Let \(M\) be a complex manifold, \(\mathcal D_M\) the sheaf of holomorphic differential operators on \(M\), \(D^b(\mathcal D_M)\) the bounded derived category of \(\mathcal D_M\)-modules and \(D^b_{\mathrm{coh}}(\mathcal D_M)\) the full triangulated subcategory of \(D^b(\mathcal D_M)\) containing objects with coherent cohomologies. For a holomorphic diffeomorphism \(\gamma\) on \(M\) and a \(\mathcal D_M\)-module \(\mathcal{M}\), there is a natural functor \(\gamma_\ast: D^b_{\mathrm{coh}}(\mathcal D_M)\longrightarrow D^b_{\mathrm{coh}}(\mathcal D_M)\). Given an element \(u\in \mathrm{Hom}_{\mathcal D_M}(\mathcal{M},\gamma_\ast(\mathcal{M}))\) the authors introduce a Hochschild Lefschetz class \(hh^\gamma(\mathcal{M},u)\in H^0(X,\mathcal{HH}(\hat{\mathcal{E}}_X,\hat{\mathcal{E}}^\gamma_X))\), where \(X=T^\ast M\), \(\hat{\mathcal{E}}_X\) is the sheaf of formal microdifferential operators on \(X\), and \(\mathcal{HH}\) stands for Hochschild homology. They prove that it satisfies nice formulas under the direct image functor thus generalizing the Kashiwara-Schapira Hochschild class (case \(\gamma=\mathrm{id}\)).
Let then \(\Gamma\) be a finite group acting on \(M\) by holomorphic diffeomorphisms. Every \(\gamma\in\Gamma\) defines an element \(\gamma\in \mathrm{Hom}_{\mathcal D_M}(\mathcal{M},\gamma_\ast(\mathcal{M}))\). Using the expression \(\displaystyle \frac{1}{|\Gamma|}\sum_{\gamma\in\Gamma}hh^\gamma(\mathcal{M},\gamma)\) the authors introduce the orbifold Hochschild class of \(\mathcal{M}\) on the quotient orbifold \(Q_X=X/\Gamma\) and prove a Riemann-Roch formula for the Euler class of a good \(\Gamma\)-equivariant coherent \(\mathcal D_M\)-module \(\mathcal{M}\) on \(M\), for \(M\) compact.