Twisted coadmissible equivariant D-modules on rigid analytic spaces

Thesis

We aim to extend the results of Ardakov and Wadsley on representations of p-adic Lie groups and sheaves of equivariant D-modules on rigid analytic spaces. Our main results are a canonical dimension estimate for coadmissible representations of a semisimple p-adic Lie group in a p-adic Banach space, and a Beilinson-Bernstein-type localisation for coadmissible equivariant D^λ -modules, where λ is a ρ-dominant ρ-regular infinitesimal central character.

This thesis is split into two principal sections. The first section contains results on the structure of the dual nilpotent cone of a semisimple Lie algebra g over an algebraically closed field K of positive characteristic p. When p is small, the structure of the adjoint and coadjoint orbits of G on g and g∗ respectively changes. We relax the hypotheses on p and prove that, under certain technical conditions, the dual nilpotent cone is a normal variety. Using this, we show 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.

The second section extends the work of Ardakov on coadmissible equivariant D-modules on rigid analytic spaces, by defining the category of coadmissible equivariant twisted D-modules. We classify sheaves of differential operators on a rigid analytic variety X and use this to define the category C^λ_X/G in the case where λ is a regular ρ-dominant ρ-integral central character. In this particular case, we show that C^λ_X/G is equivalent to Ardakov’s category C_X/G and use this to prove a twisted analogue of locally analytic equivariant Beilinson-Bernstein localisation on rigid analytic spaces.

The final part of this thesis is concerned with removing the integrality condition. We construct the enhanced completed skew-group algebra D(X, G) and realise the twisted completed skew-group algebra D^λ (X, G) as a quotient, providing a natural framework to view coadmissible Gequivariant D^λ_X-modules as coadmissible modules over this algebra. We give a general definition of C ^λ_X/G for arbitrary ρ-dominant ρ-regular central character and discuss how to check this agrees with the definition given for the integral case. Following this, we show this category is equivalent to the category of U(g, G)-modules with fixed infinitesimal central character λ.

Authors: Mathers, R

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Geometric representation theory
• Subjects: Mathematics