Branching problems ask how irreducible representations ƒÎ of groups G ''decompose” when restricted to subgroups G'.For real reductive groups, branching problems include various important special cases, however, it is notorious that ''infinite multiplicites” and ''continuous spectra” may well happen in general even if (G,G') are natural pairs such as symmetric pairs.
By using analysis on (real) spherical varieties, we give a necessary and sufficient condition on the pair of reductie groups for the multiplicities to be always finite (and also to be of uniformly bounded). Further, we discuss ''discretely decoposable restrictions” which allows us to apply algebraic tools in branching problems. Some classification results will be also presented.
If time permits, I will discuss some applications of branching laws of Zuckerman's derived functor modules to analysis on locally symmetric spaces with indefinite metric.
© Toshiyuki Kobayashi