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An Estimate of the Growth of Cohomology with Coefficients
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For a connected reductive algebraic group over an arbitrary number eld, we consider a nite dimensional algebraic, irreducible representation of the group of its real points. Each adelic locally symmetric space corresponding to a level structure constructed using the group has an associated sheaf induced by this representation. The purpose of this note is to estimate the rate of growth of the total dimension of the pertinent cohomology with coecients as either of the level structure or the associated sheaf varies. We obtain an upper bound on this total dimension. We also obtain a lower bound under certain topological conditions. Both the bounds are consistent with several classical dimension formulae as well as other known results.
Keywords
Group Cohomology, algebraic Groups, Adelic locally symmetric space, sheaf cohomology
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