| Summary: | <p>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<sup>λ</sup>
-modules, where λ is a ρ-dominant ρ-regular infinitesimal central character.</p>
<p>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.</p>
<p>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<sup>λ</sup><sub>X/G</sub> in the case where λ is a regular ρ-dominant ρ-integral
central character. In this particular case, we show that
C<sup>λ</sup><sub>X/G</sub> is equivalent to Ardakov’s category C<sub>X/G</sub> and use this to prove a twisted analogue
of locally analytic equivariant Beilinson-Bernstein localisation on rigid
analytic spaces.</p>
<p>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<sup>λ</sup>
(X, G)
as a quotient, providing a natural framework to view coadmissible Gequivariant D<sup>λ</sup><sub>X</sub>-modules as coadmissible modules over this algebra. We
give a general definition of C
<sup>λ</sup><sub>X/G</sub> 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.</p>
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