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D-modules and spherical representations. (English) Zbl 0723.22014
This book is a study on the unitary representations of complex connected reductive Lie groups $${\mathbb{G}}$$, appearing in the space of $$C^{\infty}$$- functions on the indefinite symmetric space $$C^{\infty}({\mathbb{G}}_ 0/{\mathbb{H}}_ 0)$$. Here $${\mathbb{G}}_ 0$$ is a real form of $${\mathbb{G}}$$ and $${\mathbb{H}}_ 0$$ is a real form of a fixed point subgroup $${\mathbb{H}}$$ of an involution of $${\mathbb{G}}$$. The main tool of this book is the theory of $${\mathcal D}$$-modules on the flag variety. Namely, the author considers the sheaf of the twisted differential operators $${\mathcal D}_{\lambda}$$ on the flag variety X and takes the space of the global sections of $${\mathcal D}_{\lambda}$$-module as the representation space. He simplifies the result on the discrete series representation by Oshima-Matsuki; he proves that the discrete series appearing in the space on the indefinite symmetric space is “almost” all multiplicity free and presents a cohomological formula for the multiplicities of the standard representation. Next, he studies the irreducibility of the representation by using microdifferential operators. Namely, he gives a criterion for irreducibility of $$\Gamma$$ (X,M) as a module of the universal enveloping algebra of the Lie algebra of $${\mathbb{G}}$$, where M is a $${\mathcal D}_{\lambda}$$-module on X. In the last chapter, the author returns to the problem of the multiplicity. He obtains the multiplicity of the principal series representation appearing in $$C^{\infty}({\mathbb{G}}_ 0/{\mathbb{H}}_ 0)$$.

##### MSC:
 22E46 Semisimple Lie groups and their representations 22-02 Research exposition (monographs, survey articles) pertaining to topological groups 32C38 Sheaves of differential operators and their modules, $$D$$-modules 14M15 Grassmannians, Schubert varieties, flag manifolds 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials