Gelfand-Kirillov bound for GL_2

Abstract: Let G be a p-adic Lie group. In this talk I will introduce the Gelfand-Kirillov dimension of p-adic representations of G, which is a non-commutative generalization of the Krull dimension in this setting. For this, one uses Schneider-Teitelbaum's duality theory which allows one to think of p-adic Banach representations of G as (duals of) modules over a completed group ring of G. The ``Miracle Flatness'' observation Gee-Newton shows how knowledge of this dimension can have strong structural consequences, with potential applications to completed cohomology and patching. I will discuss the example of such an application found in the work of Breuil-Herzig-Hu-Morra-Schraen: as a consequence of their GK-dim computation they deduce the non-vanishing of the candidates via patching for the p-adic Langlands correspondence for GL_2 of an unramified p-adic field. I will then discuss the following result (arXiv:2602.08856): let p>2 and K be a p-adic field; an admissible p-adic Banach representation of GL_2K whose locally analytic vectors admit an infinitesimal character has GK-dimension at most [K:Q_p]. This bound is optimal and improves the previous bound <2[K:Q_p] of Dospinescu-Paškūnas-Schraen. In my thesis I have generalized this result to families of p-adic Banach representation with an infinitesimal character in families (in the sense of Dospinescu-Paškūnas-Schraen) and I will explain how this leads to a generalization of the GK-dim computation and non-vanishing of candidates result of Breuil-Herzig-Hu-Morra-Schraen to GL_2K where K now can have arbitrary ramification.

algebraic geometrynumber theoryrepresentation theory

Audience: researchers in the topic

KIAS HCMC Algebra Seminar

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