some natural quasi-finite sequences of the groups.
Let G be a reductive algebraic group over a field of positive characteristic and denote by C(G) the category of rational G-modules. In this note we investigate the subcategory of C(G) consisting of those modules whose composition factors all have highest weights linked to the Steinberg weight. This subcategory is denoted ST and called the Steinberg component. We give an explicit equivalence between ST and C(G) and we derive some consequences. In particular, our result allows us to relate the Frobenius contracting functor to the projection functor from C(G) onto ST .
We prove an isomorphism for simple perverse sheaves on the affine Grassmannian of a connected reductive algebraic group that is a geometric counterpart (in light of the Finkelberg-Mirković conjecture) of the Steinberg tensor product formula for simple representations of reductive groups over fields of positive characteristic.
We generalize results of P. Schneider and U. Stuhler for GL_l+1 to a reductive algebraic group G defined and split over a non-archimedean local field K. Following their lines, we prove that the generalized Steinberg representations of G with coefficients in any abelian group are cyclic. When G is semi-simple of adjoint type, we give an expression of these representations, whenever it is possible and in particular for those that are of maximal degree, in terms of the parahoric subgroups of G.
Let F be a local non-archimedean field and let G be the group of F-valued points of a reductive algebraic group over F. In this paper we compute the Ext-groups of generalized Steinberg representations in the category of smooth G-representations with coefficients in a certain self-injective ring.
For a reductive group G defined over an algebraically closed field of positive characteristic, we show that the Frobenius contraction functor of G-modules is right adjoint to the Frobenius twist of the modules tensored with the Steinberg module twice. It follows that the Frobenius contraction functor preserves injectivity, good filtrations, but not semisiplicity.
We describe the equivariant K-groups of a family of generalized Steinberg varieties that interpolates between the Steinberg variety of a reductive, complex algebraic group and its nilpotent cone in terms of the extended affine Hecke algebra and double cosets in the extended affine Weyl group. As an application, we use this description to define Kazhdan-Lusztig "bar" involutions and Kazhdan-Lusztig bases for these equivariant K-groups.
We prove Dipendra Prasad's conjecture on the distinction of the Steinberg representation for symmetric spaces of the form G(E)/G(F), where G is a split reductive group defined over F and E/F an unramified quadratic extension of non-archimedean local fields.
Let G be a complex, connected, reductive algebraic group with Weyl group W and Steinberg variety Z. We show that the graded Borel-Moore homology of Z is isomorphic to the smash product of the coinvariant algebra of W and the group algebra of W.
In this paper we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps. In particular, a few classical results of Steinberg and Deligne & Lusztig on complex representations of finite groups of Lie type are extended to reductive algebraic groups with Frobenius maps.
Let $G/H$ be a reductive symmetric space over a $p$-adic field $F$, the algebraic groups $G$ and $H$ being assumed semisimple of relative rank $1$. One of the branching problems for the Steinberg representation $\St_G$ of $G$ is the determination of the dimension of the intertwining space ${\rm Hom}_H (\St_G ,π)$, for any irreducible representation $π$ of $H$. In this work we do not compute this dimension, but show how it is related to the dimensions of some other intertwining spaces ${\rm Hom}_{K_i} ({\tilde π} ,1)$, for a certain finite family $K_i$, $i=1,...,r$, of anisotropic subgroups of …
Let $G$ be the $F$-points of a connected reductive group over a non-archimedean local field $F$ of residue characteristic $p$ and $R$ be a commutative ring. Let $P=LU$ be a parabolic subgroup of $G$ and $Q$ be a parabolic subgroup of $G$ containing $P$. We study the functor $\mathrm{St}_Q^G$ taking a smooth $R$-representation $σ$ of $L$ which extends to a representation $\mathrm{e}_G(σ)$ of $G$ trivial on $U$ to the smooth $R$-representation $\mathrm{e}_G(σ) \otimes_R \mathrm{St}_Q^G(R)$ of $G$ where $\mathrm{St}_Q^G(R)$ is the generalised Steinberg representation.
Let $G$ be a complex, connected, reductive algebraic group. In this paper we show analogues of the computations by Borho and MacPherson of the invariants and anti-invariants of the cohomology of the Springer fibres of the cone of nilpotent elements, $\mathcal N$, of $\operatorname{Lie}(G)$ for the Steinberg variety $Z$ of triples. Using a general specialization argument we show that for a parabolic subgroup $W_P \times W_Q$ of $W \times W$ the space of $W_P \times W_Q$-invariants and the space of $W_P \times W_Q$-anti-invariants of $H_{4n}(Z)$ are isomorphic to the top Borel-Moore homology gro…
We investigate the structure of root data by considering their decomposition as a product of a semisimple root datum and a torus. Using this decomposition we obtain a parameterisation of the isomorphism classes of all root data. By working at the level of root data we introduce the notion of a smooth regular embedding of a connected reductive algebraic group, which is a refinement of the commonly used regular embeddings introduced by Lusztig. In the absence of Steinberg endomorphisms such embeddings were constructed by Benjamin Martin. In an unpublished manuscript Asai proved three key reducti…
We discuss some connections between the closure $\bar F$ of a Steinberg fiber in the wonderful compactification of an adjoint group and the affine Deligne-Lusztig varieties $X_w(1)$ in the affine flag variety. Among other things, we describe the emptiness/nonemptiness pattern of $X_w(1)$ if the translation part of $w$ is quasi-regular. As a by-product, we give a new proof of the explicit description of $\bar F$, first obtained in \cite{H1}.
The work of Bernstein-Zelevinsky and Zelevinsky gives a good understanding of irreducible subquotients of a reducible principal series representation of $GL_n(F)$, $F$ a $p$-adic field (without specifying their multiplicities which is done by a Kazhdan-Lusztig type conjecture). In this paper we make a proposal of a similar kind for principal series representations of $GL_n({\mathbb R})$. Our investigation on principal series representations naturally led us to consider the Steinberg representation for real groups, which has curiuosly not been paid much attention to in the subject (unlike the $…
The commuting variety of a reductive Lie algebra ${\goth g}$ is the underlying variety of a well defined subscheme of $\gg g{}$. In this note, it is proved that this scheme is normal. In particular, its ideal of definition is a prime ideal.
For a reductive Lie algbera over an algbraically closed field of charasteristic zero,we consider a borel subgroup $B$ of its adjoint group, a Cartan subalgebra contained inthe Lie algebra of $B$ and the closure $X$ of its orbit under $B$ in the Grassmannian.The variety $X$ plays an important role in the study of the commuting variety. In thisnote, we prove that $X$ is Gorenstein with rational singularities.
In 1965, Steinberg's study of conjugacy classes in connected reductive groups led him to construct an affine subspace parametrising regular conjugacy classes, which he noticed is also a cross section for the conjugation action by the unipotent radical of a Borel subgroup on another affine subspace. Recently, generalisations of this slice and its cross section property have been obtained by Sevostyanov in the context of quantum group analogues of W-algebras and by He-Lusztig in the context of Deligne-Lusztig varieties. Such slices are often thought of as group analogues of Slodowy slices. In th…
We calculate the rational cohomology of the commuting variety $X_{G, n}$ consisting of $n$-tuples of commuting elements of a compact reductive group $G$. This is done by studying a map from a related variety $Y_{G, n}$, which has easily calculated cohomology. The proof studies the fibers of the map and uses the Vietoris-Begle theorem to prove that the induced map on rational cohomology is an isomorphism.
The aim of this paper is to describe the irregular locus of the commuting variety of a reductive symmetric Lie algebra. More precisely, we want to enlighten a remark of Popov. In [Po], the irregular locus of the commuting variety of any reductive Lie algebra is described and its codimension is computed. This provides a bound for the codimension of the singular locus of this commuting variety. [Po, Remark 1.13] suggests that the arguments and methods of [Po] are suitable for obtaining analogous results in the symmetric setting. We show that some difficulties arise in this case and we obtain som…
We define a class of transversal slices in spaces which are quasi-Poisson for the action of a complex semisimple group G. This is a multiplicative analogue of Whittaker reduction. One example is the multiplicative universal centralizer of G, which is equipped with the usual symplectic structure in this way. We construct a smooth relative compactification by taking the closure of each centralizer fiber in the wonderful compactification of G. By realizing this partial compactification as a transversal in a larger quasi-Poisson variety, we show that it is smooth and log-symplectic.
We define and study the variety of reductions for a reductive symmetric pair (G,theta), which is the natural compactification of the set of the Cartan subspaces of the symmetric pair. These varieties generalize the varieties of reductions for the Severi varieties studied by Iliev and Manivel, which are Fano varieties. We develop a theoretical basis to the study these varieties of reductions, and relate the geometry of these variety to some problems in representation theory. A very useful result is the rigidity of semi-simple elements in deformations of algebraic subalgebras of Lie algebras. We…
We prove the exactness of the reduction map from étale $(ϕ,Γ)$-modules over completed localized group rings of compact open subgroups of unipotent $p$-adic algebraic groups to usual étale $(ϕ,Γ)$-modules over Fontaine's ring. This reduction map is a component of a functor from smooth $p$-power torsion representations of $p$-adic reductive groups (or more generally of Borel subgroups of these) to $(ϕ,Γ)$-modules. Therefore this gives evidence for this functor---which is intended as some kind of $p$-adic Langlands correspondence for reductive groups---to be exact. We also show that the correspon…
Let $F$ be a non-Archimedean locally compact field, let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\subset G$ and a ring $L$ we consider the $G$-representation on the $L$-module$$(*)\quad\quad\quad\quad C^{\infty}(G/Q,L)/\sum_{Q'\supsetneq Q}C^{\infty}(G/Q',L).$$Let $I\subset G$ denote an Iwahori subgroup. We define a certain free finite rank $L$-module ${\mathfrak M}$ (depending on $Q$; if $Q$ is a Borel subgroup then $(*)$ is the Steinberg representation and ${\mathfrak M}$ is of rank one) and construct an $I$-equivariant embedding of $(*)$ into $C^{\infty}…
The nilpotent cone of a reductive Lie algebra has a desingularization given by thecotangent bundle of the flag variety. Analogously, the nullcone of a cartesianpower of the algebra has a desingularization given by a vector bundle over theflag variety. As for the nullcone, the subvariety of elements whose componentsare in a same Borel subalgebra, has a desingularization given by a vector bundle overthe flag variety. In this note, some properties of these varieties are given. Forthe study of the commuting variety, the analogous variety to the flag variety isthe closure in the Grassmannian of the…
We give an algorithm to compute the associated variety of a Harish- Chandra module for a real reductive group $G({\mathbb R})$. The algorithm is implemented in the atlas software package.
According to Kazhdan-Lusztig and Ginzburg, the Hecke algebra of an affine Weyl group is identified with the equivariant $K$-group of Steinberg's triple variety. The $K$-group is equipped with a filtration indexed by closed $G$-stable subvarieties of the nilpotent variety, where $G$ is the corresponding reductive algebraic group over $\mathbb{C}$. In this paper we will show in the case of type $A$ that the filtration is compatible with the Kazhdan-Lusztig basis of the Hecke algebra.
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 $\bar{G}$.
We study the K-stability of a polarised variety with non-reductive automorphism group. We associate a canonical filtration of the co-ordinate ring to each variety of this kind, which destabilises the variety in several examples which we compute. We conjecture this holds in general. This is an algebro-geometric analogue of Matsushima's theorem regarding the existence of constant scalar curvature Kähler metrics. As an application, we give an example of an orbifold del Pezzo surface without a Kähler-Einstein metric.
In this paper we establish a connection between the associated variety of a representation and the existence of certain degenerate Whittaker functionals, for both smooth and K-finite vectors, for all quasi-split real reductive groups, thereby generalizing results of Kostant, Matumoto and others.
In this paper, we show that, under certain assumptions, a parabolic induction $Ind_B^Gλ$ from the Borel subgroup $B$ of a (real or $p$-adic) reductive group $G$ decomposes into a direct sum of the form: \[ Ind_B^Gλ= \left(Ind_P^G St_M\otimes χ_0\right) \oplus \left(Ind_P^G \mathbf{1}_M\otimes χ_0\right), \] where $P$ is a parabolic subgroup of $G$ with Levi subgroup $M$ of semi-simple rank $1$, $\mathbf{1}_M$ is the trivial representation of $M$, $St_M$ is the Steinberg representation of $M$ and $χ_0$ is a certain character of $M$. We construct examples of this phenomenon for all simply-connec…
Let G_R be a real reductive Lie group acting on a manifold M. M.Kashiwara and W.Schmid in [KaSchm] constructed representations of G_R using sheaves and quasi-G_R-equivariant D-modules on M. In this article we prove an integral character formula for these representations (Theorem 1). Our main tools will be the integral localization formula recently proved in [L3] and the integral character formula proved by W.Schmid and K.Vilonen in [SchV2] (originally established by W. Rossmann in [Ro]) in the important special case when the manifold M is the flag variety of the complexified Lie algebra of G_R…
Modular reduction of complex representations of finite reductive groups
Massachusetts Institute of Technology（麻省理工学院）；Einstein Institute of Mathematics, The Hebrew University of Jerusalem（希伯来大学爱因斯坦数学研究所）；National Research University Higher School of Economics（国立研究型高等经济大学）；Stanford University（斯坦福大学）
