Branching problems of representations of real reductive Lie groups. Representations of reductive groups: A conference dedicated to David Vogan on his 60th birthday (organized by Roman Bezrukavnikov, Pavel Etingof, George Lusztig, Monica Nevins, and Peter Trapa). MIT, USA, 19-23 May 2014.

Branching problems ask how irreducible representations π 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.

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© Toshiyuki Kobayashi