DOMAINS OF HOLOMORPHY FOR IRREDUCIBLE ADMISSIBLE UNIFORMLY BOUNDED BANACH REPRESENTATIONS OF SIMPLE LIE GROUPS

In this note, we address a question raised by B. Krötz on the classification of G-invariant domains of holomorphy for irreducible admissible Banach representations of connected non-compact simple real linear Lie groups G. When G is not of Hermitian type, we give a complete description of such G-invariant domains for irreducible admissible uniformly bounded representations on reflexive Banach spaces and, in particular, for all irreducible uniformly bounded Hilbert representations. When the group G is Hermitian, we determine such G-invariant domains only when the representations considered are highest or lowest weight representations.