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Index theorem for homogenous spaces of Lie groups

Hang Wang, Zijing Wang

Abstract

We study index theory on homogeneous spaces associated to an almost connected Lie group in terms of the topological aspect and the analytic aspect. On the topological aspect, we obtain a topological formula as a result of the Riemann-Roch formula for proper cocompact actions of the Lie group, inspired by the work of Paradan and Vergne. On the analytic aspect, we apply heat kernel methods to obtain a local index formula representing the higher indices of equivariant elliptic operators with respect to a proper cocompact actions of the Lie group.

Index theorem for homogenous spaces of Lie groups

Abstract

We study index theory on homogeneous spaces associated to an almost connected Lie group in terms of the topological aspect and the analytic aspect. On the topological aspect, we obtain a topological formula as a result of the Riemann-Roch formula for proper cocompact actions of the Lie group, inspired by the work of Paradan and Vergne. On the analytic aspect, we apply heat kernel methods to obtain a local index formula representing the higher indices of equivariant elliptic operators with respect to a proper cocompact actions of the Lie group.
Paper Structure (14 sections, 48 theorems, 288 equations)

This paper contains 14 sections, 48 theorems, 288 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivariant $K$-theory for proper actions
  4. Equivariant cohomology
  5. Equivariant Chern character
  6. Equivariant Riemann-Roch formula
  7. $G$-invariant Riemann-Roch formula on homogeneous spaces
  8. Local index formula and Getzler’s symbolic calculus
  9. Pairing between $C^{*}_{v}(M)^{G}$ and $S^{u}_{G}(M,E)$
  10. Schwartz kernel estimates for smoothing operators
  11. Higher index formula and Pairing between $K$-theory and cohomology
  12. Higher index formula
  13. Pairing between $K$-theory and cohomology
  14. Equivalence of topological index and analytic index

Key Result

Theorem 1.1

Let $G$ be an almost-connected Lie group, $M$ be a proper $G$-compact space satisfying conditions in Theorem sk, and $V$ be a $G$-vector bundle (even rank) over $M$ with an equivariant spin structure. The following diagram commutes: \xymatrix{ K^{0}_{G}(V)\ar[r]^{\pi!}\ar[d]_{\mathrm{ch_{G}}} &

Theorems & Definitions (139)

  • Theorem 1.1: Theorem \ref{['err']}
  • Theorem 1.2: Theorem \ref{['mthm2']}
  • Theorem 1.3: Theorem \ref{['mth']}
  • Definition 2.1: ref7 $G$-Hilbert bundle
  • Definition 2.2: ref7
  • Definition 2.3: p1
  • Definition 2.4: Finite dimensional (smooth) $K$-cocycle
  • Definition 2.5: equivariant (smooth) $K$ theory
  • Remark 2.6
  • Lemma 2.7
  • ...and 129 more