The orbit and \(\theta\) correspondence for some dual pairs
Let \(G\) be a semisimple Lie group and \(H\), \(H'\) reductive subgroups, each being the full centralizer of the other. Let \(\omega\) be an irreducible unitary representation of \(G\) which is minimal in the sense that \(\omega\) corresponds to a coadjoint orbit of minimal dimension. In the case \(G= Sp_n\) and \(\omega\) being the Weil representation, \textit{R. Howe} has shown [J. Am. Math. Soc. 2, 535-552 (1989; Zbl 0716.22006)] that there is a homomorphism \(\psi: {\mathfrak z}\to {\mathfrak z}'\) of the center of the universal enveloping algebras such that on the smooth vectors we have \(\omega (z)= \omega (\psi (z))\) for all \(z\in {\mathfrak z}\). This is part of a more general study of the so called Howe correspondence, the representation theoretic gadget underlying the theta correspondence in the theory of automorphic forms. -- In the present paper this is generalized to simple \(G\) not of type \(A_n\) and certain standard pairs \(H\), \(H'\) which are given by the components of the Dynkin diagram when one removes a simple root. Although it only treats very special dual pairs the paper is an important step towards the generalization of Howe theory to groups other than the symplectic one.
scientific article; zbMATH DE number 880139