[Paper Review] On the structure of K_G(T_G M)

Paul-Émile Paradan|arXiv (Cornell University)|Sep 18, 2012

Advanced Operator Algebra Research10 references3 citations

TL;DR

This paper provides a comprehensive exposition and refinement of the structure of equivariant K-theory groups $\mathbf{K}^*_{G}(\mathbf{T}_G^*M)$ for compact Lie group actions on manifolds, focusing on transversally elliptic operators. It establishes that for abelian $G$, $\mathbf{K}^0_G(\mathbf{T}_G^*V)$ is isomorphic to the generalized Dahmen-Michelli space $\mathrm{DM}_G(V)$, and $\mathbf{K}^0_G(\mathbf{T}_G^*V^{\mathrm{gen}})$ to the space $\mathcal{F}_G(V)$, offering a complete algebraic description of these groups via representation-theoretic and geometric data.

ABSTRACT

In this expository paper, we revisit the results of Atiyah-Singer and de Concini-Procesi-Vergne concerning the structure of the K-theory groups K_G(T_G M).

Motivation & Objective

• To re-express and clarify the structure of equivariant K-theory groups $\mathbf{K}^*_{G}(\mathbf{T}_G^*M)$ associated with transversally elliptic symbols.
• To establish isomorphisms between $\mathbf{K}^0_G(\mathbf{T}_G^*V)$ and generalized Dahmen-Michelli spaces $\mathrm{DM}_G(V)$ when $G$ is abelian.
• To provide a geometric and representation-theoretic characterization of generators and relations in $\mathbf{K}^*_{G}(\mathbf{T}_G^*V)$ using infinitesimal stabilizers and complex structures on quotient spaces.
• To unify and extend results from Atiyah-Singer and de Concini-Procesi-Vergne on the analytic index and its image in $R^{-\infty}(G)$.

Proposed method

• Uses the pushforward map $i_!$ to relate $\mathbf{K}^0_G(\mathbf{T}_G^*M)$ to $\mathbf{K}^0_G(\mathbf{T}_G^*V)$ for an embedding $M \hookrightarrow V$ into a $G$-representation space $V$, enabling reduction to the linear case.
• Applies the Bott-Thom isomorphism and index map $\mathrm{Index}^G_V$ to identify $\mathbf{K}^0_G(\mathbf{T}_G^*V)$ with $R^{-\infty}(G)$, leveraging the inverse of the Bott-Thom isomorphism.
• Introduces the generalized Dahmen-Michelli space $\mathcal{F}_G(V)$ and the submodule $\mathrm{DM}_G(V)$ as submodules of $R^{-\infty}(G)$ defined by support and cohomological conditions on representations.
• Employs deformation techniques and homotopy invariance of $K$-theory to show that certain symbol classes are equivalent, particularly through the use of Clifford multiplication and invertible perturbations.
• Uses the tangential $\overline{\partial}$-operator and Bott symbol constructions to realize the pushforward $i_!$ as a composition involving $\sigma_{\overline{\partial}}^V \odot \mathrm{Bott}(\mathbf{T}\mathbb{R})$, linking to the Thom isomorphism.
• Analyzes the support of families of symbols $\tau_s$ and $\sigma_s$ to prove homotopy invariance, showing that their supports intersect $\mathbf{T}_G^*V$ or $\mathbf{T}_G^*(S\times\mathbb{R})$ in compact sets, thus preserving $K$-theory classes.

Experimental results

Research questions

• RQ1 What is the precise algebraic structure of $\mathbf{K}^0_G(\mathbf{T}_G^*V)$ for a real $G$-module $V$ when $G$ is abelian?
• RQ2 How do the generalized Dahmen-Michelli spaces $\mathcal{F}_G(V)$ and $\mathrm{DM}_G(V)$ relate to the equivariant $K$-theory of $\mathbf{T}_G^*V$?
• RQ3 Can the index map $\mathrm{Index}^G_M: \mathbf{K}^0_G(\mathbf{T}_G^*M) \to R^{-\infty}(G)$ be fully described via generators and relations in the abelian case?
• RQ4 What role do infinitesimal stabilizers $\mathfrak{h} \in \Delta_G(V)$ and complex structures on $V/V^\mathfrak{h}$ play in defining the $R(G)$-module structure of $\mathbf{K}^0_G(\mathbf{T}_G^*V)$?
• RQ5 How can the pushforward $i_!$ and the Thom isomorphism be explicitly realized in terms of symbol deformations and Clifford modules?

Key findings

• For an abelian compact Lie group $G$ and a real $G$-module $V$, $\mathbf{K}^0_G(\mathbf{T}_G^*V)$ is isomorphic to the generalized Dahmen-Michelli space $\mathrm{DM}_G(V)$, which is defined as the intersection of $\langle R^{-\infty}(G/H_{\min}) \rangle$ with the kernel of $\wedge^\bullet \overline{V/V^{\mathfrak{h}}} \otimes \Phi$ for all $\mathfrak{h} \neq \mathfrak{h}_{\min}$.
• The space $\mathbf{K}^0_G(\mathbf{T}_G^*V^{\mathrm{gen}})$, where $V^{\mathrm{gen}}$ is the open dense subset of maximal orbits, is isomorphic to $\mathcal{F}_G(V)$, defined as the set of $\Phi \in R^{-\infty}(G)$ such that $\wedge^\bullet \overline{V/V^{\mathfrak{h}}} \otimes \Phi \in \langle R^{-\infty}(G/H) \rangle$ for all $\mathfrak{h} \in \Delta_G(V)$.
• The index map $\mathrm{Index}^G_V: \mathbf{K}^0_G(\mathbf{T}_G^*V) \to R^{-\infty}(G)$ identifies $\mathbf{K}^0_G(\mathbf{T}_G^*V)$ with $\mathrm{DM}_G(V)$, providing a complete algebraic description of the image of the analytic index.
• The pushforward $i_!$ from $\mathbf{T}_G^*M$ to $\mathbf{T}_G^*V$ is realized via a deformation of symbols to the $\overline{\partial}$-operator and Bott symbol, showing that $i_!(\sigma) = \sigma_{\overline{\partial}}^V \odot \mathrm{Bott}(\mathbf{T}\mathbb{R})$ in $K$-theory.
• The support of the symbol families $\tau_s$ and $\sigma_s$ is compact when restricted to $\mathbf{T}_G^*V$ or $\mathbf{T}_G^*(S\times\mathbb{R})$, proving that they define homotopies of transversally elliptic symbols and preserving $K$-theory classes.
• The difference $\mathrm{Thom}_{-\beta}(V) - \mathrm{Thom}_{\beta}(V)$ is represented by the symbol $\mathrm{Cl}(\xi') + \mathrm{Cl}(t + i\mathrm{Re}(z))$, which equals $i_!(\sigma_{\overline{\partial}}^V)$, confirming the compatibility of the Thom isomorphism with the index map.