Perverse sheaves on affine flags and langlands dual group

Arkhipov, Sergey; Bezrukavnikov, Roman

doi:10.1007/s11856-009-0024-y

Cited by 66 publications

(155 citation statements)

References 19 publications

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“…The latter is a consequence of [1], Theorem 2 and Lemma 1, which identify the left-hand side with an Ext space in the derived category of l-adic complexes on Fℓ (more precisely, with Ext • (∆ w1 , ∇ w2 ) in the notations of [1]). Now letf λ : Z λ → X be some map, such that the composition Z λ → X → j w * equalsf λ .…”

Section: Let Us Check That υ Is a Full Imbeddingmentioning

confidence: 96%

“…Thus (a) is proved. (b) Theorem 7 of [1] implies that Z λ (considered as an object of f P I ) has both a standard and a costandard filtration. The top of the latter is a surjection f λ : Z λ → j λ * whose kernel, subsequently, has a costandard filtration.…”

Section: Let Us Check That υ Is a Full Imbeddingmentioning

confidence: 99%

“…The latter inclusion amounts to the fact that Ext >0 f PI (Z(V ), X) = 0, X ∈ S, V ∈ Rep(Gˇ), which follows from the tilting property of central sheaves, Theorem 7 of [1].…”

Section: 13mentioning

confidence: 99%

“…Furthermore, it follows from Theorem 7 of [1] that Ext >0 f P f I (δ e , Z λ ) = 0, thus for a complex C • of object of f P f I of the form C i = Z(V i ), the object RH(C • ) is represented by the complex (H(C i )). Thus we get a canonical isomorphism…”

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confidence: 99%

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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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Section: Let Us Check That υ Is a Full Imbeddingmentioning

confidence: 96%

Section: Let Us Check That υ Is a Full Imbeddingmentioning

confidence: 99%

“…The latter inclusion amounts to the fact that Ext >0 f PI (Z(V ), X) = 0, X ∈ S, V ∈ Rep(Gˇ), which follows from the tilting property of central sheaves, Theorem 7 of [1].…”

Section: 13mentioning

confidence: 99%

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confidence: 99%

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

Bezrukavnikov

2009

Isr. J. Math.

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“…If z ∈ C × then the affine quantum group U √ v ( gl(n)) has an n-dimensional evaluation module V z indexed by z. 1 With z = diag(z 1 , · · · , z r ) ∈ T (C), then we may take M(z) to be V z 1 ⊗ · · · ⊗ V zr . The operators A s i : M(z) −→ M(s i z) then come from the R-matrices that are intertwining operators V z i ⊗V z i+1 −→ V z i+1 ⊗V z i , reflecting the fact that U √ v ( gl(n))-modules form (almost) a braided category.…”

Section: M(wz)mentioning

confidence: 99%

Hecke modules from metaplectic ice

Brubaker

Buciumas

Bump

et al. 2017

Sel. Math. New Ser.

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Abstract. We present a new framework for a broad class of affine Hecke algebra modules, and show that such modules arise in a number of settings involving representations of padic groups and R-matrices for quantum groups. Instances of such modules arise from (possibly non-unique) functionals on p-adic groups and their metaplectic covers, such as the Whittaker functionals. As a byproduct, we obtain new, algebraic proofs of a number of results concerning metaplectic Whittaker functions. These are thus expressed in terms of metaplectic versions of Demazure operators, which are built out of R-matrices of quantum groups depending on the cover degree and associated root system.An important tool in studying representations of p-adic groups is the affine Hecke algebra. Ion [28], Brubaker, Bump and Licata [8], Brubaker, Bump and Friedberg [7,9] and other papers considered representations of Hecke algebras related to models of representations of p-adic groups. These works mainly focused on unique functionals such as the Whittaker, spherical and Bessel functionals, but, as we shall show here, the ideas are also applicable to certain non-unique functionals on central extensions of p-adic groups. The goal is to show that these functionals act as Hecke algebra module maps to various target spaces. This results in recursion formulas for Iwahori fixed vectors in the model using Demazure-like operators based on the Hecke action. It allows for effective computation, provides links to the geometry of associated Bott-Samelson varieties, and proves that these functions match important classes of symmetric functions, such as various specializations of non-symmetric Macdonald polynomials.In a separate direction, Brubaker, Bump and Friedberg [6] showed that the values of the spherical Whittaker function for unramified principal series of GL r (F ), where F is a p-adic field, may be realized as partition functions of a solvable lattice model. A similar model for the spherical Whittaker functions for covers of GL r (F ) -a situation in which the Whittaker functional is no longer unique -was given in Brubaker, Bump, Chinta, Friedberg and Gunnells [5]. However, except when the cover is trivial, this model was not solvable in the sense of Baxter [3]. The Boltzmann weights were altered to obtain a solvable model in Brubaker, Buciumas and Bump [4]. This is the "metaplectic ice" of the title. Moreover, [4] provided a new algebraic framework, relating the models to the R-matrices of a Drinfeld twist of the quantum affine Lie superalgebra U q ( gl(n|1)). As a biproduct of this discovery, a relationship between the scattering matrices of the intertwining integrals on Whittaker coinvariants and R-matrices of a Drinfeld twist of U q ( gl(n)) came to light.In the present paper, we will unify these directions by providing simultaneous foundations for both. The search for new foundations is motivated by the following consideration. In [7], we took the point of view of the universal principal series. This means that the induced character becomes C[Λ]...

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