[Paper Review] $L^2$-index formula for proper cocompact group actions

W Hang|arXiv (Cornell University)|Jun 22, 2011

Advanced Operator Algebra Research16 references3 citations

TL;DR

This paper establishes an $L^2$-index formula for $G$-invariant elliptic pseudo-differential operators on a complete Riemannian manifold $X$ with a proper, cocompact, isometric action by a unimodular locally compact group $G$. Using the heat kernel method and $K$-theory, it proves that the $L^2$-index equals the integral of the product of the $ ext{ch}( ext{symbol})$, $ ext{ch}( ext{curvature})$, and $ ext{ch}( ext{twist})$ over the tangent bundle, generalizing Atiyah’s and Connes–Moscovici’s formulas in the non-compact setting.

ABSTRACT

We study the index of the $G$-invariant elliptic pseudo-differential operator acting on a complete Riemannian manifold, where a unimodular, locally compact group $G$ acts properly and cocompactly. An $L^2$-index formula was obtained using the heat kernel method.

Motivation & Objective

• To generalize the $L^2$-index formula to non-compact group actions where $G$ acts properly, cocompactly, and isometrically on a complete Riemannian manifold $X$.
• To establish a topological formula for the $L^2$-index of $G$-invariant elliptic operators using the $G$-trace and von Neumann algebra techniques.
• To unify and extend previous results, including Atiyah’s $L^2$-index formula for free cocompact actions and Connes–Moscovici’s formula for homogeneous spaces of unimodular Lie groups.
• To clarify the distinction between $L^2$-indices and orbifold indices when $X/G$ is an orbifold, particularly in cases where the former is rational and the latter is integer-valued.

Proposed method

• Uses the $G$-trace on $L^2(X,E)$ to define the $L^2$-index as the difference of von Neumann traces of projections onto $ ext{Ker}(P_0)$ and $ ext{Ker}(P_0^*)$.
• Reduces the $L^2$-index computation to a Dirac-type operator via Kasparov’s $K$-theory framework, preserving the higher index in $K_0(C^*(G))$.
• Applies the heat kernel method to compute the $L^2$-index of the Dirac operator, leveraging the invariance of curvature and characteristic classes under group action.
• Lifts the integration over $TX$ to a $G$-invariant form on $G imes V$, using the $H$-invariant splitting $rak{g} = rak{h} imes rak{m}$ and curvature forms on the homogeneous space $G/H$.
• Uses the Chern character and $ ext{ch}( ext{symbol})$ to express the index as an integral over the tangent space, with $ ext{ch}( ext{curvature})$ computed via the representation ring $R(H)$ and the Thom isomorphism.
• Establishes equivalence between the general formula and the Connes–Moscovici formula by showing that $ ext{ch}( ext{symbol})_V = ext{ch}(a)$ and $ ext{ch}( ext{curvature})_V = ext{ch}( ext{curvature form on } V)$, using topological invariance of $ ext{ch}$ and $ ext{ch}( ext{curvature})$.

Experimental results

Research questions

• RQ1 How can the $L^2$-index of a $G$-invariant elliptic operator be computed topologically when $G$ acts properly, cocompactly, and isometrically on a complete Riemannian manifold $X$?
• RQ2 What is the precise relationship between the $L^2$-index and the higher index in $K_0(C^*(G))$ under such group actions?
• RQ3 In what sense does the $L^2$-index formula generalize Atiyah’s and Connes–Moscovici’s results in the non-compact setting?
• RQ4 Why does the $L^2$-index differ from the orbifold index when $X/G$ is an orbifold, and when do they coincide?
• RQ5 How can the heat kernel method be adapted to compute the $L^2$-index in the context of type II von Neumann algebras and $G$-invariant geometry?

Key findings

• The $L^2$-index of a $G$-invariant elliptic operator $P$ is given by the topological formula $ ext{ind}P = rac{1}{ ext{vol}(G)} imes ext{vol}(G) imes ext{int}_{TX} c ext{ch}( ext{symbol}) ext{ch}( ext{curvature}) ext{ch}( ext{twist})$, where $c$ is a $G$-invariant cutoff function with $ ext{int}_G c(g^{-1}x) dg = 1$.
• The $L^2$-index of a Dirac-type operator $D$ on $X$ is computed as $ ext{ind}D = ext{int}_{TM} c ext{ch}( ext{symbol}) ext{ch}( ext{curvature}) ext{ch}( ext{twist})$, with the integral reduced to $V$ via $G$-invariance.
• The formula $ ext{ind}D = ext{int}_V ext{ch}(a) ext{ch}( ext{curvature})$ matches the Connes–Moscovici formula when $a$ is the preimage of $V( ext{symbol})|_{V^+}$ under the Thom isomorphism $R(H) o K_H(V)$, and the $ ext{ch}$-class is independent of the choice of connection.
• The $ ext{ch}( ext{symbol})_V$ and $ ext{ch}( ext{curvature})_V$ are both $H$-invariant and equal to the Chern character of the representation $r:H o ext{GL}(E)$, showing that the index depends only on the $H$-representation and curvature.
• The $ ext{ch}( ext{curvature})$ class is topologically invariant: $ ext{ch}( ext{curvature})_V = ext{ch}( ext{curvature form on } V)$, regardless of the connection used, due to the topological nature of $ ext{ch}$ and $ ext{ch}( ext{curvature})$.
• The $L^2$-index is rational in general, while the orbifold index is integer-valued; they coincide only when $X/G$ is a smooth manifold, highlighting a key distinction in geometric and analytic index theory.