A Note on Invariant Forms on Locally Symmetric Spaces

  • Birgit Speh
Part of the Progress in Mathematics book series (PM, volume 40)


Given a discrete subgroup Г of a connected real semisimple Lie group G with finite center, there is a natural homomorphism
$$\rm{j^q_\Gamma :\ I^q_G\ \rightarrow \ H^q(\Gamma,C),\ q\ =\ 0,1,}\ldots$$
where IG 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 cG ⩽ rank G so that if q ⩽ cG then jГ 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.


Symmetric Space Parabolic Subgroup Eisenstein Series Discrete Subgroup Maximal Compact Subgroup 
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© Birkhäuser Boston, Inc. 1983

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  • Birgit Speh

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