Boundary Value Problems on Riemannian Symmetric Spaces of the Noncompact Type

Abstract

Abstract. We characterize the image of the Poisson transform on any distinguished boundary of a Riemannian symmetric space of the noncompact type by a system of differential equations. The system corresponds to a generator system of two-sided ideals of a universal enveloping algebra, which are explicitly given by analogues of minimal polynomials of matrices. IntroductionThe classical Poisson integral of a function on the unit circle in the complex plane gives a harmonic function on the unit disk. More generally, each eigenfunction of the Laplace-Beltrami operator on the Poincaré disk can be represented by a generalized Poisson integral of a hyperfunction on the unit circle.The notion of the Poisson integral has been generalized to a Riemannian symmetric space X = G/K of the noncompact type, where G is a connected real reductive Lie group and K is its maximal compact subgroup. The so-called Helgason conjecture states that every joint eigenfunction of the invariant differential operators on X has a Poisson integral representation by a hyperfunction on the Furstenberg boundary G/P of X, where P is a minimal parabolic subgroup of G. Helgason proved the conjecture for the Poincaré disk. Kashiwara et al. [K-] proved it generally by using the theory of hyperfunctions and the system of differential equations with regular singularities and their boundary value problem due to Kashiwara and Oshima [KO].The Poisson transform is an intertwining operator from the spherical principal series representation to the eigenspace representation. For generic parameter λ of the principal series representation, the Poisson transform P λ gives an isomorphism of the representations. The principal series representation is realized on the space of the sections of a homogeneous line bundle over G/P whose parameter is λ. If λ = ρ, the line bundle is trivial and the representation is realized on the space of functions on G/P . Then the image of P ρ consists of the harmonic functions, that is the functions that are annihilated by the invariant differential operators on the symmetric space that kill the constant functions. We call this the "harmonic case".It is natural to pose the problem of characterizing the image of P λ when the map is not bijective. An interesting case corresponds to the problem of characterizing the image of the Poisson transform from another distinguished boundary of X in one of the Satake compactifications of X (cf. [Sa], [O1, O2]

The Hua operators on homogeneous line bundle on Bounded Symmetric Domains of Tube Type

“…Before ending this section, we should mention that Professor Koufany informed me that he and Professor Zhang had also obtained a characterization of Poisson integrals on homogeneous line bundles over tube type symmetric domains see [10], See also [16]. Namely, they showed that the image P λ,ν (B(G/P Ξ , L λ,ν )) can be characterized by the operator H defined in(1.1).…”

Resonances for the Laplacian on Riemannian symmetric spaces: The case of 𝑆𝐿(3,ℝ)/𝕊𝕆(3)

“…But Corollary 22 says the resonances (n+ 1 2 )ρ are not generic in this sense. In this case, according to [16,Thm. 2.4], the image of the Poisson transform consists of those elements u ∈ E (n+ 1 2 )ρ (X) which satisfy the following additional differential equations: sym(h)u = 0, where h is a K-harmonic polynomial on p * C , viewed as an element of the symmetric algebra S(p C ) of p C and sym : S(p C ) → U (p C ) is the usual symmetrization map.…”

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

“…After a preliminary version of this paper was finished we were informed by Professor T. Oshima that he and N. Shimeno have obtained in [20] some similar results about Poisson transforms and Hua operators. Professor A. Koranyi communicated also his recent preprint [13] to us where he proved the necessity of Theorem 5.2 using different methods.…”