## *Symmetry Breaking Operators and Branching Problems*.
Algebraic Geometry Seminar. Zurich University, Switzerland, 6 October
2014.

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