This paper concerns character sheaves of connected reductive algebraic groups defined over non-Archimedean local fields and their relation with characters of smooth representations. Although character sheaves were devised with characters of representations of finite groups of Lie type in mind, character sheaves are perfectly well defined for reductive algebraic groups over any algebraically closed field. Nevertheless, the relation between character sheaves of an algebraic group $G$ over an algebraic closure of a field $K$ and characters of representations of $G(K)$ is well understood only when $K$ is a finite field and when $K$ is the field of complex numbers. In this paper we consider the case when $K$ is a non-Archimedean local field and explain how to match certain character sheaves of a connected reductive algebraic group $G$ with virtual representations of $G(K)$. In the final section of the paper we produce examples of character sheaves of general linear groups and matching admissible virtual representations.
Refined forms of the local Langlands correspondence seek to relate representations of reductive groups over local fields with sheaves on stacks of Langlands parameters. But what kind of sheaves? Conjectures in the spirit of Kazhdan-Lusztig theory (due to Vogan and Soergel) describe representations of a group and its pure inner forms with fixed central character in terms of constructible sheaves. Conjectures in the spirit of geometric Langlands (due to Fargues, Zhu and Hellmann) describe representations with varying central character of a large family of groups associated to isocrystals in term…
Cuspidal character sheaves on graded Lie algebras II
Institute of Mathematics, Academia Sinica（中央研究院数学研究所）；School of Mathematics and Statistics, University of Melbourne（墨尔本大学数学与统计学院）；Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计系）
We show that some types for supercuspidal representations of tamely ramified $p$-adic groups that appear in Jiu-Kang Yu's work are geometrizable. To do so, we define a function-sheaf dictionary for one-dimensional characters of arbitrary smooth group schemes over finite fields. In previous work we considered the case of commutative smooth group schemes and found that the standard definition of character sheaves produced a dictionary with a nontrivial kernel. In this paper we give a modification of the category of character sheaves that remedies this defect, and is also extensible to non-commut…
We generalize a result by Cunningham-Salmasian to a Mackey-type formula for the compact restriction of a semisimple perverse sheaf produced by parabolic induction from a character sheaf, under certain conditions on the parahoric group used to define compact restriction. This provides new tools for matching character sheaves with admissible representations.
Institute of Mathematics, Academia Sinica（中央研究院数学研究所）；Department of Applied Mathematics, National Sun Yat-Sen University（国立中山大学应用数学系）；Department of Mathematics, National Taiwan University（国立台湾大学数学系）；School of Mathematics and Statistics, University of Melbourne（墨尔本大学数学与统计学院）；Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计系）
In this paper we propose a construction of generic character sheaves on reductive groups over finite local rings at even levels, whose characteristic functions are higher Deligne--Lusztig characters when the parameters are generic. We formulate a conjecture on the simple perversity of these complexes, and we prove it in the level two case (thus generalised a result of Lusztig from the function field case). We then discuss the induction and restriction functors, as well as the Frobenius reciprocity, based on the perversity.
We study the intermediate extension of the character sheaves on an adjoint group to the semi-stable locus of its wonderful compactification. We show that the intermediate extension can be described by a direct image construction. As a consequence, we show that the ``ordinary'' restriction of a character sheaf on the compactification to a boundary piece inside the semi-stable locus is a shift of semisimple perverse sheaf and is closely related to Lusztig's restriction functor (from a character sheaf on a reductive group to a direct sum of character sheaves on a Levi subgroup). We also provide a…
For a split reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group $H$, after passing to asymptotic versions. The other is a similar relationship between representations of…
Cuspidal character sheaves on graded exceptional Lie algebras: stable gradings
School of Mathematics and Statistics, University of Melbourne（墨尔本大学数学与统计学院）；Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计系）
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 as appear in recent advances in the Geometric Langlands program. The key technical tools in o…
We define character sheaves on an ind-variety of the form G((t))/U_P where G((t)) is a loop group and U_P is the prounipotent radical of a parahoric subgroup P of G((t)).
Let G be a reductive connected group over the algebraic closure of a finite field. In this paper we give the classification of character sheaves on G in categorical terms (as a categorical centre). Previously such a classification was known for unipotent character sheaves and also in the case where the ground field is replaced by the complex numbers.
We give a block decomposition of the dg category of character sheaves on a simple and simply-connected complex reductive group $G$, similar to the one in generalized Springer correspondence. As a corollary, we identify the category of character sheaves on $G$ as the category of quasi-coherent sheaves on an explicitly defined derived stack $\widehat{G}$.
In this paper we prove a local converse theorem for GL_n over the archimedean local fields, which characterizes an infinitesimal equivalence class of irreducible admissible representations of GL_n(R) (or GL_n(C)) in terms of twisted L-factors.
These are slides for a talk given by the authors at the conference "Current developments and directions in the Langlands program" held in honor of Robert Langlands at the Northwestern University in May of 2008. The slides can be used as a short introduction to the theory of characters and character sheaves for unipotent groups in positive characteristic, developed by the authors in a series of articles written between 2006 and 2011. We give an overview of the main results of this theory along with a bit of motivation.
We prove a local converse theorem for $GL_n$ over the archimedean local fields which characterizes an infinitesimal equivalence class of irreducible admissible representations of $GL_n(\mathbb{R})$ or $GL_n(\mathbb{C})$ in terms of twisted local gamma factors.
Let G_0 be a connected unipotent algebraic group over a finite field F_q, and let G be the unipotent group over an algebraic closure F of F_q obtained from G_0 by extension of scalars. If M is a Frobenius-invariant character sheaf on G, we show that M comes from an irreducible perverse sheaf M_0 on G_0, which is pure of weight 0. As M ranges over all Frobenius-invariant character sheaves on G, the functions defined by the corresponding perverse sheaves M_0 form a basis of the space of conjugation-invariant functions on the finite group G_0(F_q), which is orthonormal with respect to the standar…
We prove exactness of parabolic restriction and induction functors for conjugation equivariant sheaves on a reductive group generalizing a well known result of Lusztig who established this property for character sheaves. We propose a conjectural (but known for character sheaves) t-exactness property of the Harish-Chandra transform and provide an evidence for that conjecture. We also present two applications generalizing some results of Gabber and Loeser on perverse sheaves on an algebraic torus to an arbitrary reductive group.
Let A be a finite dimensional central division algebra over a local non-archimedean field F. Fix any parabolic subgroup P of GL(n,A) and a Levi factor M of P. Let πbe an irreducible unitary representation of M and ϕ(not necessarily unitary) character of M. We give an explicit necessary and sufficient condition for the parabolically induced representation Ind(ϕπ) of GL(n,A) to be irreducible.
