##
**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.

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
PDF
BibTeX
XML
Cite

\textit{D. A. Vogan jun.} and \textit{G. J. Zuckerman}, Compos. Math. 53, 51--90 (1984; Zbl 0692.22008)

### References:

[1] | M.W. Baldoni-Silva and D. Barbasch : The unitary spectrum for real rank one groups . Invent. Math. 72 (1983) 27-55. · Zbl 0561.22009 |

[2] | A. Borel and N. Wallach : Continuous cohomology, discrete subgroups, representations of reductive groups , Princeton University Press, Princeton, New Jersey (1980). · Zbl 0443.22010 |

[3] | W. Casselman and M.S. Osborne : The n-cohomology of representations with an infinitesimal character . Comp. Math. 31 (1975) 219-227. · Zbl 0343.17006 |

[4] | T. Enright : Relative Lie algebra cohomology and unitary representations of complex Lie groups . Duke Math. J. 46 (1979) 513-525. · Zbl 0427.22010 |

[5] | A. Guichardet : Cohomologie des groupes topologiques et des algèbres de Lie , CEDIC-Fernand Nathan, Paris (1980). · Zbl 0464.22001 |

[6] | Harish-Chandra : Representations of semi-simple Lie groups I . Trans. Amer. Math. Soc. 75 (1953) 185-243. · Zbl 0051.34002 |

[7] | S. Helgason : Differential Geometry, Lie Groups, and Symmetric Spaces . Academic Press, New York (1978). · Zbl 0451.53038 |

[8] | R. Hotta and R. Parthasarathy : A geometric meaning of the multiplicities of integrable discrete classes in L2(\Gamma \G) . Osaka J. Math. 10 (1973) 211-234. · Zbl 0337.22016 |

[9] | J. Humphreys : Introduction to Lie algebras and representation theory . Springer-Verlag, New York Heidelberg Berlin (1972). · Zbl 0254.17004 |

[10] | S. Kumaresan : The canonical f-types of the irreducible unitary g-modules with non-zero relative cohomology . Invent. Math. 59 (1980) 1-11. · Zbl 0442.22010 |

[11] | R. Parthasarathy : Dirac operator and the discrete series . Ann. Math. 96 (1972) 1-30. · Zbl 0249.22003 |

[12] | R. Parthasarathy : A generalization of the Enright-Varadarajan modules . Comp. Math. 36 (1978) 53-73. · Zbl 0384.17005 |

[13] | R. Parthasarathy : Criteria for the unitarizability of some highest weight modules . Proc. Indian Acad. Sci. 89 (1980) 1-24.. · Zbl 0434.22011 |

[14] | K.R. Parthasarathy , R. Ranga Rao and V.S. Varadarajan : Representations of complex semi-simple Lie groups and Lie algebras . Ann. Math. 85 (1967) 383-429. · Zbl 0177.18004 |

[15] | B. Speh : Unitary representations of GL(n, R) with non-trivial (g, K ) cohomology . Invent. Math. 71 (1983) 443-465. · Zbl 0505.22015 |

[16] | B. Speh : Unitary representations of SL(n, R) and the cohomology of congruence subgroups , In. Noncommutative Harmonic Analysis and Lie Groups , Lecture Notes in Mathematics 880, Springer-Verlag, Berlin Heidelberg New York (1981). · Zbl 0516.22008 |

[17] | B. Speh and D. Vogan : Reducibility of generalized principal series representations . Acta Math. 145 (1980) 227-299. · Zbl 0457.22011 |

[18] | D. Vogan : The algebraic structure of the representations of semi-simple Lie groups I . Ann. Math. 109 (!979) 1-60. · Zbl 0424.22010 |

[19] | D. Vogan : Representations of real reductive Lie groups , Birkhäuser, Boston-Vasel-Stuttgart (1981). · Zbl 0469.22012 |

[20] | G. Warner : Harmonic analysis on semi-simple Lie groups I , Springer-Verlag, Berlin Heidelberg New York (1972). · Zbl 0265.22020 |

[21] | J. Rawnsley , W. Schmid and J. Wolf : Singular unitary representations and indefinite harmonic theory , to appear in J. Func. Anal., 1983. · Zbl 0511.22005 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.