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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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