# The Berezin form on symmetric $R$-spaces and reflection positivity

## TL;DR

This paper investigates the Berezin form on symmetric R-spaces, identifying conditions for positivity and unitarity of related representations, and explores their connection to reflection positivity in the context of symmetric spaces.

## Contribution

It introduces a new method to construct positive Berezin forms on symmetric R-spaces and links these forms to reflection positivity and unitary highest weight representations.

## Key findings

- Determined when the Berezin form is positive semidefinite.
- Identified the corresponding unitary highest weight representations.
- Connected the construction to reflection positivity.

## Abstract

For a symmetric $R$-space $K/L=G/P$ the standard intertwining operators provide a canonical $G$-invariant pairing between sections of line bundles over $G/P$ and its opposite $G/\overline{P}$. Twisting this pairing with an involution of $G$ which defines a non-compactly causal symmetric space $G/H$ we obtain an $H$-invariant form on sections of line bundles over $G/P$. Restricting to the open $H$-orbits in $G/P$ constructs the Berezin forms studied previously by G. van Dijk, S. C. Hille and V. F. Molchanov. We determine for which $H$-orbits in $G/P$ and for which line bundles the Berezin form is positive semidefinite, and in this case identify the corresponding representations of the dual group $G^c$ as unitary highest weight representations. We further relate this procedure of passing from representations of $G$ to representations of $G^c$ to reflection positivity.