The Real-Quaternionic indicator, also called the $\delta$ indicator, indicates if a self-conjugate representation is of real or quaternionic type. The Frobenius-Schur indicator, which we call the $\epsilon$ indicator, indicates if a self-dual representation has an invariant symmetric bilinear form or a skew-symmetric bilinear form. When $G$ is compact, $\delta(\pi)$ and $\epsilon(\pi)$ coincide. In general, they are not necessarily the same.
For finite-dimensional $\pi$, the Frobenius-Schur indicator $\epsilon(\pi)$ is well known to be a particular value of the central character. We would like a similar result for the $\delta$ indicator.
In this talk, I will give a relation between the two indicators for $\pi$ with real infinitesimal character. In particular, this gives a closed formula for $\delta(\pi)$ in terms of central character when $\pi$ is finite-dimensional.