ORDINARY REPRESENTATIONS OF G(Q(p)) AND FUNDAMENTAL ALGEBRAIC REPRESENTATIONS

摘要

Let G be a split connected reductive algebraic group over Q(p) such that both G and its dual group (G) over cap have connected centers. Motivated by a hypothetical p-adic Langlands correspondence for G(Q(p)) we associate to an n-dimensional ordinary (i.e., Borel-valued) representation rho: Gal((Q) over bar (p)/Q(p)) -> (G) over cap (E) a unitary Banach space representation Pi(rho)(ord) of G(Q(p)) over E that is built out of principal series representations. (Here, E is a finite extension of Q(p).) Our construction is inspired by the "ordinary part" of the tensor product of all fundamental algebraic representations of G. There is an analogous construction over a finite extension of F-p. When G = GL(n), we show under suitable hypotheses that Pi(rho)(ord) occurs in the rho-part of the cohomology of a compact unitary group.