Projection of Elliptic Orbits and Branching Laws

Abstract

Let $G$ be a Lie group, and $H\subset G$ a closed subgroup. Let $\ddot{I}€$ be an irreducible unitary representation of $G$. In this paper, we briefly discuss the orbit method and its application to the branching problem $\ddot{I}€{|}_H$. We use the Gan-Gross-Prasad branching law for $(G,H)=(U(p,q),U(p,q-1))$ as an example to illustrate the relation between $\pro_{\f u(p, q-1)}^{\f u(p,q)} \mc O(λ)$ and the branching law of the discrete series $D_{\hat{I}}»{|}_{U(p,q-1)}$ for $\hat{I}»$ an regular elliptic element. We also discuss some results regarding branching laws and wave front sets. The presentation of this paper does not follow the historical timeline of development.