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D-modules and spherical representations. (English) Zbl 0723.22014
Mathematical Notes, 39. Princeton, NJ: Princeton University Press. 131 p. $ 22.50 (1991).
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 \(C^{\infty}({\mathbb{G}}_ 0/{\mathbb{H}}_ 0)\).

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