Schur orthogonality relations for certain non square integrable representations of real semisimple Lie groups
Let G be a noncompact connected real semisimple Lie group with finite center and K its maximal compact subgroup. Denote by d(x) the distance from x to the origin in the Riemannian symmetric space G/K. Let (\(\pi\),H) be an irreducible unitary representation of G. Assume that there exists a K-finite vector \(\phi_ 0\) in H such that \[ 0<\lim_{\epsilon \to +0}\epsilon \int_{G}| (\pi (x)\phi_ 0,\phi_ 0)|^ 2 e^{- \epsilon d(x)} dx<+\infty \quad. \] Then there exists a positive constant \(d_{\pi}\) such that \[ \lim_{\epsilon \to +0}\epsilon \int_{G}(\pi (x)\phi,\psi) \overline{(\pi (x)\phi ',\psi ')} e^{-\epsilon d(x)} dx=d_{\pi}^{-1} (\phi,\phi ') \overline{(\psi,\psi ')} \] for arbitrary K-finite vectors \(\phi\), \(\phi\) ', \(\psi\) and \(\psi\) '. The character \(\theta_{\pi}\) of such representation is a tempered distribution on G.
scientific article; zbMATH DE number 3985531
scientific article