Formal equivariant $\hat A$ class, splines and multiplicities of the index of transversally elliptic operators

TL;DR

This paper introduces a new formal equivariant  class to express the multiplicities of the index of transversally elliptic operators on manifolds with compact Lie group actions, using piecewise polynomial functions.

Contribution

It develops a novel formal equivariant  class and provides a formula for multiplicities of indices in terms of piecewise polynomial functions.

Findings

Constructed finite piecewise polynomial functions on Lie(T)*.

Derived a formula for index multiplicities using these functions.

Introduced the concept of formal equivariant  class.

Abstract

Let G be a connected compact Lie group acting on a manifold M and let D be a transversally elliptic operator on M. The multiplicity of the index of D is a function on the set of irreducible representations of G. Let T be a maximal torus of G with Lie algebra Lie(T). We construct a finite number of piecewise polynomial functions on the dual vector space Lie(T)*, and give a formula for the multiplicity in term of these functions. The main new concept is the formal equivariant $\hat A$ class.