# Holonomic equivariant tempered distributions in non-commutative harmonic analysis

Date(s) : 21/02/2017   iCal
14 h 00 min - 15 h 00 min

A holonomic distribution is a distribution that satisfies many differential equations. This notion was applied by Bernstein to construct distributions that are semi-invariant under a group action. This construction in turn gave the construction of standard intertwining operators for principal series representations of real reductive groups. I will recall these classical constructions and then describe a recent generalization by Sahi, Sayag and myself.
This generalization enables to construct standard intertwining operators on spherical pairs, as well as generalized Whittaker functionals on degenerate principal series. It also simplifies the classical construction of intertwining operators and Whittaker functionals.

Then I will formulate another classical theorem, due to Bernstein and Kashiwara, that states that the space of solutions of a holonomic D-module in tempered distributions is finite-dimensional, and give applications of this theorem to dimension bounds on the spaces of invariant distributions and on multiplicity bounds for spherical spaces.

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