Domains of holomorphy and representations of SL(n,R

Abstract. For G = SL(n,R) and K = SO(n) Akhiezer and Gindikin explicitly determine a G-invariant Stein extension A of G/K in GC/KC. We give several other descriptions of A. In terms of the geometry of an arbitrary flag variety for GC, A is described as the ‘polar set ’ of the closed G-orbit. A is also the space of ‘linear cycles ’ in an arbitrary open G-orbit. We also see that A is a domain of holomorphy for Szego ̈ kernels associated to interesting irreducible representations of G. 1