The graded equivariant Todd class and the equivariant index of transversally elliptic operators
Also appears in collection : Algebraic Analysis in honor of Masaki Kashiwara's 70th birthday
Let $G$ be a compact connected Lie group acting on a manifold $M$. Let $\sigma\in K_G(T^*_GM)$ be a transversally elliptic symbol. Thus $Index(\sigma)=\sum_{\lambda\in\hat{G}} m(\sigma,\lambda)V_\lambda$ is a (infinite) sum of irreducible representations $V_\lambda$ of $G$. Considering $\hat{G}$ as a subset of $\mathfrak{t}^\ast$, we produce a $W$ anti invariant piecewise polynomial function $\xi\to m_{\mathrm{geo}}(\sigma,\xi)$ on $\mathfrak t^\ast$, determined by the Chern character of $\sigma$ and the equivariant Todd class, coinciding with $m(\sigma,\lambda)$ on $\hat{G}$. Furthermore, if $M$ is a spin manifold, and $\sigma_k$ is the Dirac operator twisted by a line bundle $L^k$ with proper moment map, we compute the asymptotics when $k\to\infty$ of the distribution $\sum_\lambda m(\sigma_k,\lambda)\delta_{\lambda/k}$ in terms of the formal expansion $\sum_{n=0}^{\infty}\mathrm{Todd}_n(M)$ of the equivariant Todd class in the graded equivariant cohomology ring of $M$.
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