Japan. J. Math. 12, 175--210 (2017)

The size of infinite-dimensional representations

D.A. Vogan, Jr.

This paper is offered in honor and in fond remembrance of Professor Bertram Kostant

Abstract: An infinite-dimensional representation $\pi$ of a real reductive Lie group $G$ can often be thought of as a function space on some manifold $X$. Although $X$ is not uniquely defined by $\pi$, there are ``geometric invariants'' of $\pi$, first introduced by Roger Howe in the 1970s, related to the geometry of $X$. These invariants are easy to define but difficult to compute. I will describe some of the invariants, and recent progress toward computing them.