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...
