Integral models of unitary representations of current groups with values in semidirect products

Abstract

We describe a general construction of irreducible unitary representations of the group of currents with values in the semidirect product of a locally compact subgroup P 0 by a one-parameter group R * + = {r : r > 0} of automorphisms of P 0 . This construction is determined by a faithful unitary representation of P 0 (canonical representation) whose images under the action of the group of automorphisms tend to the identity representation as r → 0. We apply this construction to the current groups of maximal parabolic subgroups in the groups of motions of the n-dimensional real and complex Lobachevsky spaces. The obtained representations of the current groups of parabolic subgroups uniquely extend to the groups of currents with values in the groups O(n, 1) and U (n, 1). This gives a new description of the representations, constructed in the 1970s and realized in the Fock space, of the current groups of the latter groups. The key role in our construction is played by the so-called special representation of the parabolic subgroup P and a remarkable σ-finite measure (Lebesgue measure) L on the space of distributions.In 1972, on the initiative of I. M. Gelfand, a series of papers by three authors (Israel Moiseevich himself and the present authors) on unitary representations of functional groups, or current groups, was started. The problem that was posed by Gelfand to the first author of this paper in spring 1972 and which initiated this series of papers was to find out whether there exists a "multiplicative integral of representations" (see below) for the group SL(2, R). Soon it became clear that the answer is positive, and the first paper [2] on the topic described several models of this representation. The main idea was to study a neighborhood of the identity representation of the group SL(2, R) itself and the so-called special (infinitesimal) representation of this group, whose first cohomology is nontrivial. A multiplicative integral of representations is, in the authors' words, a "tensor product of infinitely many infinitesimal representations." In the subsequent joint papers, various generalizations of this construction (to simple Lie groups of rank 1, for groups of diffeomorphisms, etc.) were found, and a technique for working with such representations was developed. Starting from 2004, the authors of the present paper have been carrying out a systematic study of integral models of representations of current groups on the basis of a new interpretation of the continual tensor product; this interpretation is different from the Fock one and essentially exploits a remarkable σ-finite measure in the space of distributions. A systematic presentation of the whole area will be given in the book The Representation Theory of Current Groups by the three authors, which is now in preparation.

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

“…For such groups, the corresponding theory was constructed in [2,3] and other papers in the 1970s-1980s. After some break, this research was continued in our subsequent papers; here is the complete list of them: [13,14,15,16,17,18,19,20,21,22,23,24,25]. In particular, a new method, outlined in [4], was developed to realize these representations, which instead of the Fock space uses more natural probabilistic constructions: the integral, Poisson, and quasi-Poisson models.…”

Анатолий Моисеевич Вершик (К Восьмидесятилетию Со Дня Рождения)

“…Одним из главных направлений исследований А. М. Вершика оставалась теория представлений. Длинный цикл его работ с М. И. Граевым [203], [204], [210], [229], [230], [239], [249], [252], [257] продолжает направление, начатое известной статьей [26] 2 об унитарных представлениях группы измеримых токов SL(2, R) X . Новым в этих работах является изучение реализации представлений, основанной на так называемой бесконечномерной мере Лебега L. Мера L связана с распределениями Пуассона-Дирихле и является замечательным примером сигма-конечной меры с бесконечномерной группой симметрий.…”