Unipotent elements and parabolic subgroups of reductive groups. II

1987 ◽  
pp. 265-284 ◽  
Author(s):  
Jacques Tits
Keyword(s):  
Reductive Groups ◽  
Parabolic Subgroups

scholarly journals Factorizable Module Algebras

2018 ◽  
Vol 2019 (21) ◽  
pp. 6711-6764
Author(s):  
Arkady Berenstein ◽  
Karl Schmidt
Keyword(s):  
Large Class ◽  
Reductive Groups ◽  
Parabolic Subgroups ◽  
Module Algebra ◽  
Lie Bialgebra ◽  
Affine Spaces ◽  
Enveloping Algebra ◽  
Quantized Enveloping Algebra ◽  
Module Algebras ◽  
Basic Affine

Abstract The aim of this paper is to introduce and study a large class of $\mathfrak{g}$-module algebras that we call factorizable by generalizing the Gauss factorization of square or rectangular matrices. This class includes coordinate algebras of corresponding reductive groups G, their parabolic subgroups, basic affine spaces, and many others. It turns out that products of factorizable algebras are also factorizable and it is easy to create a factorizable algebra out of virtually any $\mathfrak{g}$-module algebra. We also have quantum versions of all these constructions in the category of $U_{q}(\mathfrak{g})$-module algebras. Quite surprisingly, our quantum factorizable algebras are naturally acted on by the quantized enveloping algebra $U_{q}(\mathfrak{g}^{\ast })$ of the dual Lie bialgebra $\mathfrak{g}^{\ast }$ of $\mathfrak{g}$.

Eisenstein Series for Reductive Groups Over Global Function Fields II: The General Case

1982 ◽  
Vol 34 (5) ◽  
pp. 1112-1182 ◽  
Author(s):  
L. E. Morris
Keyword(s):  
Eisenstein Series ◽  
Invariant Subspaces ◽  
Rational Functions ◽  
Cusp Forms ◽  
Intertwining Operators ◽  
Reductive Groups ◽  
Parabolic Subgroups ◽  
Reductive Algebraic Group ◽  
Global Function ◽  
Global Function Fields

This paper is a continuation of [5]. As stated there, the problem is to explicitly decompose the space L2 = L2(G(F)\G(A)) into simpler invariant subspaces, and to deal with the associated continuous spectrum in case G is a connected reductive algebraic group defined over a global function field. In that paper the solution was begun by studying Eisenstein series associated to cusp forms on Levi components of parabolic subgroups; these Eisenstein series and the associated intertwining operators were shown to be rational functions satisfying functional equations. To go further it is necessary to consider more general Eisenstein series and intertwining operators, and to show that they have similar properties. Such Eisenstein series arise from the cuspidal ones by a residue taking process, which is detailed in a disguised form suitable for induction in the first part of this paper: the main result is a preliminary form of the spectral decomposition.

scholarly journals STEENROD OPERATIONS ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC STACKS

2021 ◽  
pp. 1-48
Author(s):  
Federico Scavia
Keyword(s):  
Finite Groups ◽  
Reductive Groups ◽  
Orthogonal Groups ◽  
De Rham Cohomology ◽  
Algebraic Stacks ◽  
Steenrod Operations ◽  
De Rham

Abstract Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$ . We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when $p=2$ .

scholarly journals MULTIPLE FLAG IND-VARIETIES WITH FINITELY MANY ORBITS

2021 ◽  
Author(s):  
LUCAS FRESSE ◽  
IVAN PENKOV
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Interesting Case ◽  
Analogous Problem ◽  
Restrictive Condition ◽  
Dimensional Case ◽  
Essential Difference ◽  
Parabolic Subgroups ◽  
Finite Dimensional ◽  
Finite Dimensional Case

AbstractLet G be one of the ind-groups GL(∞), O(∞), Sp(∞), and let P1, ..., Pℓ be an arbitrary set of ℓ splitting parabolic subgroups of G. We determine all such sets with the property that G acts with finitely many orbits on the ind-variety X1 × × Xℓ where Xi = G/Pi. In the case of a finite-dimensional classical linear algebraic group G, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar–Weyman–Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for ℓ = 2, the condition that G acts on X1 × X2 with finitely many orbits is a rather restrictive condition on the pair P1, P2. We describe this condition explicitly. Using the description we tackle the most interesting case where ℓ = 3, and present the answer in the form of a table. For ℓ ≥ 4 there always are infinitely many G-orbits on X1 × × Xℓ.

scholarly journals On parabolic subgroups of Artin–Tits groups of spherical type

2019 ◽  
Vol 352 ◽  
pp. 572-610 ◽  
Author(s):  
María Cumplido ◽  
Volker Gebhardt ◽  
Juan González-Meneses ◽  
Bert Wiest
Keyword(s):  
Parabolic Subgroups ◽  
Spherical Type

scholarly journals Modular Invariants of Parabolic Subgroups of General Linear Groups

2000 ◽  
Vol 232 (1) ◽  
pp. 197-208 ◽  
Author(s):  
Pham Anh Minh ◽  
Võ Thanh Tùng
Keyword(s):  
General Linear ◽  
Linear Groups ◽  
Parabolic Subgroups ◽  
General Linear Groups ◽  
Modular Invariants
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