Mirabolic affine Grassmannian and character sheaves

Finkelberg, Michael; Ginzburg, Victor; Travkin, Roman

doi:10.1007/s00029-009-0509-x

Cited by 32 publications

(52 citation statements)

References 14 publications

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“…For the left pair above, the result was proved in the course of the proof of [FG1], Proposition 4.6.2 (although part (ii) of that proposition, claiming that the functor † H sends category O(A κ,ψ ) to C ψ,c , is incorrect, as stated).…”

Section: Lemma 621 (I)mentioning

confidence: 97%

“…In the paper [FG1] a partition of the variety X is given, based on a stratification of sl×V given in [GG,§4.2]. We recall this partition now.…”

Section: Lemma 222 (I)mentioning

confidence: 99%

“…The group GL acts on X and we let X//GL denote the categorical quotient. By [FG1,Lemma 3.2.1], the map induces an isomorphism X//GL ∼ → T /W . Thus, one has a diagram…”

Section: Specialization Of Mirabolic Modulesmentioning

confidence: 99%

“…INTRODUCTION 1.1. In this paper, we study mirabolic D-modules, following earlier works [GG], [FG1], and [FG2]. Mirabolic D-modules form an interesting category of regular holonomic D-modules on the variety SL n (C)×C n .…”

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confidence: 99%

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Hamiltonian reduction and nearby cycles for mirabolic D-modules

Bellamy

Ginzburg

2015

Advances in Mathematics

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ABSTRACT. We study holonomic D-modules on SLn(C) × C n , called mirabolic modules, analogous to Lusztig's character sheaves. We describe the supports of simple mirabolic modules. We show that a mirabolic module is killed by the functor of Hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric Cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus.We introduce an analogue of Verdier's specialization functor for representations of Cherednik algebras which agrees, on category O, with the restriction functor of Bezrukavnikov and Etingof. In type A, we also consider a Verdier specialization functor on mirabolic D-modules. We show that Hamiltonian reduction intertwines specialization functors on mirabolic Dmodules with the corresponding functors on representations of the Cherednik algebra. This allows us to apply known Hodge-theoretic purity results for nearby cycles in the setting considered by Bezrukavnikov and Etingof.

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Section: Lemma 621 (I)mentioning

confidence: 97%

“…In the paper [FG1] a partition of the variety X is given, based on a stratification of sl×V given in [GG,§4.2]. We recall this partition now.…”

Section: Lemma 222 (I)mentioning

confidence: 99%

“…The group GL acts on X and we let X//GL denote the categorical quotient. By [FG1,Lemma 3.2.1], the map induces an isomorphism X//GL ∼ → T /W . Thus, one has a diagram…”

Section: Specialization Of Mirabolic Modulesmentioning

confidence: 99%

mentioning

confidence: 99%

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Hamiltonian reduction and nearby cycles for mirabolic D-modules

Bellamy

Ginzburg

2015

Advances in Mathematics

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“…It is well-known that H is the specialization under q → q of a Z[q, q −1 ]-albebra H. The formulas (8) being polynomial in q, we may (and will) view R as the specialization under q → q of a Z[q, q −1 ]-bimodule R over the Z[q, q −1 ]-algebra H. We consider a new variable v, v 2 = q, and extend the scalars to Z[v, v −1 ] :…”

Section: Tate Sheavesmentioning

confidence: 99%

Mirabolic Robinson–Shensted–Knuth correspondence

Travkin

2009

Sel. math., New ser.

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Abstract. The set of orbits of GL(V ) in F l(V ) × F l(V ) × V is finite, and is parametrized by the set of certain decorated permutations in a work of Solomon. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of GL(V ) arising from F l(V ) × F l(V ) × V . We also give conjectural applications to the classification of unipotent mirabolic character sheaves on GL(V ) × V .

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Exotic symmetric space over a finite field, I

Shoji

Sorlin

2013

Transformation Groups

126

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Dedicated to the memory of T. A. SpringerAbstract. Let V be a 2n-dimensional vector space over an algebraically closed field k with ch k = 2. Let G = GL(V ) and H = Sp 2n be the symplectic group obtained as H = G θ for an involution θ on G. We also denote by θ the induced involution on g = Lie G. Consider the variety G/H ×V on which H acts naturally. Let g −θ nil be the set of nilpotent elements in the −1 eigenspace of θ in g. The role of the unipotent variety for G in our setup is played by g −θ nil × V , which coincides with Kato's exotic nilpotent cone. Kato established, in the case where k = C, the Springer correspondence between the set of irreducible representations of the Weyl group of type C n and the set of H-orbits in g −θ nil × V by applying Ginzburg theory for affine Hecke algebras. In this paper we develop a theory of character sheaves on G/H × V , and give an alternate proof for Kato's result on the Springer correspondence based on the theory of character sheaves.

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Mirabolic affine Grassmannian and character sheaves

Cited by 32 publications

References 14 publications

Hamiltonian reduction and nearby cycles for mirabolic D-modules

Hamiltonian reduction and nearby cycles for mirabolic D-modules

Mirabolic Robinson–Shensted–Knuth correspondence

Exotic symmetric space over a finite field, I

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