arXiv subjects

Pramod N. Achar

Publications and source records attributed to Pramod N. Achar.

At least 37 records · Page 2Linked to original sources

arXiv2019

Nearby cycles for parity sheaves on a divisor with simple normal crossings

The first author recently introduced a "nearby cycles formalism" in the framework of chain complexes of parity sheaves. In this paper, we compute this functor in two related settings: (i) affine space, stratified by the action of a torus, and (ii) the global Schubert variety associated to the first fundamental coweight of the group $PGL_n$. The latter is a parity-sheaf analogue of Gaitsgory's central sheaf construction.

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arXiv2019

Conjectures on tilting modules and antispherical $p$-cells

For quantum groups at a root of unity, there is a web of theorems (due to Bezrukavnikov and Ostrik, and relying on work of Lusztig) connecting the following topics: (i) tilting modules; (ii) vector bundles on nilpotent orbits; and (iii) Kazhdan-Lusztig cells in the affine Weyl group. In this paper, we propose a (partly conjectural) analogous picture for reductive algebraic groups over fields of positive characteristic, inspired by a conjecture of Humphreys.

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arXiv2019

Calculations with graded perverse-coherent sheaves

In this paper, we carry out several computations involving graded (or $\mathbb{G}_{\mathrm{m}}$-equivariant) perverse-coherent sheaves on the nilpotent cone of a reductive group in good characteristic. In the first part of the paper, we compute the weight of the $\mathbb{G}_{\mathrm{m}}$-action on certain normalized (or "canonical") simple objects, confirming an old prediction of Ostrik. In the second part of the paper, we explicitly describe all simple perverse coherent sheaves for $G = PGL_3$, in every characteristic other than 2 or 3. Applications include an explicit description of the cohomology of tilting modules for the corresponding quantum group, as well as a proof that $\mathsf{PCoh}^{\mathbb{G}_{\mathrm{m}}}(\mathcal{N})$ never admits a positive grading when the characteristic of the field is greater than 3.

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arXiv2018

Combinatorics of Fourier transforms for type A quiver representations

We describe two new combinatorial algorithms (using the language of "triangular arrays") for computing the Fourier transforms of simple perverse sheaves on the moduli space of representations of an equioriented quiver of type A. (A rather different solution to this problem was previously obtained by Knight-Zelevinsky.) Along the way, we also show that the closure partial order and the dimensions of orbits have especially concise descriptions in the language of triangular arrays.

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arXiv2018

Dualité de Koszul formelle et théorie des représentations des groupes algébriques réductifs en caractéristique positive

In this survey paper, we present the broad outlines of the proof of a character formula for tilting representations of reductive algebraic groups in positive characteristic, obtained partly in collaboration with several other authors. A unifying theme for a number of steps of this proof is the notion of "formal Koszul duality." We explain this notion and discuss some applications. ----- Dans cet article nous présentons les grandes lignes de la preuve d'une formule de caractères pour les représentations basculantes des groupes algébriques réductifs sur un corps de caractéristique positive, obtenue partiellement en collaboration avec plusieurs auteurs. Nous unissons les différentes étapes de cette preuve dans la notion de "dualité de Koszul formelle", et en présentons quelques applications.

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arXiv2018

Mixed perverse sheaves on flag varieties of Coxeter groups

In this paper we construct an abelian category of "mixed perverse sheaves" attached to any realization of a Coxeter group, in terms of the associated Elias-Williamson diagrammatic category. This construction extends previous work of the first two authors, where we worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias-Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization.

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arXiv2018

Reductive groups, the loop Grassmannian, and the Springer resolution

In 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.

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arXiv2018

On the Humphreys conjecture on support varieties of tilting modules

Let $G$ 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."

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arXiv2016

Modular generalized Springer correspondence: an overview

This 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.

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arXiv2016

Modular generalized Springer correspondence III: exceptional groups

We 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 $G_2$.

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arXiv2016

Corrigendum to "On the equivariant $K$-theory of the nilpotent cone in the general linear group," published in Represent. Theory 8 (2004)

In 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.

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arXiv2016

The affine Grassmannian and the Springer resolution in positive characteristic

An 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.

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arXiv2015

Parity sheaves on the affine Grassmannian and the Mirković-Vilonen conjecture

We 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.

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arXiv2015

Geometric Satake, Springer correspondence, and small representations II

For 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.

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arXiv2015

Modular perverse sheaves on flag varieties I: tilting and parity sheaves

In 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.

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arXiv2015

Modular generalized Springer correspondence II: classical groups

We 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.

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arXiv2014

Modular perverse sheaves on flag varieties II: Koszul duality and formality

Building 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).

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arXiv2014

Notes on exotic and perverse-coherent sheaves

Exotic 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).

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