## Unitary representations with nonzero cohomology.(English)Zbl 0692.22008

An important problem in the theory of automorphic forms is to compute cohomology of locally symmetric spaces. Matsushima’s formula [A. Borel and N. R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups (1980; Zbl 0443.22010), see p. 223] relates this problem to computations of cohomology of infinite-dimensional representations of the corresponding semisimple group. More precisely, the problem is the following: Suppose G is a reductive Lie group with Lie algebra $${\mathfrak g}$$ and maximal compact subgroup K. Find all unitary irreducible representations $$\pi$$ such that (*) $$H^*({\mathfrak g},K,\pi)\neq 0$$ or more generally $$H^*({\mathfrak g},K,\pi \otimes F)\neq 0$$ where F is finite-dimensional.
The paper under review describes all Harish-Chandra modules satisfying (*). The results are sharp in the sense that $$\pi$$ and $$H^*({\mathfrak g},K,\pi \otimes F)$$ are very explicit. The representation $$\pi$$ is obtained by what is known as the “derived functors construction” from a 1-dimensional unitary character on a Levi subgroup. Their unitarity is only conjectured (established later by D. Vogan [Ann. Math., II. Ser. 120, 141–187 (1984; Zbl 0561.22010)]). The techniques involve the Dirac inequality and its consequences obtained by S. Kumaresan [Invent. Math. 59, 1–11 (1980; Zbl 0442.22010)] and the classification of Harish-Chandra modules as in D. Vogan’s book [Representations of real reductive Lie groups (1981; Zbl 0469.22012)]. Several consequences are described, such as a vanishing theorem for cohomology.

### MSC:

 22E46 Semisimple Lie groups and their representations 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 11F70 Representation-theoretic methods; automorphic representations over local and global fields 32N10 Automorphic forms in several complex variables 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols

### Citations:

Zbl 0443.22010; Zbl 0561.22010; Zbl 0442.22010; Zbl 0469.22012
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### References:

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