Estimation of the dimension of cuspidal and total cohomology

Abstract

We consider the Weil restriction of a connected reductive algebraic group over anumber field to the rational numbers For a level structure in the group of its adelicpoints we form an adelic locally symmetric space A finite dimensional algebraic irreducible representation of the group of real points of the Weil restriction induces anassociated sheaf on this space Raghuram and Bhagwat found certain necessary conditions for non vanishing of thecuspidal part of the respective sheaf cohomology in case of the general linear groupunder some additional assumptions on the number field and the weight of therepresentation Motivated by this we estimate the growth rate of cuspidal cohomologywith varying level structure as well as weight in case of automorphic induction fromGL 1 over imaginary quadratic fields to GL 2 over the rationals and also that ofsymmetric square transfer from GL 2 to GL 3 both over the rationals We also present bounds on the dimension of the total sheaf cohomology which apply toan arbitrary connected reductive algebraic group with varying level structure or weight The bounds thus obtained are consistent with the classical dimension formulae as wellas several known results newline newline