On Vector Bundle Valued Harmonic Forms and Automorphic Forms on Symmetric Riemannian Manifolds

Matsushima, Yozô; Murakami, Shingo

doi:10.2307/1970348

Cited by 158 publications

(159 citation statements)

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“…Our goal in this section is to develop a variant of the Bochner-Weitzenböck formula for E(π)-valued p-forms on M which is analogous to the one found by Matsushima and Murakami [25]. As with all formulae of this type, this formula will be a computation of the difference between two second order operators on E(π)-valued p-forms and the difference is a zeroth order (algebraic) operator related to curvature.…”

Section: 4mentioning

confidence: 96%

“…We then use a Bochner-Matsushima-type formula to prove an estimate on the first eigenvalue of the Laplacian on one forms that implies vanishing of first cohomology. In order to define a Laplacian one requires a choice of metric on F ⊗H, and our choice here is similar to the one made by Matsushima and Murakami in [25]. In fact, the work here is very close to the work in that paper, and we eventually reduce to an estimate on eigenvalues of the same finite dimensional matrix as they do.…”

Section: On Theorem 12mentioning

confidence: 97%

“…These estimates are obtained by Raghunathan in [29]. It is possible to derive the formula we use by closely following the derivation in [25]; however, we choose a different method. From our point of view, the estimate follows by subtracting two standard Weitzenböck formulas for different bundle structures on the same vector bundle.…”

Section: On Theorem 12mentioning

confidence: 99%

“…One could give a development directly mirroring that of [25]. Instead, we give an alternate derivation which we feel is more intuitive and makes the relation to other Bochner-type formulae more explicit.…”

Section: 4mentioning

confidence: 99%

“…The key observation of Matsushima and Murakami in [25] is that the relevant vector bundle can be constructed in two distinct ways, that is, there are two (isomorphic) natural vector bundles associated to the given data. This allows us to compare the natural differential operators arising from the two constructions.…”

Section: 4mentioning

confidence: 99%

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Strengthening Kazhdan’s property (T) by Bochner methods

Fisher

Hitchman

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Section: 4mentioning

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Section: On Theorem 12mentioning

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Section: On Theorem 12mentioning

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On the Spectra of Compact Locally Symmetric Riemannian Manifolds

Marbes

1981

Mathematische Nachrichten

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In the last few years investigations about the spectra of RIEMANNian manifolds, their cohomology and the connection t o multiplicities, with which certain representations occur in others, were intensively carried out.In the sixties some papers [7], [8], [9] of the Japanese mathematicians MAT-SUSHIMA and MURAKAMI were published. There they dealt with the theory of harmonic forms and the investigations of some special cohomology groups.In 1967, the paper [12] of MATSUSHIMA was published. He considered a connected semi-simple LTE group G with finite center and only non-compact simple components, a maximal compact subgroup K of G, the symmetric space X =GIK and a discrete subgroup I' of G such that r\G is compact and r acts freely on X.Then MATSUSHIMA praved a formula for the dimension of the space of all harmonic r-invariant p-forms on X , where he related this dimension to the decomposition of the right-regular unitary representation of G into L*(I'\G). Finally he considered an example. In this paper the formula of MATSUSHIMA will be generalized to a formula for the dimension of eigensubspaces of the laplacian operator to an arbitrary eigenvalue il of T-invariant p-forms on X . Here this dimension will be related to the decomposition of the right-regular representation of G into LZ(r\G), too. This main result is formulated in Theorem 5.1 of section 5. For A = O we get the formula of MATSUSHIMA as a special case.In soction 1 the general situation, which is our starting-point, will be described. A regularity-and a finiteness-proposition for the Iaplacian operator is mentioned and tho CASIWK. operator for a unitary representation will be defined in section 2. Section 3 contains a connection between T-invariant differential p-forms on the quotient G/K and determined vector-valued functions on r\G. This connection playg an important role in the proof of our main result. In section 4 we will prove the decomposition proposition for the right-regular representation of G in L2(r\Q) (following GELFAND [l]). As mentioned, the main result will be described in section 5. Its detailed proof will be given in section 6.

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On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes

Shimura¹

2002

Collected Papers

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On Vector Bundle Valued Harmonic Forms and Automorphic Forms on Symmetric Riemannian Manifolds

Cited by 158 publications

References 4 publications

Strengthening Kazhdan’s property (T) by Bochner methods

Strengthening Kazhdan’s property (T) by Bochner methods

On the Spectra of Compact Locally Symmetric Riemannian Manifolds

On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes

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