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 |$X = H \backslash G$| have regular holonomic cohomologies when |$H, K \subset G$| are both spherical reductive subgroups. The relative Lie algebra homologies and |$\operatorname{Ext}$| -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. [ABSTRACT FROM AUTHOR]