The Real-Quaternionic Indicator for Finite-Dimensional Self-conjugate Representations of Real Reductive Lie Groups

The real-quaternionic indicator, also called the δ indicator, indicates if a self-conjugate representation is of real or quaternionic type. It is closely connected to the Frobenius-Schur indicator, which we call the ε indicator. It is interesting to compute the ε and δ indicators. The computation of the ε indicator is relatively straightforward. In fact, it has been proven in large generality that ε(π) is given by a particular value of the central character. We would like a similar result for the δ indicator. When G is compact, δ(π) and ε(π) coincide. In general, they are not necessarily the same. In this paper, we will give a relation between the two indicators when G is real reductive. We will also give a formula for δ(π) in terms of the central character when π is finite dimensional.