Soit $V$ une algèbre de Jordan simple euclidienne de dimension finie et $\Omega$ le cône symétrique associé. Nous étudions dans cet article le semi-groupe $\Gamma$, naturellement associé à $V$, formé des automorphismes holomorphes du domaine tube $T_{\Omega }:=V+i\Omega$ qui appliquent le cône $\Omega$ dans lui-même.

We consider separately radial (with corresponding group ${𝕋}^n$) and radial (with corresponding group $\mathrm{U}\left(n\right))$ symbols on the projective space ${\mathbb{P}}^n\left(ℂ\right)$, as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the $C^{*}$-algebras generated by each family of such Toeplitz operators are commutative (see R. Quiroga-Barranco and A. Sanchez-Nungaray (2011)). We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the...

We study the simultaneous linearizability of $d$–actions (and the corresponding $d$-dimensional Lie algebras) defined by commuting singular vector fields in ${ℂ}^n$ fixing the origin with nontrivial Jordan blocks in the linear parts. We prove the analytic convergence of the formal linearizing transformations under a certain invariant geometric condition for the spectrum of $d$ vector fields generating a Lie algebra. If the condition fails and if we consider the situation where small denominators occur, then...

We show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of GL(n, ℂ) is isomorphic to the deformation of the D2-singularity if n = 2, the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if n = 3, and to the Atiyah-Hitchin manifold itself if n = 4. For higher n, such slices to the sum of two orbits, each having only two distinct eigenvalues, are either empty or biholomorphic to open subsets of the Hilbert scheme of points on one of the above surfaces....

Let $X$ denote either ${\mathrm{ℂ\mathbb{P}}}^m$ or ${ℂ}^m$. We study certain analytic properties of the space ${ℰ}^n(X,gp)$ of ordered geometrically generic $n$-point configurations in $X$. This space consists of all $q=(q_1,\cdots ,q_n)\in X^n$ such that no $m+1$ of the points $q_1,\cdots ,q_n$ belong to a hyperplane in $X$. In particular, we show that for a big enough $n$ any holomorphic map $f:{ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)\to {ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)$ commuting with the natural action of the symmetric group $𝐒\left(n\right)$ in ${ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)$ is of the form $f\left(q\right)=\tau \left(q\right)q=(\tau \left(q\right)q_1,\cdots ,\tau \left(q\right)q_n)$, $q\in {ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)$, where $\tau :{ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)\to \mathrm{𝐏𝐒𝐋}(m+1,ℂ)$ is an $𝐒\left(n\right)$-invariant holomorphic map. A similar result holds true for mappings of the configuration space ${ℰ}^n({ℂ}^m,gp)$.

We study the action of a real-reductive group $G=Kexp\left(𝔭\right)$ on a real-analytic submanifold $X$ of a Kähler manifold. We suppose that the action of $G$ extends holomorphically to an action of the complexified group $G^{ℂ}$ on this Kähler manifold such that the action of a maximal compact subgroup is Hamiltonian. The moment map induces a gradient map ${\mu }_{𝔭}:X\to 𝔭$. We show that ${\mu }_{𝔭}$ almost separates the $K$–orbits if and only if a minimal parabolic subgroup of $G$ has an open orbit. This generalizes Brion’s characterization of spherical...

We consider an action of a connected compact Lie group on a Stein manifold by holomorphic transformations. We prove that the manifold is spherical if and only if there exists an antiholomorphic involution preserving each orbit. Moreover, for a spherical Stein manifold, we construct an antiholomorphic involution, which is equivariant with respect to the Weyl involution of the acting group, and show that this involution stabilizes each orbit. The construction uses some properties of spherical subgroups...

Let $M=G/K$ be a Hermitian symmetric space of the noncompact type and let $\pi$ be a discrete series representation of $G$ holomorphically induced from a unitary character of $K$. Following an idea of Figueroa, Gracia-Bondìa and Vàrilly, we construct a Stratonovich-Weyl correspondence for the triple $(G,\pi ,M)$ by a suitable modification of the Berezin calculus on $M$. We extend the corresponding Berezin transform to a class of functions on $M$ which contains the Berezin symbol of $d\pi \left(X\right)$ for $X$ in the Lie algebra $𝔤$ of $G$. This allows...

We construct and study a Stratonovich-Weyl correspondence for the holomorphic representations of the Jacobi group.