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Index theorems and discrete series representations of semisimple Lie groups. (English) Zbl 0578.22017
Let $$(G,K)$$ and $$(U,K)$$ be dual Riemannian symmetric pairs, of the non-compact and the compact type, respectively. Suppose that the set $$\widehat G_ d$$ of discrete series representations in $$\widehat G$$ is non-empty. Let $$\mathfrak g=\mathfrak k+\mathfrak p$$ be the Cartan decomposition for the Lie algebra $$\mathfrak g$$ of $$G$$. Let $$(\sigma,E)$$ and $$(\tau,F)$$ be finite dimensional representations of $$K$$. Assume that there exists a positive-homogeneous $$K$$-homomorphism between the $$K$$-homogeneous product bundles over $$\mathfrak p^*$$ corresponding, respectively, to $$\sigma$$ and $$\tau$$ which is an isomorphism outside a compact set. The author’s main result is the formula:
$\begin{split} (\text{vol } U/K)^{-1}\sum_{\rho \in \hat U}(<\rho |_K:\sigma >-<\rho |_K:\tau >) \deg \rho =\\ =(-1)^{\dim G/K}\sum_{\pi \in \widehat G_d}(<\pi |_K:\sigma >-<\pi |_K:\tau >) \deg \pi. \end{split}$ Here $$\deg \pi$$ denotes the formal degree and $$<\,:\,>$$ denotes the multiplicity number. The formula is proved by use of the $$L^2$$-index theorem of Connes and Moscovici and the Atiyah-Singer index theorem combined with the proportionality principle of Hirzebruch.
##### MSC:
 22E46 Semisimple Lie groups and their representations 58J20 Index theory and related fixed-point theorems on manifolds
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##### References:
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