# Reflection Positivity for the Complementary Series of <i>SL</i>(2<i>n</i>, <i>ℂ</i>)

### Robert Schrader

Freie Universität Berlin, Germany

## Abstract

We apply the concept of reflection positivity in euclidean quantum field theory to the complementary series of S_SL_(2_n_, *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*(2_n_, *ℂ*. The possible relation to the theory of noncommuting monodromy matrices appearing in the theory of integrable quantum systems is briefly discussed.