Two characterizations of symmetric Siegel domains

by the Laplace-Beltrami operator

and their geometric backgrounds

Takaaki NOMURA

Let D be a homogeneous Siegel domain. For some reason, we take D as a generalized right half plane rather than a more familiar upper half plane. The domain D has the Bergman metric, so that we have the corresponding Laplace-Beltrami operator L. Let B_{λ} (λ being a parameter) be the Berezin transforms on D (the definition will be given during the talk). The first characterization is:

Theorem 1.1 ([5]). The parameter λ being fixed, L commutes with B_{λ} if and only if D is symmetric.

Let Σ be the Šilov boundary of D. Let P (z, ζ) (z ∈ D, ζ ∈ Σ) be the Poisson kernel of D defined à la Hua.

Theorem 1.2 (Hua-Look [1], Korányi [2], Xu [8]). LP (·, ζ) = 0 for any ζ ∈ Σ if and only if D is symmetric.

Theorem 1.1 is a consequence of some geometric norm equality involving a Cayley transform. My proof of Theorem 1.2 is also through a geometric norm equality involving another Cayley transform. A very brief and incomplete history of the proof of Theorem 1.2 is given during the talk.

Let C_{1}, C_{2} be the Cayley transforms of D associated to the Bergman kernel and the Szegö kernel of D, respectively. Both C_{1} and C_{2} are birational mappings that send D biholomorphically onto the images C_{j}(D) (j = 1, 2). Moreover, the images are bounded domains [3], [6]. If D is symmetric, they are positive number multiples of the standard Cayley transform, so that C_{j}(D) are essentially the Harish-Chandra realization of a Hermitian symmetric space. However, no such relation and no such canonical bounded model exist for general D.

Let G be the split solvable Lie group acting simply transitively on D. Choose a specific base point e of D (if D is the right half plane in ℂ, then e is the point 1). Thus G is diffeomorphic with D via the orbit map g ↦ g · e.

Theorem 1.3 ([4]). The norm equality

∥C_{1}(g · e)∥ = ∥C_{1}(g^{−1} · e)∥

holds for any g ∈ G if and only if D is symmetric.

It is known that there is a nilpotent subgroup N_{D} (at most 2 step) of G that acts simply transitively on Σ. In fact, Σ is the N_{D}-orbit of the origin: Σ = N_{D} · 0.

Theorem 1.4 ([7]). The norm equality

∥C_{2}(n · 0)∥^{2} = ⟨Ψ, α⟩

holds for any n ∈ N_{D} if and only if D is symmetric. Here Ψ is the element of 𝔤 := Lie(G) that represents the linear form x ↦ tr ad(x) through the Bergman inner product, and α is the linear form related to the Szegö kernel (details are explained during the talk).

I also present the formulas that clarify the relation of Theorem 1.1 with Theorem 1.3, and of Theorem 1.2 with Theorem 1.4.

References

[1] L. K. Hua and K. H. Look, Theory of harmonic functions in classical domains, Scientia Sinica, 8 (1959), 1031–1094.

[2] A. Korányi, The Poisson integral for generalized half-planes and bounded symmetric domains, Ann. of Math., 82 (1965), 332–350.

[3] T. Nomura, On Penney’s Cayley transform of a homogeneous Siegel domain, J. Lie Theory, 11 (2001), 185–206.

[4] T. Nomura, A characterization of symmetric Siegel domains through a Cayley transform, to appear in Transform. Groups.

[5] T. Nomura, Berezin transforms and Laplace-Beltrami operators on homogeneous Siegel domains, to appear in Differential Geom. Appl.

[6] T. Nomura, A family of Cayley transforms of a homogeneous Siegel domain parametrized by admissible linear forms, preprint (revised version, June 2001).

[7] T. Nomura, Geometric connection of the Poisson kernel with a Cayley transform for homogeneous Siegel domains, in preparation.

[8] Xu Yichao, On the Bergman kernel function of homogeneous bounded domains, Scientia Sinica, Special Issue (II) (1979), 80–90.