Special representations of Iwasawa subgroups of simple Lie groups

A. M. Vershik ; M. I. Graev

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 96-106

Résumé

In the paper, a family of representations of maximal solvable subgroups of the simple Lie groups $O(p,q)$, $U(p,q)$, and $\mathrm{Sp}(p,q)$, where $1\leq p\leq q$, is introduced. These subgroups are called the Iwasawa subgroups of the corresponding simple groups. The main property of these representations is the existence of nontrivial $1$-cohomology with values in the representations. For groups of rank $1$, the representations from the family are unitary; for ranks greater than $1$, they are nonunitary. The paper continues a series of our previous papers and serves as an introduction to the theory of nonunitary current groups.