Projection of Elliptic Orbits and Branching Laws

Let $G$ be a Lie group, and $H\subset G$ a closed subgroup. Let $\pi$ be an irreducible unitary representation of $G$. In this paper, we briefly discuss the orbit method and its application to the branching problem $\pi {\mathrm{\mid }}_H$. 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 ${\text{pro}}_{\text{f}u(p,q-1)}^{\text{f}u(p,q)}\text{mc}O(\lambda )$ and the branching law of the discrete series $D_{\lambda }{\mathrm{\mid }}_{U(p,q-1)}$ for $\lambda$ 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.