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The Berezin form on symmetric $R$-spaces and reflection positivity | | Jan Möllers
; Gestur Ólafsson
; Bent Ørsted
; | | Date: |
2 May 2017 | | 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. | | Source: | arXiv, 1705.0874 | | Services: | Forum | Review | PDF | Favorites | |
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