David J. Pollack
The cohomology groups H1(Γ0(N), Vk) completely describe the space of classical cusp forms of weight k and level N. We study a generalization, Hn(Γ, Vλ), where some algebraic group G plays a role analogous to that of GL2 in the classical case.
Ash and Stevens proved that certain classes in Hn(Γ, Vλ) may be lifted through the natural map ρλ : Hn(Γ, Dλ) → Hn(Γ, Vλ) to overconvergent classes in Hn(Γ, Dλ). Pollack and Pollack were able to prove this result constructively in the case of G = GL3, by providing a filtration on the distribution space Dλ.
We construct a general filtration FilNDλ, for a split reductive algebraic group G. Using this filtration, we are able to lift classes in Hn(Γ, Vλ) to the finite dimensional spaces Hn(Γ, Dλ / FilNDλ). These lifts approximate the lifts into Hn(Γ, Dλ) and improve as N → ∞.
Graham, Bonita Lynn, "A Construction of Rigid Analytic Cohomology Classes for Split Reductive Algebraic Groups" (2014). Dissertations. 36.
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