scholarly journals Duality for representations of a reductive group over a finite field

1982 ◽  
Vol 74 (1) ◽  
pp. 284-291 ◽  
Author(s):  
Pierre Deligne ◽  
George Lusztig
Keyword(s):  
Finite Field ◽  
Reductive Group

scholarly journals ENDOSCOPY FOR HECKE CATEGORIES, CHARACTER SHEAVES AND REPRESENTATIONS

2020 ◽  
Vol 8 ◽  
Author(s):  
GEORGE LUSZTIG ◽  
ZHIWEI YUN
Keyword(s):  
Finite Field ◽  
Reductive Group ◽  
The Other ◽  
Torus Action ◽  
Similar Relationship ◽  
Unipotent Representations ◽  
Character Sheaves ◽  
Trivial Monodromy ◽  
Unipotent Character ◽  
Endoscopic Group

For a 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 $G(\mathbb{F}_{q})$ with a fixed semisimple parameter and unipotent representations of $H(\mathbb{F}_{q})$ .

scholarly journals Automorphic vector bundles on the stack of G-zips

2021 ◽  
Vol 9 ◽  
Author(s):  
Naoki Imai ◽  
Jean-Stefan Koskivirta
Keyword(s):  
Vector Bundle ◽  
Finite Field ◽  
Vector Bundles ◽  
Reductive Group ◽  
Levi Subgroup ◽  
Algebraic Subgroup ◽  
Equivalence Of Categories

Abstract For a connected reductive group G over a finite field, we study automorphic vector bundles on the stack of G-zips. In particular, we give a formula in the general case for the space of global sections of an automorphic vector bundle in terms of the Brylinski-Kostant filtration. Moreover, we give an equivalence of categories between the category of automorphic vector bundles on the stack of G-zips and a category of admissible modules with actions of a 0-dimensional algebraic subgroup a Levi subgroup and monodromy operators.

scholarly journals Central elements in affine mod p Hecke algebras via perverse -sheaves

2021 ◽  
Vol 157 (10) ◽  
pp. 2215-2241
Author(s):  
Robert Cass
Keyword(s):  
Finite Field ◽  
Reductive Group ◽  
Hecke Algebra ◽  
Convolution Product ◽  
Local Models ◽  
Frobenius Splitting ◽  
Rational Singularities ◽  
Nearby Cycles ◽  
Central Elements ◽  
Affine Flag Variety

Abstract Let $G$ be a split connected reductive group over a finite field of characteristic $p > 2$ such that $G_\text {der}$ is absolutely almost simple. We give a geometric construction of perverse $\mathbb {F}_p$ -sheaves on the Iwahori affine flag variety of $G$ which are central with respect to the convolution product. We deduce an explicit formula for an isomorphism from the spherical mod $p$ Hecke algebra to the center of the Iwahori mod $p$ Hecke algebra. We also give a formula for the central integral Bernstein elements in the Iwahori mod $p$ Hecke algebra. To accomplish these goals we construct a nearby cycles functor for perverse $\mathbb {F}_p$ -sheaves and we use Frobenius splitting techniques to prove some properties of this functor. We also prove that certain equal characteristic analogues of local models of Shimura varieties are strongly $F$ -regular, and hence they are $F$ -rational and have pseudo-rational singularities.

scholarly journals Duality for representations of a reductive group over a finite field, II

1983 ◽  
Vol 81 (2) ◽  
pp. 540-545 ◽  
Author(s):  
Pierre Deligne ◽  
George Lusztig
Keyword(s):  
Finite Field ◽  
Reductive Group

CHARACTER DEGREE SUMS AND REAL REPRESENTATIONS OF FINITE CLASSICAL GROUPS OF ODD CHARACTERISTIC

2010 ◽  
Vol 09 (04) ◽  
pp. 633-658 ◽  
Author(s):  
C. RYAN VINROOT
Keyword(s):  
Finite Field ◽  
Irreducible Character ◽  
Reductive Group ◽  
Character Degree ◽  
Orthogonal Groups ◽  
Real Representation ◽  
Irreducible Characters ◽  
Connected Group ◽  
Finite Classical Groups ◽  
Bounded Below

Let 𝔽q be a finite field with q elements, where q is the power of an odd prime, and let GSp (2n, 𝔽q) and GO ±(2n, 𝔽q) denote the symplectic and orthogonal groups of similitudes over 𝔽q, respectively. We prove that every real-valued irreducible character of GSp (2n, 𝔽q) or GO ±(2n, 𝔽q) is the character of a real representation, and we find the sum of the dimensions of the real representations of each of these groups. We also show that if G is a classical connected group defined over 𝔽q with connected center, with dimension d and rank r, then the sum of the degrees of the irreducible characters of G(𝔽q) is bounded above by (q + 1)(d+r)/2. Finally, we show that if G is any connected reductive group defined over 𝔽q, for any q, the sum of the degrees of the irreducible characters of G(𝔽q) is bounded below by q(d-r)/2(q - 1)r. We conjecture that this sum can always be bounded above by q(d-r)/2(q + 1)r.

scholarly journals THE INTERSECTION MOTIVE OF THE MODULI STACK OF SHTUKAS

2020 ◽  
Vol 8 ◽  
Author(s):  
TIMO RICHARZ ◽  
JAKOB SCHOLBACH
Keyword(s):  
Finite Field ◽  
Function Fields ◽  
Level Structure ◽  
Reductive Group ◽  
Intersection Cohomology ◽  
Triangulated Categories ◽  
Langlands Correspondence ◽  
Moduli Stack

For a split reductive group $G$ over a finite field, we show that the intersection (cohomology) motive of the moduli stack of iterated $G$ -shtukas with bounded modification and level structure is defined independently of the standard conjectures on motivic $t$ -structures on triangulated categories of motives. This is in accordance with general expectations on the independence of $\ell$ in the Langlands correspondence for function fields.

Moduli spaces of shtukas

2020 ◽  
pp. 215-224
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Finite Group ◽  
Finite Field ◽  
Moduli Spaces ◽  
Reductive Group ◽  
Connected Component ◽  
Langlands Correspondence ◽  
Mixed Characteristic ◽  
Projective Scheme ◽  
Primary Means ◽  
A Finite Group

This chapter examines the moduli spaces of mixed-characteristic local G-shtukas and shows that they are representable by locally spatial diamonds. These will be the mixed-characteristic local analogues of the moduli spaces of global equal-characteristic shtukas introduced by Varshavsky. It may be helpful to briefly review the construction in the latter setting. The ingredients are a smooth projective geometrically connected curve X defined over a finite field Fq and a reductive group G/Fq. Each connected component is a quotient of a quasi-projective scheme by a finite group. From there, it is possible to add level structures to the spaces of shtukas, to obtain a tower of moduli spaces admitting an action of the adelic group. The cohomology of these towers of moduli spaces is the primary means by which V. Lafforgue constructs the “automorphic to Galois” direction of the Langlands correspondence for G over F.

scholarly journals Cuspidal cohomology of stacks of shtukas

2020 ◽  
Vol 156 (6) ◽  
pp. 1079-1151
Author(s):  
Cong Xue
Keyword(s):  
Finite Field ◽  
Compact Support ◽  
Function Field ◽  
Automorphic Forms ◽  
Finite Dimension ◽  
Reductive Group ◽  
Constant Term ◽  
Cuspidal Cohomology

Let $G$ be a connected split reductive group over a finite field $\mathbb{F}_{q}$ and $X$ a smooth projective geometrically connected curve over $\mathbb{F}_{q}$. The $\ell$-adic cohomology of stacks of $G$-shtukas is a generalization of the space of automorphic forms with compact support over the function field of $X$. In this paper, we construct a constant term morphism on the cohomology of stacks of shtukas which is a generalization of the constant term morphism for automorphic forms. We also define the cuspidal cohomology which generalizes the space of cuspidal automorphic forms. Then we show that the cuspidal cohomology has finite dimension and that it is equal to the (rationally) Hecke-finite cohomology defined by V. Lafforgue.

Beta dynamical systems over the formal fields

2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing
Keyword(s):  
Dynamical Systems ◽  
Finite Field ◽  
Haar Measure ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Almost All

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.

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