Determinant Type Differential Operators on Homogeneous Siegel Domains

In harmonic analysis on classical domains of matrices, the differential operator whose symbol is the determinant polynomial plays important roles. Particularly, the operator is substantial in study of invariant Hilbert spaces of holomorphic functions on the domain [1, 2, 7, 14, 15, 21]. Considering the Siegel domain realization of a certain symmetric domain with Fourier-analytic methods, Jakobsen and Vergne [15] show that a unitary representation of a semisimple Lie group is realized on a Hilbert space in the kernel of the wave operator, which corresponds to the determinant of a Hermitian 2_2 matrix, and that this differential operator intertwines some unitary representations of the group. In [14] this kind of equivariance property is investigated for powers of the differential operators corresponding to the determinants of n_n symmetric and Hermitian matrices. Arazy and Upmeier [2] (see also [1]) attain more general results from another approach, the Peter Weyl theory for the maximal compact subgroups with the bounded realizations of symmetric domains. In this paper, following the Fourier-analytic approach, we obtain analogues of these results in the framework of analysis on a homogeneous (not necessarily symmetric) Siegel domain D on which a split solvable Lie doi:10.1006 jfan.2001.3755, available online at http: www.idealibrary.com on