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Higher localization and higher branching laws for Harish-Chandra modules

Abstract 
The higher branching laws proposed by Dipendra Prasad concern the Ext groups for admissible representations under restriction, but only p-adic groups were considered. I will formulate an algebraic version of this problem for the Archimedean setting, namely for Harish-Chandra (g, K)-modules restricted to reductive spherical subgroups. By localization to the corresponding homogeneous space, those Ext groups can be interpreted in terms of the equivariant derived category of D-modules. In particular, they turn out to be finite-dimensional, which partly recovers the recent results of Kitagawa. The main ingredient include: (1) a special flavor of equivariant derived categories due to Beilinson-Ginzburg, and (2) regularity of the cohomologies of derived localization, which might be of independent interest. I will also discuss the analytic aspect of this problem.