Construction of central elements in the affine Hecke algebra via nearby cycles

Gaitsgory, Dennis

doi:10.1007/s002220100122

Cited by 134 publications

(227 citation statements)

References 7 publications

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“…Here the first equality follows from (8). The second one is a consequence of the existence of a standard filtration on Z λ (considered as an object of f P I ), and the equality…”

Section: Let Us Check That υ Is a Full Imbeddingmentioning

confidence: 96%

Perverse sheaves on affine flags and nilpotent cone of the langlands dual group

Bezrukavnikov

2009

Isr. J. Math.

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Contents 1. Statements 1 1.1. Recollection of notations and set-up 1 1.2. Category f P f I and the nil-cone 2 1.3. Description of the t-structure on D Gˇ( N) 3 1.4. Dualities 4 1.5. Cells and nilpotent orbits 4 1.6. Duflo involutions 4 1.7. Truncated convolution categories 5 2. Proofs 6 2.1. Proof of Theorem 1 6 2.2. Proof of Theorem 2 8 2.3. Proof of Theorem 3 9 2.4. Proof of Theorem 4 2.5. Proof of Proposition 1 2.6. Proof of Proposition 2 ReferencesAbstract. This paper is a continuation of [1]. In [1] we constructed an equivalence between the derived category of equivariant coherent sheaves on the cotangent bundle to the flag variety of a simple algebraic group and a (quotient of) a category of constructible sheaves on the affine flag variety of the Langlands dual group. Below we prove certain properties of this equivalence related to cells in the affine Weyl group; provide a similar "Langlands dual" description for the category of equivariant coherent sheaves on the nilpotent cone, and link it to perverse coherent sheaves; and deduce some conjectures by Lusztig and Ostrik.

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“…Here the first equality follows from (8). The second one is a consequence of the existence of a standard filtration on Z λ (considered as an object of f P I ), and the equality…”

Section: Let Us Check That υ Is a Full Imbeddingmentioning

confidence: 96%

Perverse sheaves on affine flags and nilpotent cone of the langlands dual group

Bezrukavnikov

2009

Isr. J. Math.

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“…The result is fundamental in the geometric Langlands program [BD,FGV01,BG02]. It also leads to a construction of canonical bases [MV00, BG01,MV04], and a deeper understanding of the Satake isomorphism [Gai01], among other applications [BFGM02]. It also may be interpreted as providing a down-to-earth perspective on the duality itself.…”

Section: Introductionmentioning

confidence: 99%

Perverse sheaves on real loop Grassmannians

Nadler

2004

Invent. math.

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Abstract. The aim of this paper is to identify a certain tensor category of perverse sheaves on the loop Grassmannian Gr R of a real form G R of a connected reductive complex algebraic group G with the category of finite-dimensional representations of a connected reductive complex algebraic subgroupȞ of the dual groupǦ. The root system ofȞ is closely related to the restricted root system of G R . The fact thať H is reductive implies that an interesting family of real algebraic maps satisfies the conclusion of the Decomposition Theorem of Beilinson-Bernstein-Deligne.

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“…Proposition 4.5.12 The map p : Remark 4.5.13 While the "associativity constraints" of the convolution product are almost immediate from its definition, the commutativity constraints are far subtler (see [145], and also [80]). …”

Section: Theorem 453 (The Classical Satake Isomorphism) There Is Anmentioning

confidence: 99%

The decomposition theorem, perverse sheaves and the topology of algebraic maps

Andrea

Migliorini

2009

Bull. Amer. Math. Soc.

162

152

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We give a motivated introduction to the theory of perverse sheaves, culminating in the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the decomposition theorem, indicate some important applications and examples.

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Construction of central elements in the affine Hecke algebra via nearby cycles

Cited by 134 publications

References 7 publications

Perverse sheaves on affine flags and nilpotent cone of the langlands dual group

Perverse sheaves on affine flags and nilpotent cone of the langlands dual group

Perverse sheaves on real loop Grassmannians

The decomposition theorem, perverse sheaves and the topology of algebraic maps

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