scholarly journals The topology of Baumslag–Solitar representations

2019 ◽  
pp. 1-13
Author(s):  
Maxime Bergeron ◽  
Lior Silberman
Keyword(s):  
Algebraic Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Reductive Algebraic Group ◽  
Strong Deformation ◽  
Strong Deformation Retraction ◽  
Deformation Retraction

Let [Formula: see text] be a Baumslag–Solitar group and let [Formula: see text] be a complex reductive algebraic group with maximal compact subgroup [Formula: see text]. We show that, when [Formula: see text] and [Formula: see text] are relatively prime with distinct absolute values, there is a strong deformation retraction of Hom([Formula: see text]) onto Hom([Formula: see text]).

scholarly journals Spherical functions on orthogonal groups

1996 ◽  
Vol 141 ◽  
pp. 157-182 ◽  
Author(s):  
Yasuhiro Kajima
Keyword(s):  
Algebraic Group ◽  
Root System ◽  
Explicit Formula ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Spherical Functions ◽  
Orthogonal Groups ◽  
Reductive Algebraic Group

Let G be a p-adic connected reductive algebraic group and K a maximal compact subgroup of G. In [4], Casselman obtained the explicit formula of zonal spherical functions on G with respect to K on the assumption that K is special. It is known (Bruhat and Tits [3]) that the affine root system of algebraic group which has good but not special maximal compact subgroup is A1 C2, or Bn (n > 3), and all Bn-types can be realized by orthogonal groups. Here the assumption “good” is necessary for the Satake’s theory of spherical functions.

scholarly journals Yang–Mills connections on compact complex tori

2015 ◽  
Vol 07 (02) ◽  
pp. 293-307
Author(s):  
Indranil Biswas
Keyword(s):  
Algebraic Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Structure Group ◽  
Kähler Structure ◽  
Yang Mills ◽  
Complex Torus ◽  
Complex Tori

Let G be a connected reductive complex affine algebraic group and K ⊂ G a maximal compact subgroup. Let M be a compact complex torus equipped with a flat Kähler structure and (EG, θ) a polystable Higgs G-bundle on M. Take any C∞ reduction of structure group EK ⊂ EG to the subgroup K that solves the Yang–Mills equation for (EG, θ). We prove that the principal G-bundle EG is polystable and the above reduction EK solves the Einstein–Hermitian equation for EG. We also prove that for a semistable (respectively, polystable) Higgs G-bundle (EG, θ) on a compact connected Calabi–Yau manifold, the underlying principal G-bundle EG is semistable (respectively, polystable).

Canonical Extensions of Harish-Chandra Modules to Representations of G

1989 ◽  
Vol 41 (3) ◽  
pp. 385-438 ◽  
Author(s):  
W. Casselman
Keyword(s):  
Representation Theory ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Finite Length ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Joint Work ◽  
Canonical Extensions ◽  
Category Of Smooth Representations ◽  
Current Representation

Let G be the group of R-rational points on a reductive, Zariskiconnected, algebraic group defined over R, let K be a maximal compact subgroup, and let g be the corresponding complexified Lie algebra of G. It is a curious fault of the current representation theory of G that for technical reasons one very rarely works with representations of G itself, but rather with a certain category of simultaneous representations of g and K. The reasons for this are, roughly speaking, that for a given (g,K)-module of finite length there are clearly any number of overlying rather distinct continuous G-representations, whose ‘essence’ is captured by the (g, K)-module alone. At any rate, this paper will propose a remedy for this inconvenience, and define a category of smooth representations of G of finite length which will, I hope, turn out to be as easy to work with as representations of (g, K) and occasionally much more convenient. It is to be considered a report on what has been to a great extent joint work with Nolan Wallach, and is essentially a sequel to [38].

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 On the Steinberg Character of a Finite Simple Group of Lie Type

1971 ◽  
Vol 12 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Bhama Srinivasan
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Simple Groups ◽  
Closed Field ◽  
Finite Simple Groups ◽  
Reductive Algebraic Group ◽  
Simple Algebraic Group ◽  
Simply Connected ◽  
Groups Of Lie Type ◽  
Steinberg Character

Let K be an algebraically closed field of characteristic ρ >0. If G is a connected, simple connected, semisimple linear algebraic group defined over K and σ an endomorphism of G onto G such that the subgroup Gσ of fixed points of σ is finite, Steinberg ([6] [7]) has shown that there is a complex irreducible character χ of Gσ with the following properties. χ vanishes at all elements of Gσ which are not semi- simple, and, if x ∈ G is semisimple, χ(x) = ±n(x) where n(x)is the order of a Sylow p-subgroup of (ZG(x))σ (ZG(x) is the centraliser of x in G). If G is simple he has, in [6], identified the possible groups Gσ they are the Chevalley groups and their twisted analogues over finite fields, that is, the ‘simply connected’ versions of finite simple groups of Lie type. In this paper we show, under certain restrictions on the type of the simple algebraic group G an on the characteristic of K, that χ can be expressed as a linear combination with integral coefficients of characters induced from linear characters of certain naturally defined subgroups of Gσ. This expression for χ gives an explanation for the occurence of n(x) in the formula for χ (x), and also gives an interpretation for the ± 1 occuring in the formula in terms of invariants of the reductive algebraic group ZG(x).

scholarly journals Distance to the convex hull of an orbit under the action of a compact Lie group

1999 ◽  
Vol 66 (3) ◽  
pp. 331-357 ◽  
Author(s):  
Randall R. Holmes ◽  
Tin-Yau Tam
Keyword(s):  
Convex Hull ◽  
Analogous Result ◽  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Adjoint Action ◽  
Reflection Group ◽  
Compact Lie Group ◽  
Real Vector ◽  
Finite Reflection Group

AbstractFor a real vector space V acted on by a group K and fixed x and y in V, we consider the problem of finding the minimum (respectively, maximum) distance, relative to a K-invariant convex function on V, between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V. Then we use this result to obtain an analogous result in the case where K is a maximal compact subgroup of a reductive group G with adjoint action on the vector component ρ of a Cartan decomposition of Lie G. Our results generalize results of Li and Tsing and of Cheng concerning distances to the convex hulls of matrix orbits.

𝒟-modules arithmétiques sur la variété de drapeaux

2019 ◽  
Vol 2019 (754) ◽  
pp. 1-15
Author(s):  
Christine Huyghe ◽  
Tobias Schmidt
Keyword(s):  
Algebraic Group ◽  
Valuation Ring ◽  
Differential Operators ◽  
Discrete Valuation Ring ◽  
Flag Variety ◽  
Nous Obtenons ◽  
Localization Theorem ◽  
Reductive Algebraic Group ◽  
Rigid Cohomology ◽  
Complete Discrete Valuation Ring

Abstract Soient p un nombre premier, V un anneau de valuation discrète complet d’inégales caractéristiques (0,p) , et G un groupe réductif et deployé sur \operatorname{Spec}V . Nous obtenons un théorème de localisation, en utilisant les distributions arithmétiques, pour le faisceau des opérateurs différentiels arithmétiques sur la variété de drapeaux formelle de G. Nous donnons une application à la cohomologie rigide pour des ouverts dans la variété de drapeaux en caractéristique p. Let p be a prime number, V a complete discrete valuation ring of unequal characteristics (0,p) , and G a connected split reductive algebraic group over \operatorname{Spec}V . We obtain a localization theorem, involving arithmetic distributions, for the sheaf of arithmetic differential operators on the formal flag variety of G. We give an application to the rigid cohomology of open subsets in the characteristic p flag variety.

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