scholarly journals FROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF -ADIC TORI

2015 ◽  
Vol 17 (1) ◽  
pp. 1-37
Author(s):  
Clifton Cunningham ◽  
David Roe
Keyword(s):  
Cohomology Group ◽  
Monoidal Category ◽  
Algebraic Groups ◽  
Group Scheme ◽  
Local Fields ◽  
Locally Finite ◽  
Isomorphism Classes ◽  
Algebraic Tori ◽  
Component Group ◽  
Character Sheaves

We consider the rigid monoidal category of character sheaves on a smooth commutative group scheme $G$ over a finite field $k$, and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We find the group of isomorphism classes of character sheaves on $G$, and show that it is an extension of the group of characters of $G(k)$ by a cohomology group determined by the component group scheme of $G$. We also classify all morphisms in the category character sheaves on $G$. As an application, we study character sheaves on Greenberg transforms of locally finite type Néron models of algebraic tori over local fields. This provides a geometrization of quasicharacters of $p$-adic tori.

scholarly journals CHABAUTY LIMITS OF ALGEBRAIC GROUPS ACTING ON TREES THE QUASI-SPLIT CASE

2018 ◽  
Vol 19 (4) ◽  
pp. 1031-1091
Author(s):  
Thierry Stulemeijer
Keyword(s):  
Algebraic Group ◽  
Closed Subset ◽  
Algebraic Groups ◽  
Local Fields ◽  
Locally Finite ◽  
Simple Algebraic Group ◽  
Groups Acting On Trees ◽  
Residue Characteristic

Given a locally finite leafless tree $T$, various algebraic groups over local fields might appear as closed subgroups of $\operatorname{Aut}(T)$. We show that the set of closed cocompact subgroups of $\operatorname{Aut}(T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $\operatorname{Aut}(T)$. This is done via a study of the integral Bruhat–Tits model of $\operatorname{SL}_{2}$ and $\operatorname{SU}_{3}^{L/K}$, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic $2$, the Tits index of simple algebraic subgroups of $\operatorname{Aut}(T)$ is not always preserved under Chabauty limits.

scholarly journals The cohomology of semi-infinite Deligne–Lusztig varieties

2020 ◽  
Vol 2020 (768) ◽  
pp. 93-147
Author(s):  
Charlotte Chan
Keyword(s):  
Large Class ◽  
Finite Type ◽  
Cohomology Group ◽  
Geometric Realization ◽  
Local Fields ◽  
Irreducible Representations ◽  
Division Algebras ◽  
Full Generality ◽  
Homology Groups ◽  
Supercuspidal Representations

AbstractWe prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes {X_{h}}. Boyarchenko’s two conjectures are on the maximality of {X_{h}} and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invariant {1/n} in the case {h=2} (the “lowest level”) by the work of Boyarchenko–Weinstein on the cohomology of a special affinoid in the Lubin–Tate tower. We prove that the number of rational points of {X_{h}} attains its Weil–Deligne bound, so that the cohomology of {X_{h}} is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group {H_{c}^{i}(X_{h})} are irreducible representations and are supported in exactly one cohomological degree. Finally, we give a complete description of the homology groups of the semi-infinite Deligne–Lusztig varieties attached to any division algebra, thus giving a geometric realization of a large class of supercuspidal representations of these groups. Moreover, the correspondence {\theta\mapsto H_{c}^{i}(X_{h})[\theta]} agrees with local Langlands and Jacquet–Langlands correspondences. The techniques developed in this paper should be useful in studying these constructions for p-adic groups in general.

Derivations and Automorphism Group of Completed Witt Lie Algebra

2012 ◽  
Vol 19 (03) ◽  
pp. 581-590 ◽  
Author(s):  
Yongping Wu ◽  
Ying Xu ◽  
Lamei Yuan
Keyword(s):  
Finite Element ◽  
Automorphism Group ◽  
Lie Algebra ◽  
Cohomology Group ◽  
Locally Finite ◽  
Simple Lie Algebra ◽  
Derivation Algebra ◽  
First Cohomology Group

In this paper, a simple Lie algebra, referred to as the completed Witt Lie algebra, is introduced. Its derivation algebra and automorphism group are completely described. As a by-product, it is obtained that the first cohomology group of this Lie algebra with coefficients in its adjoint module is trivial. Furthermore, we completely determine the conjugate classes of this Lie algebra under its automorphism group, and also obtain that this Lie algebra does not contain any nonzero ad -locally finite element.

scholarly journals Shintani descent for algebraic groups and almost characters of unipotent groups

2016 ◽  
Vol 152 (8) ◽  
pp. 1697-1724 ◽  
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.

scholarly journals Random groups and nonarchimedean lattices

2014 ◽  
Vol 2 ◽  
Author(s):  
SYLVAIN BARRÉ ◽  
MIKAËL PICHOT
Keyword(s):  
Algebraic Groups ◽  
Local Fields ◽  
Density Model ◽  
Rank 2 ◽  
Higher Rank ◽  
Random Groups

AbstractWe consider models of random groups in which the typical group is of intermediate rank (in particular, it is not hyperbolic). These models are parallel to Gromov’s well-known constructions, and include for example a ‘density model’ for groups of intermediate rank. The main novelty is the higher rank nature of the random groups. They are randomizations of certain families of lattices in algebraic groups (of rank 2) over local fields.

scholarly journals On the classification of Schreier extensions of monoids with non-abelian kernel

2020 ◽  
Vol 32 (3) ◽  
pp. 607-623
Author(s):  
Nelson Martins-Ferreira ◽  
Andrea Montoli ◽  
Alex Patchkoria ◽  
Manuela Sobral
Keyword(s):  
Abelian Group ◽  
Cohomology Group ◽  
Isomorphism Classes ◽  
The Third ◽  
Abelian Kernel ◽  
Invertible Elements ◽  
Abstract Kernel

AbstractWe show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel {\Phi\colon M\to\frac{\operatorname{End}(A)}{\operatorname{Inn}(A)}}. If an abstract kernel factors through {\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}}, where {\operatorname{SEnd}(A)} is the monoid of surjective endomorphisms of A, then we associate to it an obstruction, which is an element of the third cohomology group of M with coefficients in the abelian group {U(Z(A))} of invertible elements of the center {Z(A)} of A, on which M acts via Φ. An abstract kernel {\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. {\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}}) is induced by a regular weakly homogeneous (resp. homogeneous) Schreier extension of M by A if and only if its obstruction is zero. We also show that the set of isomorphism classes of regular weakly homogeneous (resp. homogeneous) Schreier extensions inducing a given abstract kernel {\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. {\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}}), when it is not empty, is in bijection with the second cohomology group of M with coefficients in {U(Z(A))}.

scholarly journals Locally finite groups with finitely many isomorphism classes of derived subgroups

2013 ◽  
Vol 393 ◽  
pp. 102-119 ◽  
Author(s):  
Patrizia Longobardi ◽  
Mercede Maj ◽  
Derek J.S. Robinson
Keyword(s):  
Finite Groups ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Isomorphism Classes

scholarly journals Invariant Hopf 2-Cocycles for Affine Algebraic Groups

2018 ◽  
Vol 2020 (2) ◽  
pp. 344-366
Author(s):  
Pavel Etingof ◽  
Shlomo Gelaki
Keyword(s):  
Finite Groups ◽  
Cohomology Group ◽  
Algebraic Groups ◽  
Closed Field ◽  
Algebraically Closed Field

Abstract We generalize the theory of the second invariant cohomology group $H^{2}_{\textrm{inv}}(G)$ for finite groups G, developed in [3, 4, 14], to the case of affine algebraic groups G, using the methods of [9, 10, 12]. In particular, we show that for connected affine algebraic groups G over an algebraically closed field of characteristic 0, the map Θ from [14] is bijective (unlike for some finite groups, as shown in [14]). This allows us to compute $H^{2}_{\textrm{inv}}(G)$ in this case, and in particular show that this group is commutative (while for finite groups it can be noncommutative, as shown in [14]).

Sign in / Sign up

Export Citation Format

Share Document