Branching Laws for Infinite Dimensional Representations of Real Reductive Lie Groups. (5 lectures), Mathematical Panorama Lectures in celebration of 125th birthday of Srinivasa Ramanujan. Tata Institute, India, 18-22 February 2013.

Branching problems ask how irreducible representations decompose when restricted to subgroups. They are very difficult in general noncompact settings. I will discuss how to single out ''nice frameworks” in which we could expect fruitful studies of branching problems.

Description of Contents:
Branching problems ask how irreducible representations decompose when restricted to subgroups. Recently, obstructions for studying branching problems have been fairly well understood, in particular, wild phenomena such as infinite multiplicities or continuous spectra in dealing with noncompact subgroups. In this talk I would like to give a general intruduction to various techniques for branching problems with emphasis on the question: ''How to single out a nice framework where we could expect further detailed study?” Three aspects will be highlighted:

1. Multiplicity-free Theorems.
In this talk, I will discuss an application of the original theory of ''visible actions” on complex manifolds to ''multiplicity-free” theorems, in partucular, to branching laws for reductive symmetric pairs.

2. Theory of Discretely Decomposable Restrictions.
An algebraic and analytic approach is introduced to give a criterion for branching laws to be discretely decomposable.

3. Estimates of Multiplicities.
I plan to discuss geometric conditions that control the multiplicities in branching laws (restriction) and Plancherel formulas (induction).

If time permits, I will discuss some applications of branching problems to other areas of mathematics such as modular varieties in automorphic forms, geometric analysis, and parabolic geometry.

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