K-HOMOGENEOUS TUPLE OF OPERATORS ON BOUNDED SYMMETRIC DOMAINS

Let Omega be an irreducible bounded symmetric domain of rank r in C-d. Let K be the maximal compact subgroup of the identity component G of the biholomorphic automorphism group of the domain Omega. The group K consisting of linear transformations acts naturally on any d-tuple T = (T-1, ... , T-d) of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that T is K-homogeneous. In this paper, we obtain a model for a certain class of K-homogeneous d-tuple T as the operators of multiplication by the coordinate functions z(1), ... , z(d) on a reproducing kernel Hilbert space of holomorphic functions defined on Omega. Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class, (iii) unitary equivalence and similarity of these d-tuples. In particular, we show that the adjoint of the d-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class B-1(Omega). For an irreducible bounded symmetric domain Omega of rank 2, an explicit description of the operator Sigma(d)(i=1) T*T-i(i) is given. In general, based on this formula, we make a conjecture giving the form of this operator.