# Introduction to Representations of Real Semisimple Lie Groups

@article{Libine2012IntroductionTR, title={Introduction to Representations of Real Semisimple Lie Groups}, author={Matvei Libine}, journal={arXiv: Representation Theory}, year={2012} }

These are lecture notes for a one semester introductory course I gave at Indiana University. The goal was to make this exposition as clear and elementary as possible. A particular emphasis is given on examples involving SU(1,1). These notes are in part based on lectures given by my graduate advisor Wilfried Schmid at Harvard University and PQR2003 Euroschool in Brussels as well as other sources.

## 3 Citations

### Lie Group Machine Learning and Gibbs Density on Poincaré Unit Disk from Souriau Lie Groups Thermodynamics and SU(1, 1) Coadjoint Orbits

- MathematicsGSI
- 2019

Souriau method could be applied on Lie Groups to define a covariant maximum entropy density by Kirillov representation theory and illustrated for homogeneous Siegel domains and more especially for Poincare unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau’s moment map.

### Lie Group Statistics and Lie Group Machine Learning Based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation

- MathematicsEntropy
- 2020

A new geometric definition of Entropy is proposed that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Ent entropy by Legendre transform, is proposed.

### Probing holography in $p$-adic CFT

- Mathematics
- 2019

We holographically calculate the partition functions of CFTs dual to Bruhat-Tits trees and $p$-adic BTZ black holes. Along the way, we propose new spectral decompositions of the Laplacian operator…

## References

SHOWING 1-10 OF 25 REFERENCES

### An Introduction to Harmonic Analysis on Semisimple Lie Groups

- Mathematics
- 1989

Preface 1. Introduction 2. Compact groups: the work of Weyl 3. Unitary representations of locally compact groups 4. Parabolic induction, principal series representations, and their characters 5.…

### Lie groups beyond an introduction

- Mathematics
- 1988

Preface to the Second Edition Preface to the First Edition List of Figures Prerequisites by Chapter Standard Notation Introduction: Closed Linear Groups Lie Algebras and Lie Groups Complex Semisimple…

### Lie Groups

- MathematicsNature
- 1970

Lectures on Lie GroupsBy J. Frank Adams. (Mathematics Lecture Note Series.) Pp. xii + 182. (W. A. Benjamin: New York and Amsterdam, 1969.) n.p.

### Two geometric character formulas for reductive Lie groups

- Mathematics
- 1998

In this paper we prove two formulas for the characters of representations of reductive groups. Both express the character of a representation π in terms of the same geometric data attached to π. When…

### Poisson geometry, deformation quantisation and group representations

- Mathematics
- 2005

Poisson geometry lies at the cusp of noncommutative algebra and differential geometry, with natural and important links to classical physics and quantum mechanics. This book presents an introduction…

### Canonical Extensions of Harish-Chandra Modules to Representations of G

- MathematicsCanadian Journal of Mathematics
- 1989

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…

### Representations of real reductive groups

- Mathematics
- 2012

Let g be a real Lie algebra with the property that gC := g ⊗R C is reductive in the usual sense of complex Lie algebras. You can rig it so that g is a subalgebra of a matrix algebra over the…

### Differential Geometry, Lie Groups, and Symmetric Spaces

- Mathematics
- 1978

Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric…