Minimal and maximal invariant spaces of holomorphic functions on bounded symmetric domains

Let D be a Cartan domain in ℂ^d and let G = Aut(D) be the group of all biholomorphic automorphisms of G. Consider the projective representation of G on spaces of holomorphic functions on D (U_v(g)f)(z) := {j(g^-1)(z)}^v/pf(g^-1)(z)) g∈G, z∈D, where p is the genus of D and v is in the Wallach set D. We identify the minimal and the maximal U_v (G)-invariant Banach spaces of holomorphic functions on D in a very explicit way: The minimal space m_v is a Besov-1 space, and the maximal space M_v is a weighted H^∞-space. Moreover, with respect to the pairing under the (unique) U_v(G)- invariant inner product we have M_v* =M_v. In the second part of the paper we consider invariant Banach spaces of vector-valued holomorphic functions and obtain analogous descriptions of the unique maximal and minimal space, in particular for the important special case of "constant" partitions which arises naturally in connection with nontube type domains.