Lie Groups and Representation Theory

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Date, time & place Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.)

2021/06/01

17:30-18:30   Room #Online (Graduate School of Math. Sci. Bldg.)
Joint with Tuesday Seminar on Topology. Online.
Masatoshi KITAGAWA (Waseda University)
On the discrete decomposability and invariants of representations of real reductive Lie groups (Japanese)
[ Abstract ]
A problem to determine the behavior of the restriction of an irreducible group representation to a subgroup is called the branching problem. The restriction of an irreducible representation is not irreducible in general, and if the representation is unitary, the restriction has an irreducible decomposition described by a direct integral. The decomposition can be regarded as a generalization of spectral decomposition of unitary operators, and has continuous spectrum and discrete spectrum in general. If the decomposition has no continuous spectrum, the representation is said to be discretely decomposable.

Discretely decomposable branching laws are technically easy to deal with, and in the setting, it is relatively easy to extract information about representations of a small subgroup from that of a large group. The following applications are known. It is known that the operators, called the Rankin--Cohen brackets which make a new automorphic form from a automorphic form, can be obtained as
intertwiner from discretely decomposable representations to irreducible representations. Many generalizations of the operators are obtained recently. The discrete decomposability is used to construct discrete spectrum of the space of L^2 functions on homogeneous spaces (T. Kobayashi, J.
Funct. Anal. ('98)).

In this talk, I will give several criterion about the discrete decomposability and G'-admissibility based on the general theory and criterion given by T. Kobayashi (Invent. math. '94, Annals of Math. '98, Invent. math. '98).
The criterion are written by associated varieties (algebraic invariants), wave front sets (analytic invariants) and topological structure of representations.