Index of transversally elliptic \({\mathcal D}\)-modules (Q2731039)

The author studies the action of a complex Lie group \(G\) on a complex manifold \(X\). For a \(G\)-quasi-equivariant \({\mathcal D}_X\)-module \({\mathcal M}\) and an \(\mathbb{R}\)-constructible sheaf \(F\) on \(X\), equivariant under the action of a real form \(G_\mathbb{R}\) of \(G\), a transversal ellipticity assumption on the characteristic varieties of \({\mathcal M}\) and \(F\) allows the construction of a hyperfunction on \(G_\mathbb{R}\), \(\chi=\chi (\varphi, {\mathcal M},F,u,v)\), by a microlocal product of characteristic classes. Here, generally, \(\varphi:Z\times X\to X\), \(X\) and \(Z\) complex manifolds with \(Z\) a complexification of a real submanifold \(Z_\mathbb{R}\), and \(\varphi_z:X\to X\), \(x\mapsto\varphi (z,x)\), smooth and proper. \(u\) and \(v\) are ``liftings'' of \(\varphi\) to sheaves. If \(Z_\mathbb{R}\) is compact, given an analytic form \(\omega\) on \(Z_\mathbb{R}\), then under suitable conditions on \({\mathcal M}\) and strong transversal ellipticity of \(({\mathcal M},F)\), a nuclear morphism \(S(u,v) (\omega\otimes.)\) is constructed with trace \(\int_{ Z_\mathbb{R}} \omega\cdot \chi(\varphi, {\mathcal M},F,u,v)\).NEWLINENEWLINENEWLINEIn particular, for actions of compact Lie groups, the cohomological index considered here is Atiyah's index of transversally elliptic operators. The given treatment leads also to a character formula of Kashiwara.