# Vanishing theorems on cohomology associated to hermitian symmetric spaces

Annales de l'institut Fourier (1987)

- Volume: 37, Issue: 4, page 225-233
- ISSN: 0373-0956

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topMurakami, Shingo. "Vanishing theorems on cohomology associated to hermitian symmetric spaces." Annales de l'institut Fourier 37.4 (1987): 225-233. <http://eudml.org/doc/74777>.

@article{Murakami1987,

abstract = {We consider the cohomoly groups of compact locally Hermitian symmetric spaces with coefficients in the sheaf of germs of holomorphic sections of those vector bundles over the spaces which are defined by canonical automorphic factors. We give a quick survey of the research on these cohomology groups, and then discuss vanishing theorems of the cohomology groups.},

author = {Murakami, Shingo},

journal = {Annales de l'institut Fourier},

keywords = {cohomology groups of compact locally Hermitian symmetric spaces; coefficients in the sheaf of germs of holomorphic sections; canonical automorphic factors},

language = {eng},

number = {4},

pages = {225-233},

publisher = {Association des Annales de l'Institut Fourier},

title = {Vanishing theorems on cohomology associated to hermitian symmetric spaces},

url = {http://eudml.org/doc/74777},

volume = {37},

year = {1987},

}

TY - JOUR

AU - Murakami, Shingo

TI - Vanishing theorems on cohomology associated to hermitian symmetric spaces

JO - Annales de l'institut Fourier

PY - 1987

PB - Association des Annales de l'Institut Fourier

VL - 37

IS - 4

SP - 225

EP - 233

AB - We consider the cohomoly groups of compact locally Hermitian symmetric spaces with coefficients in the sheaf of germs of holomorphic sections of those vector bundles over the spaces which are defined by canonical automorphic factors. We give a quick survey of the research on these cohomology groups, and then discuss vanishing theorems of the cohomology groups.

LA - eng

KW - cohomology groups of compact locally Hermitian symmetric spaces; coefficients in the sheaf of germs of holomorphic sections; canonical automorphic factors

UR - http://eudml.org/doc/74777

ER -

## References

top- [1] G. W. ANDERSON, Theta functions and holomorphic differential forms on compact quotients of bounded symmetric domains, Duke Math. J., 50 (1983), 1137-1170. Zbl0557.32006MR85i:32047
- [2] A. BOREL and N. WALLACH, Continuous cohomology, discrete subgroups, and representations of reductive groups, Princeton, Princeton University Press, 1980. (Annals of mathematical studies, 94.) Zbl0443.22010
- [3] G. FALTINGS, On the cohomology of locally symmetric Hermitian symmetric spaces, Paul Dubriel and Marie-Paul Malliavin algebra seminar, 25th year (Paris, 1982), 55-98, Lecture Notes in Math., 1029 (1983). Zbl0539.22008
- [4] R. HOTTA and S. MURAKAMI, On a vanishing theorem for certain cohomology groups, Osaka J. Math., 12 (1975), 555-564. Zbl0328.57020MR57 #6291
- [5] R. HOTTA and N. WALLACH, On Matsushima's formula for the Betti numbers of a locally symmetric space, Osaka J. Math., 12 (1975), 419-431. Zbl0327.53032MR52 #661
- [6] J. L. KOSZUL, Formes harmoniques vectorielles sur espaces localement symétriques, Geometry of Homogeneous Bounded Domains, C.I.M.E. 3 Ciclo, 1967, 197-260. Zbl0183.50903
- [7] S. KUMARESAN, On the canonical k-types in the irreducible unitary g-modules with non-zero relative cohomology, Inventiones Math., 59 (1980), 1-11. Zbl0442.22010MR83c:17011
- [8] Y. MATSUSHIMA, On the first Betti number of compact quotient spaces of higher dimensional symmetric spaces, Ann. of Math., 75 (1962), 312-330. Zbl0118.38303MR28 #1629
- [9] Y. MATSUSHIMA, On Betti numbers of compact, locally symmetric Riemannian manifolds, Osaka J. Math., 14 (1982), 312-330. Zbl0118.38401
- [10] Y. MATSUSHIMA, A formula for the Betti numbers of locally symmetric Riemannian manifolds, J. Differential Geometry, 1 (1987), 99-109. Zbl0164.22101MR36 #5958
- [11] Y. MATSUSHIMA and S. MURAKAMI, On vector bundle valued harmonic forms and automorphic forms on symmetric Riemannian manifolds, Ann. of Math., 78 (1963), 365-416. Zbl0125.10702MR27 #2997
- [12] Y. MATSUSHIMA and S. MURAKAMI, On certain cohomology groups attached to hermitian symmetric spaces, Osaka J. Math., 2 (1965), 1-35. Zbl0142.19503MR32 #1728
- [13] Y. MATSUSHIMA and S. MURAKAMI, On certain cohomology groups attached to Hermitian symmetric spaces (II), Osaka J. Math., 5 (1968), 223-241. Zbl0183.26103MR42 #1145
- [14] S. MURAKAMI, Cohomology of vector-valued forms on compact, locally symmetric Riemannian manifolds, Proceeding Symposia in Pure Mathematics, vol. 9, Algebraic groups and discontinuous subgroups, 1966, 387-399. Zbl0208.24402MR34 #6803
- [15] S. MURAKAMI, Cohomology groups of vector-valued forms on symmetric spaces, Lecture Notes, University of Chicago, 1966.
- [16] S. MURAKAMI, Facteur d'automorphie associé à un espace hermitien symétrique, Geometry of Homogeneous Bounded Domains, C.I.M.E. 3 Ciclo, (1967), 281-287. Zbl0187.02902
- [17] S. MURAKAMI, Certain cohomology groups attached to Hermitian symmetric spaces and unitary representations, Southeast Asian Bull. Math., 5 (1981), 39-44. Zbl0496.22020MR83e:32034
- [18] S. MURAKAMI, Laplacians and cohomologies associated to locally symmetric Hermitian manifolds, Spectra of Riemannian Manifolds (Proceedings of the France-Japan Seminar, Kyoto, 1981), Kaigai Publ. Tokyo, (1983), 73-78.
- [19] R. PARTHASARATHY, A note of the vanishing of L²-cohomologies, J. Math. Soc. Japan, 22 (1971), 1-30.
- [20] R. PARTHASARATHY, Criteria for the unitarizability of some highest weight modules, Proc. Indian Acad. Sci., 89 (1980), 1-24. Zbl0434.22011MR82c:22020
- [21] D. A. VOGAN and G. J. ZUCKERMAN, Unitary representations with non-zero cohomology, Compositio Mathematica, 53 (1984), 51-90. Zbl0692.22008MR86k:22040
- [22] F. L. WILLIAMS, Vanishing theorems for type (0,q)-cohomology of locally symmetric spaces, Osaka J. Math., 18 (1981), 147-160. Zbl0469.32011MR82k:22013
- [23] F. L. WILLIAMS, Remarks on the unitary representations appearing in the Matsushima-Murakami formula, Proceeding of the Conference on Non-commutative Harmonic Analysis, Marseille-Luminy, France, Lecture Notes in Math., 880 (1981), 536-553. Zbl0508.22015MR83e:22017
- [24] F. L. WILLIAMS, Vanishing theorems for type (0,q)-cohomology of locally symmetric spaces II, Osaka J. Math., 20 (1983), 95-108. Zbl0523.22016MR84h:22031
- [25] S. ZUCKER, Locally homogeneous variations of Hodge structure, L'Enseignement Mathématiques, IIe série, 27 (1981), 243-276. Zbl0584.14003MR83m:32034

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