results for au:Achar_P in:math

- Jul 26 2017 math.RT arXiv:1707.07740v2Let $G$ be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic $p$, assumed to be larger than the Coxeter number. The "support variety" of a $G$-module $M$ is a certain closed subvariety of the nilpotent cone of $G$, defined in terms of cohomology for the first Frobenius kernel $G_1$. In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for $G = \mathrm{SL}_n$ in earlier work of the second author. In this paper, we show that for any $G$, the support variety of a tilting module always contains the variety predicted by Humphreys, and that they coincide (i.e., the Humphreys conjecture is true) when $p$ is sufficiently large. We also prove variants of these statements involving "relative support varieties."
- Jun 02 2017 math.RT arXiv:1706.00183v1We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic p in terms of p-Kazhdan-Lusztig polynomials, for p>h the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if p>2h-3. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun.
- Mar 20 2017 math.RT arXiv:1703.05843v1In this paper we propose a construction of a monoidal category of "free-monodromic" tilting perverse sheaves on (Kac-Moody) flag varieties in the setting of the "mixed modular derived category" introduced by the first and third authors. This category shares most of the properties of their counterpart in characteristic 0, defined by Bezrukavnikov-Yun using certain pro-objects in triangulated categories. This construction is the main new ingredient in the forthcoming construction of a "modular Koszul duality" equivalence for constructible sheaves on flag varieties.
- Feb 16 2016 math.RT arXiv:1602.04412v2In this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic $p$ bigger than the Coxeter number and a derived category of equivariant coherent sheaves on the Springer resolution (or a parabolic counterpart). In the case of the principal block, combined with previous results, this provides a modular version of celebrated constructions due to Arkhipov-Bezrukavnikov-Ginzburg for Lusztig's quantum groups at a root of unity. As an application, we prove a "graded version" of a conjecture of Finkelberg-Mirković describing the principal block in terms of mixed perverse sheaves on the dual affine Grassmannian.
- Feb 10 2016 math.RT arXiv:1602.02853v1In the paper [P. Achar, "On the equivariant $K$-theory of the nilpotent cone in the general linear group," Represent. Theory 8 (2004), 180-211], the author gave a combinatorial algorithm for computing the Lusztig-Vogan bijection for $GL(n,\mathbb{C})$. However, that paper failed to mention one easy case that may sometimes arise, making the description of the algorithm incomplete. This note fills in that gap.
- Nov 02 2015 math.RT arXiv:1510.08962v2This is an overview of our series of papers on the modular generalized Springer correspondence. It is an expansion of a lecture given by the second author in the Fifth Conference of the Tsinghua Sanya International Mathematics Forum, Sanya, December 2014, as part of the Master Lecture `Algebraic Groups and their Representations' Workshop honouring G. Lusztig. The material that has not appeared in print before includes some discussion of the motivating idea of modular character sheaves, and heuristic remarks about geometric functors of parabolic induction and restriction.
- Jul 24 2015 math.RT arXiv:1507.06581v1We study some aspects of modular generalized Springer theory for a complex reductive group $G$ with coefficients in a field $\mathbb k$ under the assumption that the characteristic $\ell$ of $\mathbb k$ is rather good for $G$, i.e., $\ell$ is good and does not divide the order of the component group of the centre of $G$. We prove a comparison theorem relating the characteristic-$\ell$ generalized Springer correspondence to the characteristic-$0$ version. We also consider Mautner's characteristic-$\ell$ `cleanness conjecture'; we prove it in some cases; and we deduce several consequences, including a classification of supercuspidal sheaves and an orthogonal decomposition of the equivariant derived category of the nilpotent cone.
- Jul 03 2015 math.RT arXiv:1507.00401v3We complete the construction of the modular generalized Springer correspondence for an arbitrary connected reductive group, with a uniform proof of the disjointness of induction series that avoids the case-by-case arguments for classical groups used in previous papers in the series. We show that the induction series containing the trivial local system on the regular nilpotent orbit is determined by the Sylow subgroups of the Weyl group. Under some assumptions, we give an algorithm for determining the induction series associated to the minimal cuspidal datum with a given central character. We also provide tables and other information on the modular generalized Springer correspondence for quasi-simple groups of exceptional type, including a complete classification of cuspidal pairs in the case of good characteristic, and a full determination of the correspondence in type $G_2$.
- Sep 26 2014 math.RT arXiv:1409.7346v1Exotic sheaves are certain complexes of coherent sheaves on the cotangent bundle of the flag variety of a reductive group. They are closely related to perverse-coherent sheaves on the nilpotent cone. This expository article includes the definitions of these two categories, applications, and some structure theory, as well as detailed calculations for SL(2).
- Sep 01 2014 math.RT arXiv:1408.7050v2An important result of Arkhipov-Bezrukavnikov-Ginzburg relates constructible sheaves on the affine Grassmannian to coherent sheaves on the dual Springer resolution. In this paper, we prove a positive-characteristic analogue of this statement, using the framework of "mixed modular sheaves" recently developed by the first author and Riche. As an application, we deduce a relationship between parity sheaves on the affine Grassmannian and Bezrukavnikov's "exotic t-structure" on the Springer resolution.
- Aug 20 2014 math.RT arXiv:1408.4189v1We further develop the general theory of the "mixed modular derived category" introduced by the authors in a previous paper in this series. We then use it to study positivity and Q-Koszulity phenomena on flag varieties.
- Apr 07 2014 math.RT arXiv:1404.1096v3We construct a modular generalized Springer correspondence for any classical group, by generalizing to the modular setting various results of Lusztig in the case of characteristic-$0$ coefficients. We determine the cuspidal pairs in all classical types, and compute the correspondence explicitly for $\mathrm{SL}(n)$ with coefficients of arbitrary characteristic and for $\mathrm{SO}(n)$ and $\mathrm{Sp}(2n)$ with characteristic-$2$ coefficients.
- Jan 29 2014 math.RT arXiv:1401.7256v2Building on the theory of parity sheaves due to Juteau-Mautner-Williamson, we develop a formalism of "mixed modular perverse sheaves" for varieties equipped with a stratification by affine spaces. We then give two applications: (1) a "Koszul-type" derived equivalence relating a given flag variety to the Langlands dual flag variety, and (2) a formality theorem for the modular derived category of a flag variety (extending a previous result of Riche-Soergel-Williamson).
- Jan 29 2014 math.RT arXiv:1401.7245v2In this paper we prove that the category of parity complexes on the flag variety of a complex connected reductive group is a "graded version" of the category of tilting perverse sheaves on the flag variety of the dual group, for any field of coefficients whose characteristic is good. We derive some consequences on Soergel's modular category O, and on multiplicities and decomposition numbers in the category of perverse sheaves.
- Jul 11 2013 math.RT arXiv:1307.2702v3We define a generalized Springer correspondence for the group GL(n) over any field. We also determine the cuspidal pairs, and compute the correspondence explicitly. Finally we define a stratification of the category of equivariant perverse sheaves on the nilpotent cone of GL(n) satisfying the `recollement' properties, and with subquotients equivalent to categories of representations of a product of symmetric groups.
- May 09 2013 math.RT arXiv:1305.1684v4We prove the Mirković-Vilonen conjecture: the integral local intersection cohomology groups of spherical Schubert varieties on the affine Grassmannian have no p-torsion, as long as p is outside a certain small and explicitly given set of prime numbers. (Juteau has exhibited counterexamples when p is a bad prime.) The main idea is to convert this topological question into an algebraic question about perverse-coherent sheaves on the dual nilpotent cone using the Juteau-Mautner-Williamson theory of parity sheaves.
- Apr 10 2013 math.RT arXiv:1304.2642v2We show that two Weyl group actions on the Springer sheaf with arbitrary coefficients, one defined by Fourier transform and one by restriction, agree up to a twist by the sign character. This generalizes a familiar result from the setting of l-adic cohomology, making it applicable to modular representation theory. We use the Weyl group actions to define a Springer correspondence in this generality, and identify the zero weight spaces of small representations in terms of this Springer correspondence.
- Sep 07 2012 math.RT arXiv:1209.1172v2Let W be a complex reflection group, acting on a complex vector space H. Kato has recently introduced the notion of a "Kostka system," which is a certain collection of finite-dimensional W-equivariant modules for the symmetric algebra on H. In this paper, we show that Kostka systems can be used to construct "exotic" t-structures on the derived category of finite-dimensional modules, and we prove a derived-equivalence result for these t-structures.
- Jul 31 2012 math.RT arXiv:1207.7044v1Given the nilpotent cone of a complex reductive Lie algebra, we consider its equivariant constructible derived category of sheaves with coefficients in an arbitrary field. This category and its subcategory of perverse sheaves play an important role in Springer theory and the theory of character sheaves. We show that the composition of the Fourier--Sato transform on the Lie algebra followed by restriction to the nilpotent cone restricts to an autoequivalence of the derived category of the nilpotent cone. In the case of $GL_n$, we show that this autoequivalence can be regarded as a geometric version of Ringel duality for the Schur algebra.
- May 24 2012 math.RT arXiv:1205.5089v4For a split reductive group scheme $G$ over a commutative ring $k$ with Weyl group $W$, there is an important functor $Rep(G,k) \to Rep(W,k)$ defined by taking the zero weight space. We prove that the restriction of this functor to the subcategory of small representations has an alternative geometric description, in terms of the affine Grassmannian and the nilpotent cone of the Langlands dual group to $G$. The translation from representation theory to geometry is via the Satake equivalence and the Springer correspondence. This generalizes the result for the $k=\mathbb{C}$ case proved by the first two authors, and also provides a better explanation than in that earlier paper, since the current proof is uniform across all types.
- In characteristic zero, Bezrukavnikov has shown that the category of perverse coherent sheaves on the nilpotent cone of a simply connected semisimple algebraic group is quasi-hereditary, and that it is derived-equivalent to the category of (ordinary) coherent sheaves. We prove that graded versions of these results also hold in good positive characteristic.
- Aug 26 2011 math.RT arXiv:1108.4999v2For a simply-connected simple algebraic group $G$ over $\C$, we exhibit a subvariety of its affine Grassmannian that is closely related to the nilpotent cone of $G$, generalizing a well-known fact about $GL_n$. Using this variety, we construct a sheaf-theoretic functor that, when combined with the geometric Satake equivalence and the Springer correspondence, leads to a geometric explanation for a number of known facts (mostly due to Broer and Reeder) about small representations of the dual group.
- For a certain class of abelian categories, we show how to make sense of the "Euler characteristic" of an infinite projective resolution (or, more generally, certain chain complexes that are only bounded above), by passing to a suitable completion of the Grothendieck group. We also show that right-exact functors (or their left-derived functors) induce continuous homomorphisms of these completed Grothendieck groups, and we discuss examples and applications coming from categorification.
- We prove that on a certain class of smooth complex varieties (those with "affine even stratifications"), the category of mixed Hodge modules is "almost" Koszul: it becomes Koszul after a few unwanted extensions are eliminated. We also give an equivalence between perverse sheaves on such a variety and modules for a certain graded ring, obtaining a formality result as a corollary. For flag varieties, these results were proved earlier by Beilinson-Ginzburg-Soergel using a rather different construction.
- A fundamental result of Beilinson-Ginzburg-Soergel states that on flag varieties and related spaces, a certain modified version of the category of l-adic perverse sheaves exhibits a phenomenon known as Koszul duality. The modification essentially consists of discarding objects whose stalks carry a nonsemisimple action of Frobenius. In this paper, we prove that a number of common sheaf functors (various pull-backs and push-forwards) induce corresponding functors on the modified category or its triangulated analogue. In particular, we show that these functors preserve semisimplicity of the Frobenius action.
- Feb 15 2011 math.RT arXiv:1102.2821v1In this paper we study the derived category of sheaves on the affine Grassmannian of a complex reductive group G, contructible with respect to the stratification by G(C[[x]])-orbits. Following ideas of Ginzburg and Arkhipov-Bezrukavnikov-Ginzburg, we describe this category (and a mixed version) in terms of coherent sheaves on the nilpotent cone of its Langlands dual reductive group. We also show, in the mixed case, that restriction to the nilpotent cone of a Levi subgroup corresponds to hyperbolic localization on affine Grassmannians.
- We generalize a result by Cunningham-Salmasian to a Mackey-type formula for the compact restriction of a semisimple perverse sheaf produced by parabolic induction from a character sheaf, under certain conditions on the parahoric group used to define compact restriction. This provides new tools for matching character sheaves with admissible representations.
- Aug 09 2010 math.RT arXiv:1008.1117v1In this note, we point out an error in the proof of Theorem 4.7 of [P. Achar and A.~Henderson, `Orbit closures in the enhanced nilpotent cone', Adv. Math. 219 (2008), 27-62], a statement about the existence of affine pavings for fibres of a certain resolution of singularities of an enhanced nilpotent orbit closure. We also give independent proofs of later results that depend on that statement, so all other results of that paper remain valid.
- Apr 27 2010 math.RT arXiv:1004.4412v1Let G be a reductive algebraic group, with nilpotent cone N and flag variety G/B. We construct an exact functor from perverse sheaves on N to locally constant sheaves on G/B, and we use it to study Ext-groups of simple perverse sheaves on N in terms of the cohomology of G/B. As an application, we give new proofs of some known results on stalks of perverse sheaves on N.
- We continue the study of the closures of $GL(V)$-orbits in the enhanced nilpotent cone $V\times\cN$ begun by the first two authors. We prove that each closure is an invariant-theoretic quotient of a suitably-defined enhanced quiver variety. We conjecture, and prove in special cases, that these enhanced quiver varieties are normal complete intersections, implying that the enhanced nilpotent orbit closures are also normal.
- We compare orbits in the nilpotent cone of type $B_n$, that of type $C_n$, and Kato's exotic nilpotent cone. We prove that the number of $\F_q$-points in each nilpotent orbit of type $B_n$ or $C_n$ equals that in a corresponding union of orbits, called a type-$B$ or type-$C$ piece, in the exotic nilpotent cone. This is a finer version of Lusztig's result that corresponding special pieces in types $B_n$ and $C_n$ have the same number of $\F_q$-points. The proof requires studying the case of characteristic 2, where more direct connections between the three nilpotent cones can be established. We also prove that the type-$B$ and type-$C$ pieces of the exotic nilpotent cone are smooth in any characteristic.
- This note is an expository account of the theory of staggered sheaves, based on a series of lectures given by the author at RIMS (Kyoto) in October 2008.
- Oct 29 2008 math.RT arXiv:0810.5030v1Nous obtenons une formule pour les valeurs de la fonction caractéristique d'un faisceau caractère en fonction de la théorie des représentations de certains groupes finis, liés au groupe de Weyl. Cette formule, qui généralise des résultats antérieurs de M\oeglin et de Waldspurger, dépend de la connaissance de certains sous-groupes réductifs admettant un faisceau caractère cuspidal. Dans un second temps, afin de rendre la formule plus explicite dans le cas d'un groupe quasi-simple, nous déterminons ces sous-groupes à conjugaison près. We obtain a formula for the values of the characteristic function of a character sheaf, in terms of the representation theory of certain finite groups related to the Weyl group. This formula, a generalization of previous results due to M\oeglin and Waldspurger, depends on knowledge of certain reductive subgroups that admit cuspidal character sheaves. For quasi-simple groups, we make the formula truly explicit by determining all such subgroups upto conjugation.
- Let G be an algebraic group over an algebraically closed field, acting on a variety X with finitely many orbits. "Staggered sheaves" are certain complexes of G-equivariant coherent sheaves on X that seem to possess many remarkable properties. In this paper, we construct "standard" and "costandard" objects in the category of staggered sheaves, and we prove that that category has enough projectives and injectives.
- Two major results in the theory of l-adic mixed constructible sheaves are the purity theorem (every simple perverse sheaf is pure) and the decomposition theorem (every pure object in the derived category is a direct sum of shifts of simple perverse sheaves). In this paper, we prove analogues of these results for coherent sheaves. Specificially, we work with staggered sheaves, which form the heart of a certain t-structure on the derived category of equivariant coherent sheaves. We prove, under some reasonable hypotheses, that every simple staggered sheaf is pure, and that every pure complex of coherent sheaves is a direct sum of shifts of simple staggered sheaves.
- We introduce the notion of a "baric structure" on a triangulated category, as an abstraction of S. Morel's weight truncation formalism for mixed l-adic sheaves. We study these structures on the derived category D_G(X) of G-equivariant coherent sheaves on a G-scheme X. Our main result shows how to endow this derived category with a family of nontrivial baric structures when G acts on X with finitely many orbits. We also describe a general construction for producing a new t-structure on a triangulated category equipped with given t- and baric structures, and we prove that the staggered t-structures on D_G(X) introduced by the first author arise in this way.
- Dec 12 2007 math.RT arXiv:0712.1615v1Staggered $t$-structures are a class of $t$-structures on derived categories of equivariant coherent sheaves. In this note, we show that the derived category of coherent sheaves on a partial flag variety, equivariant for a Borel subgroup, admits an artinian staggered $t$-structure. As a consequence, we obtain a basis for its equivariant $K$-theory consisting of simple staggered sheaves.
- We study the orbits of $G=\mathrm{GL}(V)$ in the enhanced nilpotent cone $V\times\mathcal{N}$, where $\mathcal{N}$ is the variety of nilpotent endomorphisms of $V$. These orbits are parametrized by bipartitions of $n=\dim V$, and we prove that the closure ordering corresponds to a natural partial order on bipartitions. Moreover, we prove that the local intersection cohomology of the orbit closures is given by certain bipartition analogues of Kostka polynomials, defined by Shoji. Finally, we make a connection with Kato's exotic nilpotent cone in type C, proving that the closure ordering is the same, and conjecturing that the intersection cohomology is the same but with degrees doubled.
- Let X be a scheme, and let G be an affine group scheme acting on X. Under reasonable hypotheses on X and G, we construct a t-structure on the derived category of G-equivariant coherent sheaves that in many ways resembles the perverse coherent t-structure, but which incorporates additional information from the G-action. Under certain circumstances, this t-structure, called the "staggered t-structure," has an artinian heart, and its simple objects are particularly easy to describe. We also exhibit two small examples in which the staggered t-structure is better-behaved than the perverse coherent t-structure.
- Recent work by a number of people has shown that complex reflection groups give rise to many representation-theoretic structures (e.g., generic degrees and families of characters), as though they were Weyl groups of algebraic groups. Conjecturally, these structures are actually describing the representation theory of as-yet undescribed objects called ''spetses'', of which reductive algebraic groups ought to be a special case. In this paper, we carry out the Lusztig--Shoji algorithm for calculating Green functions for the dihedral groups. With a suitable set-up, the output of this algorithm turns out to satisfy all the integrality and positivity conditions that hold in the Weyl group case, so we may think of it as describing the geometry of the ''unipotent variety'' associated to a spets. From this, we determine the possible ''Springer correspondences'', and we show that, as is true for algebraic groups, each special piece is rationally smooth, as is the full unipotent variety.
- Jul 06 2007 math.RT arXiv:0707.0836v2To a spetsial complex reflection group, equipped with a root lattice in the sense of Nebe, we attach a certain finite set playing a role which is analogous to the role of the set of unipotent classes of an algebraic group. In the case of imprimitive groups, we give a combinatoric parametrization of it in terms of Malle-Shoji generalized symbols. This result provides a link between the works of Shoji on Green functions for complex reflection groups and of Broue, Kim, Malle, Rouquier, et. al. on the cyclotomic Hecke algebras and their families of characters. ----- A un groupe de reflexions complexe spetsial, muni d'un reseau radiciel au sens de Nebe, nous associons un certain ensemble fini qui doit jouer un role analogue a celui de l'ensemble des classes unipotentes d'un groupe algebrique. Dans le cas des groupes imprimitifs, nous en donnons un parametrage combinatoire en termes des symboles generalises de Malle et Shoji. Ce resultat fournit un lien entre les travaux de Shoji sur les fonctions de Green pour les groupes de reflexions complexes et ceux de Broue, Kim, Malle, Rouquier, et al. sur les algebres de Hecke cyclotomiques et leurs familles de caracteres.
- Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U be an open subset whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent middle extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent middle extension functor. Under suitable hypotheses, we introduce a construction (called "S2-extension") in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical "S2-ification" of appropriate X. The construction also has applications to the "Macaulayfication" problem, and it is particularly well-behaved when X is Gorenstein. Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown in the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S2-extension to give a uniform construction of the desired variety.
- Jun 06 2006 math.RT arXiv:math/0606075v2Let 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.
- Apr 25 2003 math.RT arXiv:math/0304363v1Soit G un groupe algebrique reductif sur la cloture algebrique d'un corps fini F_q et defini sur ce dernier. L'existence du support unipotent d'un caractere irreductible du groupe fini G(F_q), ou d'un faisceau caractere de G, a ete etablie dans differents cas par Lusztig, Geck et Malle, et le second auteur. Dans cet article, nous demontrons que toute classe unipotente sur laquelle la restriction du faisceau caractere ou du caractere donne est non nulle est contenue dans l'adherence de Zariski de son support unipotent. Pour etablir ce resultat, nous etudions certaines representations des groupes de Weyl, dites "bien supportees". Let G be a reductive algebraic group over the algebraic closure of a finite field F_q that is defined over F_q. Under various conditions, Lusztig, Geck and Malle, and the second author have shown that an irreducible character of the finite group G(F_q), or a character sheaf on G, has a unipotent support. In this paper, we show that every unipotent class on which a given character or character sheaf has nonzero restriction is contained in the Zariski closure of its unipotent support. This result is obtained by studying the class of so-called "well-supported" representations of Weyl groups.
- Mar 11 2002 math.RT arXiv:math/0203082v3We define a partial order on the set of pairs (O,C), where O is a nilpotent orbit and C is a conjugacy class in Lusztig's canonical quotient of A(O). We then show that there is a unique order-reversing duality map on this set that has certain properties analagous to those of the original Lusztig-Spaltenstein duality map. This generalizes work of E. Sommers. In this revised version, a new approach leads to a strengthened version of Theorem 2, and applications of the results are discussed in greater detail.
- Jan 28 2002 math.RT arXiv:math/0201248v1We study the Lusztig-Vogan bijection for the case of a local system. We compute the bijection explicitly in type A for a local system and then show that the dominant weights obtained for different local systems on the same orbit are related in a manner made precise in the paper. We also give a conjecture (putatively valid for all groups) detailing how the weighted Dynkin diagram for a nilpotent orbit in the dual Lie algebra should arise under the bijection.