Bushnell-Kutzko types for P-ordinary automorphic representations on unitary groups

2023/10/13 by David Marcil, Marcil, David · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2310.09110

Abstract

This paper generalizes a theorem of Hida on the structure of ordinary representations on unitary groups to P-ordinary representations, where P is a general parabolic subgroup of some general linear group. When P is minimal, we recover Hida's theorem which asserts that ordinary subspaces are 1-dimensional. While analogous P-ordinary subspaces are infinite-dimensional in general, we use the theory of Bushnell-Kutzko types to canonically associate a finite-dimensional type to the representation (under minor assumptions) that has multiplicity one in its P-ordinary subspace. We simultaneous develop the theory of modular forms on unitary groups with P-Iwahoric level structure whose nebentypus is a type (instead of a character) and construct lattices of P-ordinary modular forms inside P-ordinary automorphic representations. We also obtain direct consequences for the dual notion of P-anti-ordinary forms and representations.

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• Test Schwartz functions at p for theta lifts of Hida families

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