## Regular Representations on Homogeneous Spaces.Journées SL2 R - Théorie des représentations et analyse harmonique, in honor of Jacques Faraut's 80th birthday. Nancy, France, 12-13 March 2020. * Conference was cancelled.

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.