Almost-commuting variety, D-modules, and Cherednik algebras

Gan, Wee Liang; Ginzburg, Victor

doi:10.1155/imrp/2006/26439

Cited by 47 publications

(101 citation statements)

References 17 publications

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“…Further, let O κ be the category of finitely generated left U κ -modules which are locally finite as (Sym t) W -modules, see §6. Then, it was shown in [GG,sectcion 6] that the functor M → H c (M ) restricts to an exact functor H c : C q → O κ and, moreover, this yields an equivalence C q / Ker(H c ) ∼ → O κ . In this paper, we give a description of the kernel of the functor H c .…”

Section: The Functor Of Hamiltonian Reductionmentioning

confidence: 99%

“…To see this, let M(sl) be the zero fiber of the moment map T * X = sl × sl × V × V * → gl. According to [GG,Proposition 8.2.1] one has an algebra isomorphism C[M(sl)] GL ∼ = C[t×t * ] W . The variety M nil (sl) is a GL-stable closed subvariety of M(sl).…”

Section: Lemma 621 (I)mentioning

confidence: 99%

“…The variety M nil (sl) is a GL-stable closed subvariety of M(sl). Furthermore, the proof in [GG,Proposition 8.2.1] shows that the restriction map C[M(sl)] → C[M nil (sl)] induces, via the isomorphism above, an algebra isomorphism…”

Section: Lemma 621 (I)mentioning

confidence: 99%

“…The fact that M ∈ C N L ,c is C × -monodromic with respect to the C × -action on l ⊂ N L means that one can use the Fourier transform, as in [GG,Proposition 5.3.2], to show that the…”

Section: Lemma 933 the Following Diagram Commutesmentioning

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: The Functor Of Hamiltonian Reductionmentioning

confidence: 99%

Section: Lemma 621 (I)mentioning

confidence: 99%

Section: Lemma 621 (I)mentioning

confidence: 99%

“…The fact that M ∈ C N L ,c is C × -monodromic with respect to the C × -action on l ⊂ N L means that one can use the Fourier transform, as in [GG,Proposition 5.3.2], to show that the…”

Section: Lemma 933 the Following Diagram Commutesmentioning

confidence: 99%

See 2 more Smart Citations

Hamiltonian reduction and nearby cycles for mirabolic D-modules

Bellamy

Ginzburg

2015

Advances in Mathematics

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“…Since radial reduction maps the standard Laplacian operator ∆ to the Calogero-Moser operator on h [5], we can think of X and f as describing a broader, yet simpler, precursor situation to the one studied in [11]. More generally, the function f and the space X are relevant to the study of mirabolic D-modules and rational Cherednik algebras [2] - [4].…”

Section: Introductionmentioning

confidence: 99%