scholarly journals Analytic representation theory of Lie groups: general theory and analytic globalizations of Harish-Chandra modules

2011 ◽  
Vol 147 (5) ◽  
pp. 1581-1607 ◽  
Author(s):  
Heiko Gimperlein ◽  
Bernhard Krötz ◽  
Henrik Schlichtkrull
Keyword(s):  
General Theory ◽  
Representation Theory ◽  
Lie Groups ◽  
Lie Group ◽  
Analytic Functions ◽  
General Framework ◽  
Analytic Representation ◽  
Reductive Groups ◽  
Topological Properties ◽  
Rapid Decay

AbstractIn this article a general framework for studying analytic representations of a real Lie group G is introduced. Fundamental topological properties of the representations are analyzed. A notion of temperedness for analytic representations is introduced, which indicates the existence of an action of a certain natural algebra 𝒜(G) of analytic functions of rapid decay. For reductive groups every Harish-Chandra module V is shown to admit a unique tempered analytic globalization, which is generated by V and 𝒜(G) and which embeds as the space of analytic vectors in all Banach globalizations of V.

scholarly journals Corrigendum: Analytic representation theory of Lie groups: general theory and analytic globalizations of Harish-Chandra modules

2017 ◽  
Vol 153 (1) ◽  
pp. 214-217
Author(s):  
Heiko Gimperlein ◽  
Bernhard Krötz ◽  
Henrik Schlichtkrull
Keyword(s):  
General Theory ◽  
Representation Theory ◽  
Lie Groups ◽  
Analytic Representation ◽  
Corrected Proof ◽  
Intermediate Results

We correct the proof of the main result of the paper, Theorem 5.7. Our corrected proof relies on weaker versions of a number of intermediate results from the paper. The original, more general, versions of these statements are not known to be true.

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 Contractions of group representations via geometric quantization

2019 ◽  
Vol 110 (1) ◽  
pp. 43-59
Author(s):  
Rauan Akylzhanov ◽  
Alexis Arnaudon
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
General Framework ◽  
Geometric Quantization ◽  
Coadjoint Orbits ◽  
Sufficient Condition ◽  
Group Representations ◽  
Unitary Dual

AbstractWe propose a general framework to contract unitary dual of Lie groups via holomorphic quantization of their coadjoint orbits, using geometric quantization. The sufficient condition for the contractibility of a representation is expressed via cocycles on coadjoint orbits. This condition is verified explicitly for the contraction of SU$$_2$$2 into $$\mathbb {H}$$H. We construct two types of contractions that can be implemented on every matrix Lie group with diagonal contraction matrix.

Stability of discontinuous groups for reduced threadlike Lie groups

2015 ◽  
Vol 26 (08) ◽  
pp. 1550057 ◽  
Author(s):  
Fatma Khlif
Keyword(s):  
Lie Groups ◽  
Parameter Space ◽  
Lie Group ◽  
Deformation Space ◽  
Topological Properties ◽  
Local Rigidity ◽  
Connected Subgroup ◽  
The Stability ◽  
Closed Connected Subgroup ◽  
Discontinuous Groups

Let G be a reduced threadlike Lie group, H an arbitrary closed connected subgroup of G and Γ ⊂ G an abelian discontinuous subgroup for G/H. We study in this work some topological properties of the parameter space [Formula: see text] and the deformation space [Formula: see text], namely the stability and the rigidity. Instead of treating stability, we consider a weaker form by using an explicit covering of Hom (Γ, G) which we call layering and we show that the local rigidity holds if and only if Γ is finite.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

scholarly journals The Higher Order Riesz Transform andBMOType Space Associated with Schrödinger Operators on Stratified Lie Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Yu Liu ◽  
Jianfeng Dong
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Riesz Transform ◽  
Integral Operators ◽  
Higher Order ◽  
Reverse Hölder Class ◽  
Stratified Lie Group ◽  
Homogeneous Dimension ◽  
Stratified Lie Groups ◽  
Reverse Hölder

Assume thatGis a stratified Lie group andQis the homogeneous dimension ofG. Let-Δbe the sub-Laplacian onGandW≢0a nonnegative potential belonging to certain reverse Hölder classBsfors≥Q/2. LetL=-Δ+Wbe a Schrödinger operator on the stratified Lie groupG. In this paper, we prove the boundedness of some integral operators related toL, such asL-1∇2,L-1W, andL-1(-Δ) on the spaceBMOL(G).

ON THE MODEL THEORY OF THE LOGARITHMIC FUNCTION IN COMPACT LIE GROUPS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350055
Author(s):  
SONIA L'INNOCENTE ◽  
FRANÇOISE POINT ◽  
CARLO TOFFALORI
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Model Theory ◽  
Effective Model ◽  
Logarithmic Function ◽  
Compact Lie Groups ◽  
Model Completeness ◽  
Schanuel's Conjecture ◽  
Logarithm Function ◽  
Natural Expansion

Given a compact linear Lie group G, we form a natural expansion of the theory of the reals where G and the graph of a logarithm function on G live. We prove its effective model-completeness and decidability modulo a suitable variant of Schanuel's Conjecture.

THE FERMI–WALKER DERIVATIVE IN LIE GROUPS

2013 ◽  
Vol 10 (07) ◽  
pp. 1320011 ◽  
Author(s):  
FATMA KARAKUŞ ◽  
YUSUF YAYLI
Keyword(s):  
Vector Field ◽  
Lie Groups ◽  
Lie Group ◽  
Necessary Conditions ◽  
Rotating Frame ◽  
Darboux Vector

In this study, Fermi–Walker derivative, Fermi–Walker parallelism, non-rotating frame, Fermi–Walker termed Darboux vector concepts are given for Lie groups in E4. First, we get any Frénet curve and any vector field along the Frénet curve in a Lie group. Then, the Fermi–Walker derivative is defined for the Lie group. Fermi–Walker derivative and Fermi–Walker parallelism are analyzed in Lie groups. Finally, the necessary conditions for Fermi–Walker parallelism are explained.

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