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

Abstract: Given a complex manifold $M$ with a Lie group $G$ action by holomorphic transformations, it is of interest to understand associated invariant objects like the invariant Stein subdomains and the invariant plurisubharmonic functions.

A classical example of this framework is given by tube domains in complex Euclidean space, where $M={\bf C}^n$ and $G={\bf R}^n$ acts by translations.

An ${\bf R}^n$-invariant domain $D={\bf R}^n+i\Omega$ in ${\bf C}^n$ is Stein if and only if its base $\Omega$ is geometrically convex (Bochner's tube theorem). Moreover an ${\bf R}^n$-invariant function on a Stein tube domain $D$ is plurisubharmonic if and only if its restriction to $\Omega$ is convex.

In this talk, we present a generalization of the above results in the setting of a Hermitian symmetric space of the non-compact type $G/K$ under the action of a maximal unipotent subgroup $N\subset G$. As a by-product we obtain all $N$-invariant potentials of the Bergman metric of $G/K$ in a Lie theoretical fashion and an explicit formula for the moment maps $\mu\colon G/K\to {\mathfrak n}^*$ associated to such potentials.

This is work in collaboration with Andrea Iannuzzi.

differential geometry

Audience: researchers in the topic

( video )

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Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Differential geometry.

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