The theory of D-modules deals with the algebraic aspects of
differential equations. These are particularly interesting on
homogeneous manifolds, since the infinitesimal action of a Lie
algebra consists of differential operators. Hence, it is possible
to attach geometric invariants, like the support and the
characteristic variety, to representations of Lie groups. By
considering D-modules on flag varieties, one obtains a simple
classification of all irreducible admissible representations of
reductive Lie groups. On the other hand, it is natural to study the
representations realized by functions on pseudo-Riemannian
symmetric spaces, i.e., spherical representations. The problem is
then to describe the spherical representations among all
irreducible ones, and to compute their multiplicities. This is the
goal of this work, achieved fairly completely at least for the
discrete series representations of reductive symmetric spaces. The
book provides a general introduction to the theory of D-modules on
flag varieties, and it describes spherical D-modules in terms of a
cohomological formula. Using microlocalization of representations,
the author derives a criterion for irreducibility. The relation
between multiplicities and singularities is also discussed at
length.
Originally published in 1990.
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