Projection of orbits and K multiplicities
Let \({\mathfrak g}\) be a real semisimple noncompact Lie algebra with a chosen Cartan decomposition \({\mathfrak g}={\mathfrak k}\oplus {\mathfrak p}\). Any coadjoint orbit \(\Omega\) in \({\mathfrak g}^*\) carries a canonical invariant measure \(\beta_{\Omega}\). The main result is an explicit formula for the push forward \(J_*\beta_{\Omega}\) under the canonical projection \({\mathfrak g}^*\to {\mathfrak k}^*\) in the case of a regular orbit. The formula is technically complicated; it involves, of course, structural data of the algebra, besides others the root system of a Cartan subalgebra adapted to the chosen Cartan decomposition. The formula is a generalization of a result of \textit{M. Duflo}, \textit{G. Heckman} and \textit{M. Vergne} [in Mem. Soc. Math. Fr., Nouv. Sér. 15, 65-128 (1984; Zbl 0575.22014)] where the case of regular elliptic orbits was treated. At the end of the paper a heuristic interpretation for the formula is given as a ``continuous manifestation'' of the Frobenius reciprocity theorem in the representation theory of compact groups.
scientific article; zbMATH DE number 4204694