Cohomology of Classifying Spaces and Hermitian Representations

It is shown that a large part of the cohomology of the classifying space of a Lie group satisfying certain hypotheses can be obtained by a difference construction from hermitian representations of that Lie group. This result is relevant to the study of Novikov’s higher signatures. 1. Statement of the Theorem 1.1. Let G be a connected Lie group. We set H G = H(BG,C) (cohomology with complex coefficients) where BG is a classifying space of G. We set H∗ G = H G×H G× H G × . . . ; we regard this as a topological C-algebra in which a fundamental system of neighbourhoods of 0 is provided by the subspaces 0×0× . . .×0×H l G×H l+1 G × . . . for various integers l ≥ 0. For any continuous finite dimensional complex representation V ′ of G, we can form the associated complex vector bundle on BG and consider its Chern character chV ′ ∈ H∗ G. It is well known that, in the case where G is compact, the elements chV ′ (for various V ′ as above) span over C a dense subspace of H∗ G. 1.2. How to extend this result to not necessarily compact groups? Let V be a finite dimensional C-vector space with a given non-degenerate hermitian form on which G acts linearly and continuously, preserving the hermitian form. We associate to V an element c̃hV ∈ H∗ G as follows. We choose a maximal compact subgroup K of G. We can find a direct sum decomposition V = V + ⊕ V − with V , V − orthogonal to each other for the hermitian form such that V , V − are K-invariant subspaces and the hermitian form is positive definite on V + and negative definite on V −. The Chern characters chV + ∈ H∗ K, chV − ∈ H∗ K are then well defined since V , V − are representations of K. We may identify H∗ K = H∗ G since the inclusion K → G induces a homotopy equivalence BK ∼ −→ BG ; we define c̃hV = chV + − chV − ∈ H∗ G. Note that the element c̃hV is independent of the choices of K and of the decomposition V = V + ⊕ V −, since the set of these choices is a contractible space. Our main result is the following: Received by the editors August 13, 1996 and, in revised form, August 21, 1996. 1991 Mathematics Subject Classification. Primary 20G99. Supported in part by the National Science Foundation. c ©1997 American Mathematical Society