I plan to discuss some fundamental problems about regular representations onXacted by reductive Lie groupsG.We highlight ''multiplicities” and ''temperdness” for these questions, and give their geometric criteria.

- (function spaces) Does the group
Ghave a ''good control” on the spaceC(X) of function onX?- (
L^{2}theory) What can we say about ''spectrum” forL^{2}(X)?If time permits, I will explain some applications to branching problems for restriction of infinite-dimensional representations.

© Toshiyuki Kobayashi