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Algebra & Combinatorics Seminar

Title: Linkage principle and Tame cyclic base change for $GL_n(F)$
Speaker: Santosh Nadimpalli (Radboud University, Nijmegen, Netherlands)
Date: 20 September 2019
Time: 3 pm
Venue: LH-1, Mathematics Department

Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be a finite cyclic Galois extension of $F$. The theory of base change associates to an irreducible smooth $\overline{\mathbb{Q}}_l$-representation $(\pi_F, V)$ of ${\rm GL}_n(F)$ an irreducible $\overline{\mathbb{Q}}_l$-representation $(\pi_E, W)$ of ${\rm GL}_n(E)$. The ${\rm GL}_n(E)$-representation $\pi_E$ extends as a representation of ${\rm GL}_n(E)\rtimes {\rm Gal}(E/F)$. Assume that the central character of $\pi_F$ takes values in $\overline{\mathbb{Z}}_l^\times$, and $l\neq p$. When $\pi_E$ is cuspidal, for any ${\rm GL}_n(E)\rtimes {\rm Gal}(E/F)$ stable lattice $\mathcal{L}$ in $\pi_E$, Ronchetti supporting the linkage principle of Treumann and Venkatesh conjectured that the zeroth Tate cohomology of $\mathcal{L}$ with respect to ${\rm Gal}(E/F)$ is the Frobenius twist of mod-$l$ reduction of the representation $\pi_F$, i.e.,

\begin{equation} \widehat{H}^0 ({\rm Gal}(E/F), \mathcal{L}) \simeq \overline{\pi} _F^{(l)}. \end{equation}

This conjecture is verified by Ronchetti when $\pi_F$ is a depth-zero cuspidal representation using compact induction model. We will explain a proof in the case where $n=2$ and $\pi_F$ has arbitrary depth, using Kirillov model. If time permits, we will discuss the general case by local Rankin–Selberg convolutions.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 18 May 2024