Small unitary representations of the double cover of $\operatorname {SL}(m)$

Abstract:

The irreducible unitary representations of the double cover $\widetilde {\mathrm {SL}(m)}$ of the real group $\mathrm {SL}(m)$, with infinitesimal character $\frac {1}{2}\rho$, which are small in the sense that their annihilator in the universal enveloping algebra is maximal, are expressed as Langlands quotients of generalized principal series. In the case where $m$ is even we show that there are four such representations and in the case where $m$ is odd there is just one. The representations’ smallness allows them to be written as a sum of virtual representations, leading to a character formula for their $K$-types. We investigate the place of these small representations in the orbit method and, in the case of $\widetilde {\mathrm {SL}(2l+1)}$, show that the representation is attached to a nilpotent coadjoint orbit.The $K$-type spectrum for the Langlands quotients is explicitly determined and shown to be multiplicity free.