Index theorems and dicrete series representations of semisimple lie groups
Copyright policy )Index theorems and dicrete series representations of semisimple lie groups
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-\textit{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: \[ \begin{multlined} (\text{vol } U/K)^{-1}\sum_{\rho \in \hat U}(-) \deg \rho =\\ =(-1)^{\dim G/K}\sum_{\pi \in \widehat G_d}(-) \deg \pi. \end{multlined} \] 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.
- University of Warsaw Poland
Cartan decomposition, Atiyah-Singer index theorem, dual Riemannian symmetric pairs, \(L^ 2\)-index theorem, discrete series representations, proportionality principle, Semisimple Lie groups and their representations, symmetric space, semisimple algebraic Lie group, Index theory and related fixed-point theorems on manifolds, multiplicity, homogeneous product bundles
Cartan decomposition, Atiyah-Singer index theorem, dual Riemannian symmetric pairs, \(L^ 2\)-index theorem, discrete series representations, proportionality principle, Semisimple Lie groups and their representations, symmetric space, semisimple algebraic Lie group, Index theory and related fixed-point theorems on manifolds, multiplicity, homogeneous product bundles
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