Representation Theory of Reductive Groups pp 197-206 | Cite as

# A Note on Invariant Forms on Locally Symmetric Spaces

Chapter

## Abstract

Given a discrete subgroup Г of a connected real semisimple Lie group G with finite center, there is a natural homomorphism
where I

$$\rm{j^q_\Gamma :\ I^q_G\ \rightarrow \ H^q(\Gamma,C),\ q\ =\ 0,1,}\ldots$$

_{G}^{q}denotes the space of G-invariant harmonic q-forms on the symmetric space X = G/K. Here K is a maximal compact subgroup of G. If Г is cocompact, this homomorphism is injective in all dimensions. If G/Г is not compact there exists a constant c_{G}⩽ rank G so that if q ⩽ c_{G}then j_{Г}^{q}is injective (and in fact is bijective) [1]. On the other hand, the cohomological dimension of Г\X is dim X-rank G [2]. So j_{Г}^{q}is trivial for q > dim X-rank G.## Keywords

Symmetric Space Parabolic Subgroup Eisenstein Series Discrete Subgroup Maximal Compact Subgroup
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

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## Copyright information

© Birkhäuser Boston, Inc. 1983