## $${\mathcal D}$$-modules and characters of semisimple Lie groups.(English)Zbl 1057.22018

The authors study some properties of the semisimple Lie groups related to the eigendistributions invariant under the adjoint action and the relation with the Hotta-Kashiwara $$\mathcal{D}$$-module. Let $$G_{{R}}$$ be a real semisimple Lie group. The Harish-Chandra theorem asserts that all invariant eigendistributions on $$G_{{R}}$$ are locally integrable functions. The authors show that the Harish-Chandra theorem and its extension to symmetric pairs are consequences of an algebraic property of the Hotta-Kashiwara $$\mathcal{D}$$-module. If $$\mathcal{M}$$ is a holonomic $$\mathcal{D}_X$$-module on a manifold $$X$$, then to each submanifold $$Y$$ of $$X$$ a polynomial is associated that is called the $$b$$-function of $$\mathcal{M}$$ along $$Y$$. The module $$\mathcal{M}$$ is tame if there exists a locally finite stratification $$X=\bigcup X_a$$ such that for each $$a$$, the roots of the $$b$$-function of $$\mathcal{M}$$ along $$X_a$$ are greater than the opposite of the codimension of $$X_a$$.
The authors show that the Hotta-Kashiwara module is tame in the semisimple case, and in the case of symmetric pairs, they find a relation between the roots of the $$b$$-functions and some numbers introduced by Sekiguchi. Concerning the integrability of the solutions in the case of symmetric pairs, the authors obtain this property under a condition that is slightly stronger than the condition satisfied by nice pairs. The tame $$\mathcal{D}$$-modules have no quotients supported by a hypersurface. The $$L^p$$-growth of the solutions is given by the roots of the $$b$$-functions. The authors present an example of nontrivial holonomic $$\mathcal{D}$$-modules for which it is possible to calculate the $$b$$-functions explicitly.

### MSC:

 22E46 Semisimple Lie groups and their representations 17B15 Representations of Lie algebras and Lie superalgebras, analytic theory 35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs 35D10 Regularity of generalized solutions of PDE (MSC2000)
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