Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces

On the complex and convex geometry of Ol'shanskii semigroups

“…Moreover, we show in Section 7 that for each connected biinvariant domain D = GExp(D^) the envelope of holomorphy is schlicht over 5max and that it coincides with the domain D = GExp(convD^). This means in particular that every holomorphic function on D extends to the domain D C 6'max-We note that since we do not assume that P is a convex subset of zt, even in the case where Q is a compact Lie algebra our results are a generalization of those in [AL92].…”

“…-Since the mapping Exp: t + P -» D is holomorphic, the function ip is a t-invariant plurisubharmonic function on the tube domain t+ P, hence it is locally convex (cf. [AL92,p.369]). …”

“…Let t C ^ be a Cartan subalgebra. Each JC-biinvariant domain D C K<^ can be written as D = KexpVK, where V C ii is a domain which is invariant under the Weyl group of K, we call it the base of D. In [AL92] Azad and Loeb give a characterization of the JC-biinvariant plurisubharmonic functions on D under the assumption that V is convex. Moreover, they obtain a description of the holomorphy hulls of biinvariant domains which previously has been derived by Lasalle (cf.…”

“…As already mentioned above, the main result of Section 4 is that plurisubharmonic biinvariant functions on Smax correspond to convex COMPLEX AND CONVEX GEOMETRY OF OL'SHANSKII SEMIGROUPS 151 functions on IVmax-The method to obtain this result is to adopt the strategy from [AL92] in the sense that instead of using finite dimensional holomorphic representations of complex reductive groups, we use infinite dimensional holomorphic representations of semigroups on Hilbert spaces (cf. [Ne94a], [Ne95a,b]).…”

On the complex geometry of invariant domains in complexified symmetric spaces

“…Exp(a), and likewise a G-invariant domain D C Me is determined by a corresponding G-invariant domain J9q C iq with D = G.Exp(jDq). In [AL92] Azad and Loeb give a characterization of the G-invariant plurisubharmonic functions on D (under the assumption that jDq H a is convex) as those corresponding to Weyl group invariant convex functions on D^ H a. Moreover, they obtain a description of the envelopes of holomorphy of invariant domains which previously has been derived by Lasalle (cf.…”