# Reflection Positivity for the Complementary Series of $SL(2n,C)$

### Robert Schrader

Freie Universität Berlin, Germany

## Abstract

We apply the concept of reflection positivity in euclidean quantum field theory to the complementary series of $SL(2n,C)$ as given by Gelfand and Neumark for $n=1$ and by Stein for $n>1$. The result is a virtual representation in the sense of Fröhlich, Osterwalder and Seiler or equivalently a strongly continuous representation of a closed subsemigroup by contractions on a new Hilbert space. Analytic continuation gives a unitary representation of a certain dual group of $SL(2n,C)$. The possible relation to the theory of noncommuting monodromy matrices appearing in the theory of integrable quantum systems is briefly discussed.

## Cite this article

Robert Schrader, Reflection Positivity for the Complementary Series of $SL(2n,C)$. Publ. Res. Inst. Math. Sci. 22 (1986), no. 1, pp. 119–141

DOI 10.2977/PRIMS/1195178376