Ring-theoretic properties of augmented Iwasawa algebras

Thesis

Let p be a prime, G be a p-adic Lie group and k be a field of characteristic p. The augmented Iwasawa algebra kG is a noncommutative algebra extending the construction of the Iwasawa algebra of a compact p-adic Lie group. In this thesis we investigate the ring-theoretic properties of augmented Iwasawa algebras.

We prove that a split-semisimple group over a p-adic field F has a coherent augmented Iwasawa algebra if and only if its root system is of rank one, and deduce that kGL_n(F) is coherent precisely when n is at most two. We also characterise when certain solvable p-adic Lie groups have a coherent augmented Iwasawa algebra. When kG is a coherent ring, the category of finitely-presented smooth representations of G over k is abelian. We give a module-theoretic characterisation of the latter property, and hence give examples of p-adic Lie groups for which it fails. These results are achieved by developing the theory of augmented Iwasawa algebras in detail, including their relation to smooth and admissible representations of G.

The Iwasawa algebra of a compact p-adic Lie group is always an Auslander-Gorenstein ring. Thus finitely-generated modules over Iwasawa algebras have a canonical dimension. Let G be a compact p-adic Lie group whose Lie algebra is isomorphic to a split simple F-Lie algebra. We prove that whenever F is a non-trivial finite extension of the p-adic numbers, kG has no modules of canonical dimension one. Consequently kG has a module of canonical dimension one only if the Lie algebra of G is isomorphic to sl₂(Qp). As a corollary we obtain a new upper bound on the Krull dimension of kG.

When G is an arbitrary p-adic Lie group, any smooth admissible representation V is the dual of a kG-module M, finitely-generated over the Iwasawa algebra of a compact open subgroup of G. The canonical dimension of M measures the rate of growth of invariant subspaces of V. We prove a canonical dimension-theoretic criterion for V to be a finite length representation. We deduce that the finite length smooth admissible representations of GL2(Qp) with central character are precisely those whose duals have canonical dimension at most one. Finally, let F be a non-trivial finite extension of the p-adic numbers, G = GL_n(F), and V be a smooth admissible representation of G with central character. A combination of our results shows that V has finite length if its dual has canonical dimension two.

Authors: Timmins, J

• DOI: 10.5287/ora-eyq0kz8ym
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Smooth representations; Noncommutative rings; Coherent rings; Langlands programme; Completed group algebras; Canonical dimension; p-adic Lie groups
• Subjects: Mathematics; Algebraic number theory; Representations of groups; Algebra; Rings (Algebra)