Reflection Positivity for the Complementary Series of $\text{SL}(2n,C)$

Abstract

We apply the concept of reflection positivity in euclidean quantum field theory to the complementary series of $\text{SL}(2n,C)$ 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 $\text{SL}(2n,C)$. The possible relation to the theory of noncommuting monodromy matrices appearing in the theory of integrable quantum systems is briefly discussed.