Index theory for non-compact \(G\)-manifolds (Q2928470)

This lecture note surveys the Atiyah-Singer index theorem and its generalisation to the equivariant case and to transversally elliptic operators, proposes an open question regarding index theory on noncompact manifolds with compact group actions and then introduced a solution to the open problem for a special class of manifolds [\textit{M. Braverman}, \(K\)-Theory 27, No. 1, 61--101 (2002; Zbl 1020.58020)]. The content of the note is summarised as follows.NEWLINENEWLINELet \(M\) be a compact manifold admitting an action of a compact group \(G\). A transversally elliptic operator \(A\) has an analytical index in the complete ring of characters of \(G\): NEWLINE\[NEWLINE\mathrm{ind}_G^a A=[\mathrm{ker} A]-[\mathrm{coker} A]\in \oplus_{V\in \mathrm{Irr}(G)} (m_V^+-m_V^-)V\in \hat R(G).NEWLINE\]NEWLINE Here, \(m_V^+\) (resp., \(m_V^-\)) stands for the multiplicity of the irreducible representation \(V\) to \(G\) in \(\mathrm{ker} A\) (resp., \(\mathrm{coker} A\)) and is a finite integer. Note that \(m^{\pm}_V\) may not vanish for infinitely many \(V\in \mathrm{Irr}(G)\). \textit{M. F. Atiyah} [Elliptic operators and compact groups. Lect. Notes Math. 401. Berlin-Heidelberg-York: Springer-Verlag (1974; Zbl 0297.58009)] showed that the analytical index of \(A\) coincides with the topological index NEWLINE\[NEWLINE\mathrm{ind}_G^t A \in \hat R(G)NEWLINE\]NEWLINE depending only on its principal symbol \([\sigma_A]\in K^*_G(T^*_GM).\)NEWLINENEWLINEWhen the manifold \(M\) is noncompact, the symbol of a transversally elliptic operator no longer represents a class in \(K^*_G(T^*_GM)\) while the topological index map \(K^*_G(T^*_GM)\rightarrow \hat R(G)\) still makes sense [\textit{R. S. Palais}, Seminar on the Atiyah-Singer index theorem. Princeton, N. J.: Princeton University Press (1965; Zbl 0137.17002)]. In this context, the note proposes the open question: Find a \(G\)-invariant operator \(A\) where the topological index of its principal symbol \(\sigma_A\) represents a class in \(K^*_G(T^*_GM)\) and NEWLINE\[NEWLINE\mathrm{ind}_G^a(A)=\mathrm{ind}_G^t(\sigma_A)\in\hat R(G).NEWLINE\]NEWLINE When \(M\) has a vector field \({\mathbf v}\) in the image of the natural \(G\)-equivariant map \(\mathfrak{g}\rightarrow \Gamma(TM)\) satisfying the condition that \({\mathbf v}\) is nonvanishing outside some noncompact subset of \(M\), it turns out that an operator \(A\) can be explicitly constructed using the deformed Dirac operator NEWLINE\[NEWLINEA=D+ic(fv)NEWLINE\]NEWLINE where \(D\) is the Dirac operator on \(M\), \(f\) is some carefully chosen smooth nonnegative \(G\)-invariant function on \(M\) and \(c\) is the Clifford multiplication.NEWLINENEWLINEThe lecture notes is a good survey of the Atiyah-Singer index theorem and further developments of transversally elliptic operators.NEWLINENEWLINEFor the entire collection see [Zbl 1290.81004].