scholarly journals Demazure Descent and Representations of Reductive Groups

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Sergey Arkhipov ◽  
Tina Kanstrup
Keyword(s):  
Algebraic Group ◽  
Borel Subgroup ◽  
Triangulated Category ◽  
Reductive Groups ◽  
Reductive Algebraic Group ◽  
Descent Data

We introduce the notion of Demazure descent data on a triangulated category C and define the descent category for such data. We illustrate the definition by our basic example. Let G be a reductive algebraic group with a Borel subgroup B. Demazure functors form Demazure descent data on DbRepB and the descent category is equivalent to DbRepG.

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} $.

scholarly journals Commuting Varieties for Nilpotent Radicals

2018 ◽  
Vol 62 (2) ◽  
pp. 559-594
Author(s):  
Rolf Farnsteiner
Keyword(s):  
Algebraic Group ◽  
Borel Subgroup ◽  
Unipotent Radical ◽  
Closed Field ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Commuting Variety ◽  
Good For

AbstractLetUbe the unipotent radical of a Borel subgroup of a connected reductive algebraic groupG, which is defined over an algebraically closed fieldk. In this paper, we extend work by Goodwin and Röhrle concerning the commuting variety of Lie(U) for Char(k) = 0 to fields whose characteristic is good forG.

scholarly journals Gradedness of the set of rook placements in A n−1

2021 ◽  
Vol 29 (2) ◽  
pp. 171-182
Author(s):  
Mikhail V. Ignatev
Keyword(s):  
Algebraic Group ◽  
Root System ◽  
Borel Subgroup ◽  
Coadjoint Orbit ◽  
Combinatorial Structure ◽  
Natural Partial Order ◽  
Reductive Algebraic Group ◽  
Rook Placements ◽  
Positive Roots ◽  
Different Order

Abstract A rook placement is a subset of a root system consisting of positive roots with pairwise non-positive inner products. To each rook placement in a root system one can assign the coadjoint orbit of the Borel subgroup of a reductive algebraic group with this root system. Degenerations of such orbits induce a natural partial order on the set of rook placements. We study combinatorial structure of the set of rook placements in An− 1 with respect to a slightly different order and prove that this poset is graded.

scholarly journals Coordinate rings and birational charts

2022 ◽  
Vol 26 (1) ◽  
pp. 1-16
Author(s):  
Sergey Fomin ◽  
George Lusztig
Keyword(s):  
Algebraic Group ◽  
Borel Subgroup ◽  
Total Positivity ◽  
Unipotent Radical ◽  
Reductive Groups ◽  
Content Type ◽  
Coordinate Rings ◽  
Simply Connected

Let G G be a semisimple simply connected complex algebraic group. Let U U be the unipotent radical of a Borel subgroup in  G G . We describe the coordinate rings of U U (resp., G / U G/U , G G ) in terms of two (resp., four, eight) birational charts introduced by Lusztig [Total positivity in reductive groups, Birkhäuser Boston, Boston, MA, 1994; Bull. Inst. Math. Sin. (N.S.) 14 (2019), pp. 403–459] in connection with the study of total positivity.

scholarly journals On Commuting Varieties of Nilradicals of Borel Subalgebras of Reductive Lie Algebras

2014 ◽  
Vol 58 (1) ◽  
pp. 169-181 ◽  
Author(s):  
Simon M. Goodwin ◽  
Gerhard Röhrle
Keyword(s):  
Finite Number ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Borel Subgroup ◽  
Closed Field ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Commuting Variety ◽  
Irreducible Components

AbstractLet G be a connected reductive algebraic group defined over an algebraically closed field of characteristic 0. We consider the commuting variety of the nilradical of the Lie algebra of a Borel subgroup B of G. In case B acts on with only a finite number of orbits, we verify that is equidimensional and that the irreducible components are in correspondence with the distinguishedB-orbits in . We observe that in general is not equidimensional, and determine the irreducible components of in the minimal cases where there are infinitely many B-orbits in .

scholarly journals ON THE SUPPORT VARIETIES FOR DEMAZURE MODULES

2011 ◽  
Vol 91 (3) ◽  
pp. 343-363
Author(s):  
Benjamin F. Jones ◽  
Daniel K. Nakano
Keyword(s):  
Borel Subgroup ◽  
Root Systems ◽  
Natural Generalization ◽  
Bruhat Order ◽  
Reductive Groups ◽  
Reductive Algebraic Group ◽  
Frobenius Kernel ◽  
Demazure Modules ◽  
Support Varieties ◽  
Entire Picture

AbstractThe support varieties for the induced modules or Weyl modules for a reductive algebraic group G were computed over the first Frobenius kernel G1 by Nakano, Parshall and Vella. A natural generalization of this computation is the calculation of the support varieties of Demazure modules over the first Frobenius kernel, B1, of the Borel subgroup B. In this paper we initiate the study of such computations. We complete the entire picture for reductive groups with underlying root systems A1 and A2. Moreover, we give complete answers for Demazure modules corresponding to a particular (standard) element in the Weyl group, and provide results relating support varieties between different Demazure modules which depend on the Bruhat order.

scholarly journals Complete reducibility: variations on a theme of Serre

2021 ◽  
Author(s):  
Maike Gruchot ◽  
Alastair Litterick ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Spherical Building ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
General Properties ◽  
Complete Reducibility ◽  
Special Cases ◽  
Variations On A Theme

AbstractIn this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. We show that other variations of this notion, such as relative complete reducibility and $$\sigma $$ σ -complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.

scholarly journals RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
MAIKE GRUCHOT ◽  
ALASTAIR LITTERICK ◽  
GERHARD RÖHRLE
Keyword(s):  
Automorphism Group ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
Complete Reducibility ◽  
Closed Fields ◽  
Completely Reducible ◽  
Reductive Subgroup ◽  
The Absolute

We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ . We show that such a subgroup is relatively $G$ -completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$ , as well as ‘rational’ versions over nonalgebraically closed fields.

scholarly journals The Modality of a Borel Subgroup in a Simple Algebraic Group of Type E8

2018 ◽  
Vol 29 (3) ◽  
pp. 326-327
Author(s):  
Simon M. Goodwin ◽  
Peter Mosch ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Borel Subgroup ◽  
Simple Algebraic Group
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