A Note on Invariant Forms on Locally Symmetric Spaces

Abstract

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 I_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 ^q_Г is trivial for q > dim X-rank G.