讲座内容：

Let B_d be the open Euclidean ball in C_d and T:= (T1….. Td) be a commuting tuple of bounded linear operators on a complex separable Hilbert space H.Let U(d) be the linear group of unitary transformations acting on C_d by the rule: z:\rightarrow u.z,z\in C_d, where u _ z is the usual matrix product.Let u1(z),…., ud(z) be the coordinate functions of uz.We define uT tobe the operator (u1(T ),….,ud(T )) and say that T is U(d)-homogeneous if u T is unitarily equivalent to T for all u\in U(d). In this talk, we describe U(d)-homogeneous tuples M of multiplication by coordinate functions acting on some reproducing kernel Hilbert space HK(Bd;Cn) \subset Hol(Bd;Cn), where n is the dimension of the joint kernel of the d-tuple T. The case of n = 1 is well understood, here, we focus on the case of n = d. We describe a large class of U(d)-homogeneous operators and obtain explicit criterion for (i) boundedness, (ii) reducibility and (iii) mutual unitary equivalence of these operators. Finally, we classify the kernels K taking values in Mn(C), 1 \leq n \leq d, quasi-invariant under an irreducible unitary representation c of the group U(d). A crucial ingredient of this proof is that the group SU(d) has exactly two inequivalent irreducible unitary representations of dimension d andnone in dimensions 2,…, d - 1, d> 2.