Hua operators and Poisson transform for bounded symmetric domains

Abstract

Let Ω be a bounded symmetric domain of non-tube type in C n with rank r and S its Shilov boundary. We consider the Poisson transform P s f (z) for a hyperfunction f on S defined by the Poisson kernel P s (z, u) = (h(z, z) n/r /|h(z, u) n/r | 2 ) s , (z, u) × Ω × S, s ∈ C. For all s satisfying certain non-integral condition we find a necessary and sufficient condition for the functions in the image of the Poisson transform in terms of Hua operators. When Ω is the type I matrix domain in M n,m (C) (n m), we prove that an eigenvalue equation for the second order M n,n -valued Hua operator characterizes the image.

Degenerate principal series representations and their holomorphic extensions

“…[21]. (Similar formulas hold, [34], in the Siegel domain realization, for general linear functional λ on a.)…”

Hua operators, Poisson transform and relative discrete series on line bundles over bounded symmetric domains

“…Recall the following formulas in [14,Lemma 5.2]: for any fixedw ∈V and any complex number s, Thus the claim is true for general z by the invariant property (28) of H.…”

“…For tube domains with general parameters Shimeno [25] proved an analogue of Kashiwara et al theorem for P max ⊂ G. More precisely, he proved that the Poisson transform is a G-isomorphism from the space of hyperfunctions on the Shilov boundary onto the space of eigenfunctions of the Hua operator of the second order. The generalization to the non-tube bounded symmetric domains has been given in our earlier paper [14].…”

Characterization of the Poisson integrals for the non-tube bounded symmetric domains

“…For general bounded symmetric domains the Poisson integrals are not eigenfunctions of the second-order Hua operator H, see for instance [3] or [12]. However for type I r,r+b domains of non-tube type, (see [3] and [12]) there is a variant of the second-order Hua operator, H (1) , by taking the first component of H, since in this case k C is a sum of two irreducible ideals k C = k…”

“…C . It is proved, in [12] (and in [3] for the harmonic case, s = (2r + b)/r) that a smooth function f on I r,r+b is a solution of the Hua system, H (1) f = 1 4 (s 2 − (r + b) 2 )f I r if and only if it is the Poisson transform P s of a hyperfunction on the Shilov boundary.…”