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Laura Geatti

Publications and source records attributed to Laura Geatti.

9 recordsLinked to original sources
arXiv2022

Geometry of Hermitian symmetric spaces under the action of a maximal unipotent group

Let $\,G/K\,$ be a non-compact irreducible Hermitian symmetric space of rank $\,r\,$ and let $\,NAK\,$ be an Iwasawa decomposition of $\,G$. By the polydisc theorem, $\,AK/K\,$ can be regarded as the base of an $\,r$-dimensional tube domain holomorphically embedded in $\,G/K$. As every $\,N$-orbit in $\,G/K\,$ intersects $\,AK/K$ in a single point, there is a one-to-one correspondence between $\,N$-invariant domains in $\,G/K\,$ and tube domains in the product of $\,r\,$ copies of the upper half-plane in $\,\C$. In this setting we prove a generalization of Bochner's tube theorem. Namely, an $\,N$-invariant domain $\,D\,$ in $\,G/K\,$ is Stein if and only if the base $\,\Omega\,$ of the associated tube domain is convex and ``cone invariant". We also obtain a precise description of the envelope of holomorphy of an arbitrary holomorphically separable $\,N$-invariant domain over $\,G/K$.

math.CV↗
arXiv2019

Invariant plurisubharmonic functions on non-compact Hermitian symmetric spaces

Let G/K be an irreducible Hermitian symmetric space and let D be a K-invariant domain in G/K. In this paper we characterize several classes of K-invariant plurisubharmonic functions on D in terms of their restrictions to a slice intersecting all K-orbits. As applications we show that K-invariant plurisubharmonic functions on D are necessarily continuous and we reproduce the classification of Stein K-invariant domains in G/K obtained by E. Bedford and J. Dadok.

math.CV↗
arXiv2017

The adapted hyper-Kähler structure on the crown domain

Let $\,Ξ\,$ be the crown domain associated with a non-compact irreducible hermitian symmetric space $\,G/K$. We give an explicit description of the unique $\,G$-invariant adapted hyper-Kähler structure on $\,Ξ$,$\ $i.$\,$e.$\ $compatible with the adapted complex structure $\,J_{ad}\,$ and with the $\,G$-invariant Kähler structure of $\,G/K$. We also compute invariant potentials of the involved Kähler metrics and the associated moment maps.

math.CV↗
arXiv2017

Polar symplectic representations

We study polar representations in the sense of Dadok and Kac which are symplectic. We show that such representations are coisotropic and use this fact to give a classification. We also study their moment maps and prove that they separate closed orbits. Our work can also be seen as a specialization of some of the results of Knop on multiplicity free symplectic representations to the polar case.

math.RT↗
arXiv2013

Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space

In this paper we investigate invariant domains in $\, Ξ^+$, a distinguished $\,G$-invariant, Stein domain in the complexification of an irreducible Hermitian symmetric space $\,G/K$. The domain $\,Ξ^+$, recently introduced by Krötz and Opdam, contains the crown domain $\,Ξ\,$ and it is maximal with respect to properness of the $\,G$-action. In the tube case, it also contains $\,S^+$, an invariant Stein domain arising from the compactly causal structure of a symmetric orbit in the boundary of $\,Ξ$. We prove that the envelope of holomorphy of an invariant domain in $\,Ξ^+$, which is contained neither in $\,Ξ\,$ nor in $\,S^+$, is univalent and coincides with $\,Ξ^+$. This fact, together with known results concerning $\,Ξ\,$ and $\,S^+$, proves the univalence of the envelope of holomorphy of an arbitrary invariant domain in $\,Ξ^+\,$ and completes the classification of invariant Stein domains therein.

math.CV↗
arXiv2008

Polar orthogonal representations of real reductive algebraic groups

We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generalizes some results of the structural theory of real reductive Lie algebras.

math.RT↗
arXiv2006

On univalence of equivariant Riemann domains over the complexification of a non-compact, Riemannian symmetric space

Let G/K be a non-compact, rank-one, Riemannian symmetric space and let G^C be the universal complexification of G. We prove that a holomorphically separable, G-equivariant Riemann domain over G^C / K^C is necessarily univalent, provided that G is not a covering of SL(2, R). As a consequence of the above statement one obtains a univalence result for holomorphically separable, G x K -equivariant Riemann domains over G^C. Here G x K acts on G^C by left and right translations. The proof of such results involves a detailed study of the G-invariant complex geometry of the quotient G^C / K^C, including a complete classification of all its Stein G-invariant subdomains.

math.CV↗
arXiv2006

Complex extensions of semisimple symmetric spaces

Let G/H be a pseudo-Riemannian semisimple symmetric space. The tangent bundle T(G/H) contains a maximal G-invariant neighbourhood of the zero section where the adapted complex structure exists. Such neighbourhood is endowed with a canonical G-invariant pseudo-Kaehler metric of the same signature as the metric on G/H. We use the polar map from T(G/H) to the complexified symmetric space G^C/H^C to define a G-invariant pseudo-Kaehler metric on distinguished G-invariant domains in G^C/H^C or on coverings of principal orbit strata in G^C/H^C.

math.CV↗