Let $G$ be a real reductive linear Lie group, and $K$ a maximal compact subgroup of $G$. Harish-Chandra's admissibility theorem asserts that any irreducible unitary representation decomposes into a direct sum of irreducible $K$-modules with each multiplicity finite. Such a theorem does not hold if we replace the Riemannian symmetric pair $(G,K)$ by a reductive symmetric pair $(G,G')$ in general. We explore a “nice” framework for the restriction of an irreducible representation of $G$ to the subgroup $G′$ in this generality with focus on finite/uniformly bounded multiplicity property. If time permits, I also will discuss its application to analysis of locally pseudo-Riemannian symmetric spaces.
© Toshiyuki Kobayashi