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K-types of tempered representations Hochs, Peter


Let $G$ be a real semisimple Lie group, and $K < G$ a maximal compact subgroup. A tempered representation $\pi$ of G is an irreducible representation that occurs in the Plancherel decomposition of $L^2(G)$. The restriction $\pi|_K$ of $\pi$ to $K$ contains a substantial amount of information about $\pi$. (This is roughly analogous to the fact that an irreducible representation of $K$ is determined by its restriction to a maximal torus.) By realising this restriction as the geometric quantisation of a suitable space, which is a coadjoint orbit under a regularity assumption on $\pi$, we can apply a suitable version of the quantisation commutes with reduction principle to obtain geometric expressions for the multiplicities of the irreducible representations of $K$ in $\pi|_K$ (the $K$-types of $\pi$). This was done for the discrete series by Paradan in 2003. In recent joint work with Song and Yu, we extended this to arbitrary tempered representation. The resulting multiplicity formula was obtained in a different way for tempered representations with regular parameters by Duflo and Vergne in 2011. In independent work in progress with Higson and Song, we give a new proof of Blattner's formula for multiplicities of $K$-types of discrete series representations using geometric quantisation. This formula was first proved by Hecht and Schmid in 1975, and later by Duflo, Heckman and Vergne in 1984.

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