The aim of this article is to present a simple generalized Plancherel formula for a locally compact unimodular topological group G of type I. This formula applies to the functions representing c-Ind G U ψ for a unitary character ψ of a closed unimodular subgroup U of G. This specializes to the Whittaker-Plancherel formula for a split reductive p-adic group of Sakellaridis-Venkatesh and differs from that of a quasi-split p-adic group due to Delorme. Furthermore, it also applies to certain metaplectic groups and other interesting situations where the local theory of distinguished representations has been studied.
We give three necessary and sufficient conditions for a pro-p group to be p-adic analytic. We show that a noetherian pro-p group having finite chain length has a finite rank and conversely. We further deduce that a noetherian pro-p group has a finite rank precisely when it satisfies the weak descending chain condition. Using these results, we resolve a conjecture posed by Lubotzky and Mann in the affirmative within the class of noetherian groups which are countably based.Using these results, we answer a related conjecture about pro-p groups for the case of countably based pro-p groups. Namely, we prove that if every closed but non-open subgroup of a countably based pro-p group has finite rank, then the group is p-adic analytic and conversely.
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.
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