We study the extension problem for germs of holomorphic isometries $f:(D;x_0)\to (\Omega ;f\left(x_0\right))$ up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics $d{s}_D^2$ on $D$ and $d{s}_{\Omega }^2$ on $\Omega$. Our main focus is on boundary extension for pairs of bounded domains $(D,\Omega )$ such that the Bergman kernel $K_D(z,w)$ extends meromorphically in $(z,\overline{w})$ to a neighborhood of $\overline{D}\times D$, and such that the analogous statement holds true for the Bergman kernel $K_{\Omega }(\varsigma ,\xi )$ on $\Omega$. Assuming that $(D;d{s}_D^2)$ and $(\Omega ;d{s}_{\Omega }^2)$ are complete Kähler manifolds, we prove that the germ...

A Gutzmer formula for the complexification of a Riemann symmetric space. We consider a complex manifold $\mathrm{\Omega }$ and a real Lie group $G$ of holomorphic automorphisms of $\mathrm{\Omega }$. The question we study is, for a holomorphic function $f$ on $\mathrm{\Omega }$, to evaluate the integral of ${\left|f\right|}^2$ over a $G$-orbit by using the harmonic analysis of $G$. When $\mathrm{\Omega }$ is an annulus in the complex plane and $G$ the rotation group, it is solved by a classical formula which is sometimes called Gutzmer’s formula. We establish a generalization of it when $\mathrm{\Omega }$ is...

For the scalar holomorphic discrete series representations of $\mathrm{SU}(2,2)$ and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside $\mathrm{SU}(2,2)$. We construct a Cayley transform between the Ol’shanskiĭ semigroup having $\mathrm{U}(1,1)$ as Šilov boundary and an open dense subdomain of the Hermitian symmetric space for $\mathrm{SU}(2,2)$. This allows calculating the composition series in terms of harmonic analysis on $\mathrm{U}(1,1)$. In particular we show that the Ol’shanskiĭ Hardy space for $\mathrm{U}(1,1)$ is different...

Consider a complex projective space with its Fubini-Study metric. We study certain one parameter deformations of this metric on the complement of an arrangement (= finite union of hyperplanes) whose Levi-Civita connection is of Dunkl type. Interesting examples are obtained from the arrangements defined by finite complex reflection groups. We determine a parameter interval for which the metric is locally of Fubini-Study type, flat, or complex-hyperbolic. We find a finite subset of this interval for...

Let $G$ be a quasi-Hermitian Lie group with Lie algebra $𝔤$ and $K$ be a compactly embedded subgroup of $G$. Let ${\xi }_0$ be a regular element of ${𝔤}^{*}$ which is fixed by $K$. We give an explicit $G$-equivariant diffeomorphism from a complex domain onto the coadjoint orbit $𝒪\left({\xi }_0\right)$ of ${\xi }_0$. This generalizes a result of [B. Cahen, Berezin quantization and holomorphic representations, Rend. Sem. Mat. Univ. Padova, to appear] concerning the case where $𝒪\left({\xi }_0\right)$ is associated with a unitary irreducible representation of $G$ which is holomorphically...

Nous présentons quelques résultats au sujet des groupes engendrés par trois involutions antiholomorphes dans le cadre du plan hyperbolique complexe ${\mathbf{H}}_{ℂ}^2$.

Let G be the Banach-Lie group of all holomorphic automorphisms of the open unit ball $$B_{𝔄}$$ in a J*-algebra $$𝔄$$ of operators. Let $$𝔉$$ be the family of all collectively compact subsets W contained in $$B_{𝔄}$$ . We show that the subgroup F ⊂ G of all those g ∈ G that preserve the family $$𝔉$$ is a closed Lie subgroup of G and characterize its Banach-Lie algebra. We make a detailed study of F when $$𝔄$$ is a Cartan factor.