From $\mathrm{GL}(2,Q_p)$ to $\mathrm{SL}(2,Q_p)$

Dubravka Ban (Southern Illinois University)

17-Oct-2020, 14:30-15:00 (3 years ago)

Abstract: Let $E$ be a finite extension of $\mathbb{Q}_p$. The $p$-adic Langlands correspondence for $G=GL_2(\mathbb{Q}_p)$ is a bijection between the set of absolutely irreducible 2-dimensional $E$-representations of ${\rm Gal} \overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ and the set of absolutely irreducible admissible non-ordinary $E$-Banach space representations of $G$. We study the corresponding objects for $H=SL_2(\mathbb{Q}_p)$. Let $\Pi$ be an irreducible admissible Banach space representation of $G$. Then $\Pi|_H$ decomposes as a direct sum of inequivalent representations.

Let $\psi: {\rm Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p) \to GL_2(E)$ be the associated Galois representation. Assume that $\psi$ is de Rham with Hodge-Tate weights 0 and 1. We compute the centralizer in $PGL_2(\overline{E})$ of the image of the corresponding projective Galois representation $\overline{\psi}: {\rm Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p) \to PGL_2(E)$. We show that the order of the centralizer is equal to the number of components of $\Pi|_H$.

Encapsulated in the $p$-adic Langlands correspondence for $G$ is the classical smooth Langlands correspondence. The representation $\Pi$ contains a smooth representation $\pi$. In the case when $\psi$ is non-trianguline, $\pi$ is supercuspidal. We investigate the connection between $\Pi|_H$ and $\pi|_H$. This is a joint work with Matthias Strauch.

number theoryrepresentation theory

Audience: researchers in the discipline


The 2020 Paul J. Sally, Jr. Midwest Representation Theory Conference

Series comments: The 44th Midwest Representation Theory Conference will address recent progress in the theory of representations for groups over non-archimedean local fields, and connections of this theory to other areas within mathematics, notably number theory and geometry.

In order to receive information on how to participate (to be sent out closer to the conference), please register by October 14 here: forms.gle/zFAnQBnuPGRnKzMr7

Organizers: Stephen DeBacker, Jessica Fintzen*, Muthu Krishnamurthy, Loren Spice
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