I plan to discuss some basic questions about regular representations on $X$ acted algebraically by real reductive groups $G$.
1. (function space) Does the group G have a "good control" on the space $C(X)$ of function on $X$?
2. ($L^2$ theory) What can we say about "spectrum" for $L^2(X)$?
We highlight "multiplicities" and "temperdness" for these questions, and give their geometric necessary and sufficient conditions.
If time permits, I will mention some applications to branching problems for restriction of infinite-dimensional representations.
© Toshiyuki Kobayashi