Unitary representations with nonzero cohomology.

*(English)*Zbl 0692.22008An 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.

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.

Reviewer: D. Barbasch (MR 86k:22040)

##### 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 |

##### Keywords:

automorphic forms; cohomology of locally symmetric spaces; infinite- dimensional representations; semisimple group; reductive Lie group; Lie algebra; unitary irreducible representations; Harish-Chandra modules; vanishing theorem for cohomology
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\textit{D. A. Vogan jun.} and \textit{G. J. Zuckerman}, Compos. Math. 53, 51--90 (1984; Zbl 0692.22008)

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