On irreducible representations of compact $p$-adic analytic groups

TL;DR

This paper establishes a lower bound on the canonical dimension of certain $p$-adic group representations, linking it to coadjoint orbits, and extends classical algebraic results to the $p$-adic setting.

Contribution

It introduces $p$-adic analogues of Bernstein's inequality, Beilinson-Bernstein localisation, and Quillen's Lemma, advancing the understanding of $p$-adic representation theory.

Findings

Canonical dimension is either zero or at least half the dimension of a coadjoint orbit.

Established $p$-adic versions of Bernstein's inequality and Beilinson-Bernstein localisation.

Extended Quillen's Lemma to $p$-adically completed enveloping algebras.

Abstract

We prove that the canonical dimension of a coadmissible representation of a semisimple $p$-adic Lie group in a $p$-adic Banach space is either zero or at least half the dimension of a non-zero coadjoint orbit. To do this we establish analogues for $p$-adically completed enveloping algebras of Bernstein's inequality for modules over Weyl algebras, the Beilinson-Bernstein localisation theorem and Quillen's Lemma about the endomorphism ring of a simple module over an enveloping algebra.