Functional Analytic Background for a Theory of Infinite-Dimensional Reductive Lie Groups

Abstract

Motivated by the interesting and yet scattered developments in representation theory of Banach-Lie groups, we discuss several functional analytic issues which should underlie the notion of infinite-dimensional reductive Lie group: norm ideals, triangular integrals, operator factorizations, and amenability. What a reductive Lie group is supposed to beIntroduction. We approach the problem of finding an appropriate infinite-dimensional version of reductive Lie group. The discussion is motivated by the need to have a reasonably general setting where the representation theoretic properties of the classical Lie groups associated with the Schatten ideals of Hilbert space operators -in the sense of [dlH72]-can be investigated in a systematic way. Thus the theory of operator ideals (see e.g., [GK69], [GK70], [DFWW04], and [KW07]) provides the natural background for the present exposition.The ideas and methods of representation theory of finite-dimensional Lie groups cannot possibly be extended to the setting of Banach-Lie groups in a direct manner. Any attempt to do that fails because of some phenomena specific to the infinite-dimensional Lie groups: there exist Lie algebras that do not arise from Lie groups (see e.g., [Ne06] or [Be06]), closed subgroups of Lie groups need not be Lie groups in the relative topology (see e.g., [Up85]), one does not know how to construct smooth structures on homogeneous spaces unless one is able to find a direct complement of the Lie subalgebra in the ambient Lie algebra (see [Up85], [BP07], [Ga06], or [Be06]), there exists no Haar measure on topological groups which are not locally compact ([We40]), and finally every infinite-dimensional Banach space is the model space of some abelian Lie group without any non-trivial continuous representations (see [Ba83]). Nevertheless, the study of representation theoretic properties of some specific Banach-Lie groups has led to a number of interesting results; see for instance the papers [SV75], [Bo80], [Pi90], [Ne98], [NØ98], [BR07] or [BG07]. It seems reasonable to try to find out a class of Banach-Lie groups whose representations can be studied in a coherent fashion following the pattern of representation theory of finitedimensional reductive Lie groups. The aim of the present paper is to survey some of the ideas and notions that might eventually lead to the description of a class of Banach-Lie groups appropriate for the purposes of such a representation theory. The exposition is streamlined by a number of phenomena that play a central role in the classical theory of reductive Lie groups: Cartan decompositions, Iwasawa decompositions, Harish-Chandra decompositions, and existence of invariant measures. By way of describing appropriate versions of these phenomena in infinite dimensions, the paper provides a self-contained discussion of a number of functional analytic issues which should underlay the notion of infinite-dimensional reductive Lie group: triangular integrals, operator factorizations, and amenability.Finite-dimensional reductive Lie groups...

Iwasawa decompositions of some infinite-dimensional Lie groups

“…The proof can be achieved by using appropriate infinite-dimensional versions of some standard ideas from the theory of Iwasawa decompositions of reductive groups (specifically, see for instance the proofs of Theorems 6.31 and 6.46 in [Kn96]). We refer to Proposition 4.4 in [Be09] for details. Proof.…”

“…We refer to [GK69], [GK70], [Er72], [EL72], [Er78], [KW02], [We05], [DFWW], [KW06], [Be06], and [Be09] for various special topics involving symmetric norm ideals related to the circle of ideas discussed here.…”

“…[Be09]) is related to the place held by reductive structures in the geometry of many infinite-dimensional manifolds; see, for instance, [CG99] and [Nee02b]. We refer also to the recent survey [Ga06], which skillfully highlights the special relationship between the reductive structures and the idea of amenability.…”

“…In particular we address an old conjecture on the existence of such decompositions for the classical Banach-Lie groups of operators associated with the Schatten operator ideals on Hilbert spaces (see subsection 8.4 of Section II.8 in [Ha72] and Section 3 below), and we show that the corresponding question can be answered in the affirmative in many cases, even in the case of the covering groups (Corollary 7.2 below). To place these results in a proper perspective, we mention that part of our motivation comes from the problem of describing an appropriate class of infinite-dimensional reductive Lie groups, as discussed in the paper [Be09].…”

“…For the reader's convenience we recorded in Appendix A some auxiliary facts on operator ideals, in particular the factorization results suitable for our purposes. We refer to the paper [Be09] for more details. * + ξ j whenever 0 ≤ j < ω}, and N := {n ∈ G | nξ j ∈ ξ j + span {ξ l | l < j} whenever 0 ≤ j < ω}.…”