Nonunitary representations of the groups of $U(p,q)$-currents for $q\geq p>1$

TL;DR

This paper constructs nonunitary representations of current groups for semisimple groups of rank greater than one, using cohomology and Iwasawa subgroups, as unitary representations are not possible in this context.

Contribution

It introduces a novel method to build nonunitary representations of current groups for semisimple groups with no nontrivial cohomology in faithful irreducible representations.

Findings

Constructed cohomology of semisimple groups in nonunitary representations

Reduced constructions to Iwasawa subgroups for extension

Realized representations in quasi-Poisson Hilbert spaces

Abstract

The purpose of this paper is to give a construction of representations of the group of currents for semisimple groups of rank greater than one. Such groups have no unitary representations in the Fock space, since the semisimple groups of this form have no nontrivial cohomology in faithful irreducible representations. Thus we first construct cohomology of the semisimple groups in nonunitary representations. The principal method is to reduce all constructions to Iwasawa subgroups (solvable subgroups of the semisimple groups), with subsequent extension to the original group. The resulting representation is realized in the so-called quasi-Poisson Hilbert space associated with natural measures on infinite-dimensional spaces. Key words: Iwasawa subgroup, cohomology, group of currents, nonunitary representations.