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## Banach Center Publications

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

### Volume 113 / 2017

Banach Center Publications 113 (2017), 135-168 MSC: Primary 22E46; Secondary 43A85, 57S25. DOI: 10.4064/bc113-0-9

#### 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.

#### Authors

• Jan FrahmDepartment Mathematik
FAU Erlangen–Nürnberg
Cauerstr. 11
91058 Erlangen, Germany
e-mail
• Gestur ÓlafssonDepartment of Mathematics
Louisiana State University
Baton Rouge, LA 70803, USA
e-mail
• Bent ØrstedInstitut for Matematiske Fag
Aarhus Universitet