On a global realization of a discrete series for SU(n,1) as applications of Szegö operator and limits of discrete series
Let G be a connected Lie group with finite center, having real rank one, and a discrete series. The aim of this paper is to give a global (not infinitesimal) realization of discrete series representations of G. For limit of discrete series, this can be achieved by means of the Szegö operator composed with a boundary value operator, relating this representation, in a G-equivariant way, to a reducible unitary principal series representation [\textit{B. Blank}, Pac. J. Math. 122, 299-318 (1986; Zbl 0595.22012)]. Here, the author shifts this realization from a limit of discrete series to a discrete series representation, by means of Zuckerman's technique (tensor product with a finite dimensional representation). His construction is valid for all rank one semisimple Lie groups. However, in the final step, he needs the extra assumption that the associated symmetric space is hermitian, in order to prove the square-integrability of the image of the Szegö operator. Thus, the results are complete for \(G=SU(n,1)\) only. The examples of SU(1,1) and SU(2,1) are considered with some detail at the end of the paper.
scientific article; zbMATH DE number 4146301
scientific article