Frobenius Conjugacy Classes

2012 ◽  
Author(s):  
Nicholas M. Katz
Keyword(s):  
Conjugacy Class ◽  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Conjugacy Classes ◽  
Jordan Decomposition ◽  
Semisimple Part ◽  
Good For

This chapter analyzes Frobenius conjugacy classes. It shows that in either the split or nonsplit case, when χ‎ is good for N, the conjugacy class FrobE,X has unitary eigenvalues in every representation of the reductive group Garith,N. Now fix a maximal compact subgroup K of the complex reductive group Garith,N (ℂ). The semisimple part (in the sense of Jordan decomposition) of FrobE,X gives rise to a well-defined conjugacy class θE,X in K.

scholarly journals Stable base change for spherical functions

1987 ◽  
Vol 106 ◽  
pp. 121-142 ◽  
Author(s):  
Yuval Z. Flicker
Keyword(s):  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Base Change ◽  
Conjugacy Classes ◽  
Natural Homomorphism ◽  
Convolution Algebra ◽  
Orbital Integral ◽  
Twisted Conjugacy ◽  
Complex Valued

Let E/F be an unramified cyclic extension of local non-archimedean fields, G a connected reductive group over F, K(F) (resp. K(E)) a hyper-special maximal compact subgroup of G(F) (resp. G(E)), and H(F) (resp. H(E)) the Hecke convolution algebra of compactly-supported complex-valued K(F) (resp. G(E))-biinvariant functions on G(F) (resp. G(E)). Then the theory of the Satake transform defines (see § 2) a natural homomorphism H(E) → H(F), θ→f. There is a norm map N from the set of stable twisted conjugacy classes in G(E) to the set of stable conjugacy classes in G(F); it is an injection (see [Ko]). Let Ω‱(x, f) denote the stable orbital integral of f in H(F) at the class x, and Ω‱(y, θ) the stable twisted orbital integral of θ in H(E) at the class y.

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.

scholarly journals Algorithms for representation theory of real reductive groups

2009 ◽  
Vol 8 (2) ◽  
pp. 209-259 ◽  
Author(s):  
Jeffrey Adams ◽  
Fokko du Cloux
Keyword(s):  
Representation Theory ◽  
Lie Groups ◽  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Flag Variety ◽  
Irreducible Representations ◽  
Structure Theory ◽  
Effective Algorithm ◽  
Reductive Groups

AbstractThe admissible representations of a real reductive groupGare known by work of Langlands, Knapp, Zuckerman and Vogan. This paper describes an effective algorithm for computing the irreducible representations ofGwith regular integral infinitesimal character. The algorithm also describes structure theory ofG, including the orbits ofK(ℂ) (a complexified maximal compact subgroup) on the flag variety. This algorithm has been implemented on a computer by the second author, as part of the ‘Atlas of Lie Groups and Representations’ project.

scholarly journals Stratifications with respect to actions of real reductive groups

2008 ◽  
Vol 144 (1) ◽  
pp. 163-185 ◽  
Author(s):  
Peter Heinzner ◽  
Gerald W. Schwarz ◽  
Henrik Stötzel
Keyword(s):  
Critical Points ◽  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Kähler Manifold ◽  
Reductive Groups ◽  
Cartan Decomposition ◽  
Kahler Manifold ◽  
Real Submanifold

AbstractWe study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of G extends holomorphically to an action of the complexified group $G^{\mathbb {C}}$ and that with respect to a compatible maximal compact subgroup U of $G^{\mathbb {C}}$ the action on Z is Hamiltonian. There is a corresponding gradient map $\mu _{\mathfrak {p}}\colon X\to \mathfrak {p}^*$ where $\mathfrak {g}=\mathfrak {k}\oplus \mathfrak {p}$ is a Cartan decomposition of $\mathfrak {g}$. We obtain a Morse-like function $\eta _{\mathfrak {p}}:=\Vert \mu _{\mathfrak {p}}\Vert ^2$ on X. Associated with critical points of $\eta _{\mathfrak {p}}$ are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds Sβ of X which are called pre-strata. In cases where $\mu _{\mathfrak {p}}$ is proper, the pre-strata form a decomposition of X and in cases where X is compact they are the strata of a Morse-type stratification of X. Our results are generalizations of results of Kirwan obtained in the case where $G=U^{\mathbb {C}}$ and X=Z is compact.

On the restriction of cuspidal representations to unipotent elements

2002 ◽  
Vol 132 (1) ◽  
pp. 35-56 ◽  
Author(s):  
DIPENDRA PRASAD ◽  
NILABH SANAT
Keyword(s):  
Conjugacy Class ◽  
Maximal Torus ◽  
Reductive Group ◽  
Conjugacy Classes ◽  
Irreducible Representations ◽  
Cuspidal Representation ◽  
Linear Character ◽  
Maximal Tori ◽  
Complex Linear ◽  
Cuspidal Representations

Let G be a connected split reductive group defined over a finite field [ ]q, and G([ ]q) the group of [ ]q-rational points of G. For each maximal torus T of G defined over [ ]q and a complex linear character θ of T([ ]q), let RGT(θ) be the generalized representation of G([ ]q) defined in [DL]. It can be seen that the conjugacy classes in the Weyl group W of G are in one-to-one correspondence with the conjugacy classes of maximal tori defined over [ ]q in G ([C1, 3·3·3]). Let c be the Coxeter conjugacy class of W, and let Tc be the corresponding maximal torus. Then by [DL] we know that πθ = (−1)nRGTc(θ) (where n is the semisimple rank of G and θ is a character in ‘general position’) is an irreducible cuspidal representation of G([ ]q). The results of this paper generalize the pattern about the dimensions of cuspidal representations of GL(n, [ ]q) as an alternating sum of the dimensions of certain irreducible representations of GL(n, [ ]q) appearing in the space of functions on the flag variety of GL(n, [ ]q) as shown in the table below.

scholarly journals A Satake isomorphism in characteristic p

2010 ◽  
Vol 147 (1) ◽  
pp. 263-283 ◽  
Author(s):  
Florian Herzig
Keyword(s):  
Galois Representations ◽  
Maximal Compact Subgroup ◽  
Algebraic Closure ◽  
Residue Field ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Reductive Groups ◽  
Characteristic P ◽  
Satake Isomorphism

AbstractSuppose that G is a connected reductive group over a p-adic field F, that K is a hyperspecial maximal compact subgroup of G(F), and that V is an irreducible representation of K over the algebraic closure of the residue field of F. We establish an analogue of the Satake isomorphism for the Hecke algebra of compactly supported,K-biequivariant functions f:G(F)→End   V. These Hecke algebras were first considered by Barthel and Livné for GL 2. They play a role in the recent mod p andp-adic Langlands correspondences for GL 2 (ℚp) , in generalisations of Serre’s conjecture on the modularity of mod p Galois representations, and in the classification of irreducible mod p representations of unramified p-adic reductive groups.

Relative Kloosterman Integrals for GL(3): II

1992 ◽  
Vol 44 (6) ◽  
pp. 1220-1240 ◽  
Author(s):  
Hervé Jacquet
Keyword(s):  
Characteristic Function ◽  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Quadratic Extension ◽  
Maximal Dimension ◽  
Unipotent Subgroup ◽  
Orbital Integral ◽  
Generic Character ◽  
Relative Version

AbstractLet G′ be a quasi–split reductive group over a local field F, ƒ′ the characteristic function of a maximal compact subgroup K′ of G′, N′ a maximal unipotent subgroup of G′. We consider the orbits of maximal dimension for the action of N′ × N′ on G′ and the weighted orbital integral of f′ on such an orbit, the weight being a generic character. The resulting integral, we call a Kloosterman integral. A relative version of this construction is to consider a symmetric space S associated to a quasi-split group G, a maximal unipotent subgroup N of G, a maximal compact K of G and the orbits of maximal dimension for the action of N on S. The weighted orbital integral of the characteristic function f of K ∩ S on such an orbit is what we call a relative Kloosterman integral; the weight is an appropriate character of N. We conjecture that a relative Kloosterman integral is actually a Kloosterman integral for an appropriate group G′. We prove the conjecture in a simple case: E is an unramified quadratic extension of F,G is GL(3, E), S is the set of 3 × 3 matrices s such that the group G′ is then the quasi-split unitary group in three variables.

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

2021 ◽  
pp. 1-5
Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Regular Graph ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

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).

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