scholarly journals Corrigendum: On the cuspidal cohomology of S-arithmetic subgroups of reductive groups over number fields

2021 ◽  
Vol 157 (6) ◽  
pp. 1207-1210
Author(s):  
Jean-Pierre Labesse ◽  
Joachim Schwermer
Keyword(s):  
Algebraic Group ◽  
Number Fields ◽  
Reductive Groups ◽  
Cartan Type ◽  
Semisimple Algebraic Group ◽  
Cuspidal Cohomology

The aim of this corrigendum is to correct an error in Corollary 10.7 to Theorem 10.6, one of the main results in the paper ‘On the cuspidal cohomology of $S$ -arithmetic subgroups of reductive groups over number fields’. This makes necessary a thorough investigation of the conditions under which a Cartan-type automorphism exists on $G_1=\mathrm {Res}_{\mathbb {C}/\mathbb {R}}G_0$ , where $G_0$ is a connected semisimple algebraic group defined over $\mathbb {R}$ .

scholarly journals Residues of Eisenstein series and the automorphic cohomology of reductive groups

2013 ◽  
Vol 149 (7) ◽  
pp. 1061-1090 ◽  
Author(s):  
Harald Grobner
Keyword(s):  
Algebraic Group ◽  
Eisenstein Series ◽  
Number Field ◽  
Automorphic Forms ◽  
Reductive Groups ◽  
Algebraic Representation ◽  
Reductive Algebraic Group ◽  
Automorphic Representations ◽  
Residual Spectrum ◽  
Eisenstein Cohomology

AbstractLet $G$ be a connected, reductive algebraic group over a number field $F$ and let $E$ be an algebraic representation of ${G}_{\infty } $. In this paper we describe the Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ of $G$ below a certain degree ${q}_{ \mathsf{res} } $ in terms of Franke’s filtration of the space of automorphic forms. This entails a description of the map ${H}^{q} ({\mathfrak{m}}_{G} , K, \Pi \otimes E)\rightarrow { H}_{\mathrm{Eis} }^{q} (G, E)$, $q\lt {q}_{ \mathsf{res} } $, for all automorphic representations $\Pi $ of $G( \mathbb{A} )$ appearing in the residual spectrum. Moreover, we show that below an easily computable degree ${q}_{ \mathsf{max} } $, the space of Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ is isomorphic to the cohomology of the space of square-integrable, residual automorphic forms. We discuss some more consequences of our result and apply it, in order to derive a result on the residual Eisenstein cohomology of inner forms of ${\mathrm{GL} }_{n} $ and the split classical groups of type ${B}_{n} $, ${C}_{n} $, ${D}_{n} $.

An Explicit Cell Decomposition of the Wonderful Compactification of a Semisimple Algebraic Group

2003 ◽  
Vol 46 (1) ◽  
pp. 140-148 ◽  
Author(s):  
Lex E. Renner
Keyword(s):  
Algebraic Group ◽  
Cell Decomposition ◽  
Bruhat Decomposition ◽  
Semisimple Algebraic Group ◽  
Reductive Monoid

AbstractWe determine an explicit cell decomposition of the wonderful compactification of a semisimple algebraic group. To do this we first identify the B × B-orbits using the generalized Bruhat decomposition of a reductive monoid. From there we show how each cell is made up from B × B orbits.

scholarly journals On Cocharacters Associated to Nilpotent Elements of Reductive Groups

2008 ◽  
Vol 190 ◽  
pp. 105-128 ◽  
Author(s):  
Russell Fowler ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Nilpotent Element ◽  
Maximal Rank ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Reductive Groups ◽  
Algebraically Closed Field ◽  
Reductive Subgroup ◽  
Good For

Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we consider particular classes of connected reductive subgroups H of G and show that the cocharacters of H that are associated to a given nilpotent element e in the Lie algebra of H are precisely the cocharacters of G associated to e that take values in H. In particular, we show that this is the case provided H is a connected reductive subgroup of G of maximal rank; this answers a question posed by J. C. Jantzen.

scholarly journals Hopf-theoretic approach to motives of twisted flag varieties

2021 ◽  
Vol 157 (5) ◽  
pp. 963-996
Author(s):  
Victor Petrov ◽  
Nikita Semenov
Keyword(s):  
Hopf Algebra ◽  
Algebraic Group ◽  
Cohomology Theory ◽  
Algebra Structure ◽  
Flag Varieties ◽  
Theoretic Approach ◽  
Hopf Algebra Structure ◽  
Uniform Approach ◽  
Semisimple Algebraic Group ◽  
Oriented Cohomology

Let $G$ be a split semisimple algebraic group over a field and let $A^*$ be an oriented cohomology theory in the Levine–Morel sense. We provide a uniform approach to the $A^*$ -motives of geometrically cellular smooth projective $G$ -varieties based on the Hopf algebra structure of $A^*(G)$ . Using this approach, we provide various applications to the structure of motives of twisted flag varieties.

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