scholarly journals Associated symmetric pair and multiplicities of admissible restriction of discrete series

2016 ◽  
Vol 27 (12) ◽  
pp. 1650100
Author(s):  
Jorge A. Vargas
Keyword(s):  
Spectral Analysis ◽  
Maximal Compact Subgroup ◽  
Discrete Series ◽  
Compact Subgroup ◽  
Representation Formula ◽  
Symmetric Pair ◽  
Semisimple Lie Group ◽  
Integrable Representation ◽  
Square Integrable Representation ◽  
Associated Pair

Let [Formula: see text] be a symmetric pair for a real semisimple Lie group [Formula: see text] and [Formula: see text] its associated pair. For each irreducible square integrable representation [Formula: see text] of [Formula: see text] so that its restriction to [Formula: see text] is admissible, we find an irreducible square integrable representation [Formula: see text] of [Formula: see text] which allows us to compute the Harish-Chandra parameter of each irreducible [Formula: see text]-subrepresentation of [Formula: see text] as well as its multiplicity. The computation is based on the spectral analysis of the restriction of [Formula: see text] to a maximal compact subgroup of [Formula: see text]

Minimal actions of semisimple groups

1996 ◽  
Vol 16 (4) ◽  
pp. 821-831 ◽  
Author(s):  
Garrett Stuck
Keyword(s):  
Parabolic Subgroup ◽  
Lie Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Semisimple Lie Group ◽  
Semisimple Groups ◽  
Open Problems ◽  
Minimal Action ◽  
Orbit Structure ◽  
Low Dimension

AbstractWe show that a C0 minimal action of a semisimple Lie group without compact factors is either locally free or induced from a minimal action of a proper parabolic subgroup. We describe the orbit structure of the action restricted to a maximal compact subgroup, and then apply this to minimal actions in low dimension. We give some examples, applications, and open problems.

scholarly journals AnLp−Lqversion of Hardy's theorem for spherical Fourier transform on semisimple Lie groups

2004 ◽  
Vol 2004 (33) ◽  
pp. 1757-1769 ◽  
Author(s):  
S. Ben Farah ◽  
K. Mokni ◽  
K. Trimèche
Keyword(s):  
Fourier Transform ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Semisimple Lie Groups ◽  
Semisimple Lie Group ◽  
Positive Real ◽  
Finite Center ◽  
Hardy’S Theorem ◽  
Spherical Fourier Transform ◽  
Almost Everywhere

We consider a real semisimple Lie groupGwith finite center andKa maximal compact subgroup ofG. We prove anLp−Lqversion of Hardy's theorem for the spherical Fourier transform onG. More precisely, leta,bbe positive real numbers,1≤p,q≤∞, andfaK-bi-invariant measurable function onGsuch thatha−1f∈Lp(G)andeb‖λ‖2ℱ(f)∈Lq(𝔞+*)(hais the heat kernel onG). We establish that ifab≥1/4andporqis finite, thenf=0almost everywhere. Ifab<1/4, we prove that for allp,q, there are infinitely many nonzero functionsfand ifab=1/4withp=q=∞, we havef=const ha.

scholarly journals Asymptotics for the Radon transform on hyperbolic spaces

2019 ◽  
pp. 241-251
Author(s):  
Nils Byrial ANDERSEN ◽  
Mogens FLENSTED-JENSEN
Keyword(s):  
Differential Operator ◽  
Maximal Compact Subgroup ◽  
Discrete Series ◽  
Compact Subgroup ◽  
Hyperbolic Spaces ◽  
Schwartz Function ◽  
Invariant Differential Operator ◽  
Abel Transform ◽  
Invariant Functions ◽  
Invariant Differential

Let G/H be a hyperbolic space over R; C or H; and let K be a maximal compact subgroup of G. Let D denote a certain explicit invariant differential operator, such that the non-cuspidal discrete series belong to the kernel of D. For any L^2-Schwartz function f on G/H we prove that the Abel transform A(Df) of Df is a Schwartz function. This is an extension of a result established in [2] for K-finite and K∩H-invariant functions.

scholarly journals Character formulas for discrete series on semisimple Lie groups

1976 ◽  
Vol 64 ◽  
pp. 47-61 ◽  
Author(s):  
Rebecca A. Herb
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Maximal Compact Subgroup ◽  
Discrete Series ◽  
Compact Subgroup ◽  
Semisimple Lie Groups ◽  
Analytic Group ◽  
Finite Center ◽  
Real Analytic ◽  
Analytic Subgroup

Let G be a connected, semisimple real Lie group with finite center, K a maximal compact subgroup of G. Assume rank G = rank K. Let be the Lie algebra of G, its complexification. If Gc is the simplyconnected complex analytic group corresponding to assume G is the real analytic subgroup of Gc corresponding to .

scholarly journals Differential equations and an analog of the Paley-Wiener theorem for linear semisimple Lie groups

1976 ◽  
Vol 64 ◽  
pp. 17-29 ◽  
Author(s):  
Kenneth D. Johnson
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Nilpotent Subgroup ◽  
Iwasawa Decomposition ◽  
Semisimple Lie Groups ◽  
Semisimple Lie Group ◽  
Simply Connected

Let G be a noncompact linear semisimple Lie group. Fix G = KAN an Iwasawa decomposition of G. That is, K is a maximal compact subgroup of G, A is a vector subgroup with AdA consisting of semisimple transformations and A normalizes N, a simply connected nilpotent subgroup of G.

scholarly journals Square-integrable representations of non-unimodular groups

1976 ◽  
Vol 15 (1) ◽  
pp. 1-12 ◽  
Author(s):  
A.L. Carey
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Hilbert Algebra ◽  
Technical Tool ◽  
Integrable Representation ◽  
Hilbert Algebras ◽  
Orthogonality Relations ◽  
Square Integrable Representation ◽  
Main Technical Tool

In the last three years a number of people have investigated the orthogonality relations for square integrable representations of non-unimodular groups, extending the known results for the unimodular case. The results are stated in the language of left (or generalized) Hilbert algebras. This paper is devoted to proving the orthogonality relations without recourse to left Hilbert algebra techniques. Our main technical tool is to realise the square integrable representation in question in a reproducing kernel Hilbert space.

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).

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.

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