Higher localization and higher branching laws

For a connected reductive group $G$ and an affine smooth $G$-variety $X$ over the complex numbers, the localization functor takes $\mathfrak{g}$-modules to $D_X$-modules. We extend this construction to an equivariant and derived setting using the formalism of h-complexes due to Beilinson-Ginzburg, and show that the localizations of Harish-Chandra $(\mathfrak{g},K)$-modules onto $X=H\mathrm{\backslash }G$ have regular holonomic cohomologies when $H,K\subset G$ are both spherical reductive subgroups. The relative Lie algebra homologies and $\mathrm{E}\mathrm{x}\mathrm{t}$-branching spaces for $(\mathfrak{g},K)$-modules are interpreted geometrically in terms of equivariant derived localizations. As direct consequences, we show that they are finite-dimensional under the same assumptions, and relate Euler-Poincaré characteristics to local index theorem; this recovers parts of the recent results of M. Kitagawa. Examples and discussions on the relation to Schwartz homologies are also included.