Character Sheaves and Generalizations

We present a Langlands dual realization of the putative category of affine character sheaves. Namely, we calculate the categorical center and trace (also known as the Drinfeld center and trace, or categorical Hochschild cohomology and homology) of the affine Hecke category starting from its spectral presentation. The resulting categories comprise coherent sheaves on the commuting stack of local systems on the two-torus satisfying prescribed support conditions, in particular singular support conditions, which appear in recent advances in the geometric Langlands program. The key technical tools in our arguments are a new descent theory for coherent sheaves or ${\mathcal{D}}$-modules with prescribed singular support and the theory of integral transforms for coherent sheaves developed in the companion paper by Ben-Zvi et al. [Integral transforms for coherent sheaves, J. Eur. Math. Soc. (JEMS), to appear].

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Shintani descent for algebraic groups and almost characters of unipotent groups

Compositio Mathematica ◽

10.1112/s0010437x16007429 ◽

2016 ◽

Vol 152 (8) ◽

pp. 1697-1724 ◽

Cited By ~ 1

Author(s):

Tanmay Deshpande

Keyword(s):

Finite Field ◽

Algebraic Group ◽

Algebraic Groups ◽

Unipotent Group ◽

Connected Component ◽

Unipotent Groups ◽

Character Sheaves ◽

Treat All ◽

Trace Of Frobenius

In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_{q}$. For this, it is essential to treat all the pure inner $\mathbb{F}_{q}$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the ‘trace of Frobenius’ functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko and Drinfeld’s theory of character sheaves on neutrally unipotent groups are in fact positive integral, confirming a conjecture due to Drinfeld.

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Le Groupe GLn Tordu, Sur un Corps Fini

Nagoya Mathematical Journal ◽

10.1017/s002776300002691x ◽

2006 ◽

Vol 182 ◽

pp. 313-379 ◽

Cited By ~ 2

Author(s):

J.-L. Waldspurger

Keyword(s):

Irreducible Representation ◽

Finite Field ◽

Complex Number ◽

Outer Automorphism ◽

The Other ◽

Characteristic Functions ◽

Connected Component ◽

Connected Group ◽

Unipotent Representations ◽

Character Sheaves

AbstractLet q be a finite field, G = GLn(q), θ be the outer automorphism of G, suitably normalized. Consider the non-connected group G ⋊ {1, θ} and its connected component = Gθ. We know two ways to produce functions on , with complex values and invariant by conjugation by G: on one hand, let π be an irreducible representation of G we can and do extend to a representation π+ of G ⋊ {1, θ}, then the restriction trace to of the character of π+ is such a function; on the other hand, Lusztig define character-sheaves a, whose characteristic functions ϕ(a) are such functions too. We consider only “quadratic-unipotent” representations. For all such representation π, we define a suitable extension π+, a character-sheave f(π) and we prove an identity trace = γ(π)ϕ(f(π)) with an explicit complex number γ(π).

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On the Deligne–Lusztig involution for character sheaves

Selecta Mathematica ◽

10.1007/s00029-019-0495-6 ◽

2019 ◽

Vol 25 (3) ◽

Author(s):

Alexander Yom Din

Keyword(s):

Character Sheaves

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Character sheaves and generalized Springer correspondence

Nagoya Mathematical Journal ◽

10.1017/s0027763000008539 ◽

2003 ◽

Vol 170 ◽

pp. 47-72 ◽

Cited By ~ 2

Author(s):

Anne-Marie Aubert

Keyword(s):

Finite Field ◽

Algebraic Group ◽

Algebraic Closure ◽

Reductive Algebraic Group ◽

Good For ◽

Character Sheaves ◽

Springer Correspondence

AbstractLetGbe a connected reductive algebraic group over an algebraic closure of a finite field of characteristicp. Under the assumption thatpis good forG, we prove that for each character sheafAonGwhich has nonzero restriction to the unipotent variety ofG, there exists a unipotent classCAcanonically attached toA, such thatAhas non-zero restriction onCA, and any unipotent classCinGon whichAhas non-zero restriction has dimension strictly smaller than that ofCA.

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