Analytic deformations of equivariant genera, universal symmetry blocks and Witten's rigidity

Abstract

This paper is an extension of [9], where we have established a fixed-point formula for the formal t-deformations ind,(D, M) of the G-index ind(D, M) (G being a compact Lie group) of a basic differential operator D on a closed G-manifold M. The most important examples of D are the Signature, Dirac, and Euler-Todd operators. This deformation ind,(D, M) takes value in the ring %!(G)[ [t]], g’(G) denoting the algebra of central functions on G, @-spanned by the characters of the complex G-representations. The deformation is produced with the help of a stable exponential operation 4: K,( -) + @ @ KG( -)[ [t]] in the equivariant complex K-theory. In fact, these operations are in one-to-one correspondence with the elements cpt(u) E C [u, u- ‘1 [[t]], cpO(u) being invertible in C [u, u- ‘1. In the text we can also use notation Q(M) for ind,(D, M). One can view ind,(D, M) as a formal sum C, ~ o Xjt’, where xj is a G-index of the operator D twisted by certain tensor fields on M. The type of the twist is prescribed by q,(u), j, and dim M (cf. (1.3) and (1.6) from Section 1).