Cet exposé est en distanciel.

Abstract: Let \( K \) be a number field and let \( G \) be a connected reductive algebraic group defined over \( K \). Let \( \sigma \) be an automorphism of \( G \) of prime order \( l \). In the foundational work [TV16], Treumann and Venkatesh established a functoriality lifting of mod-\( l \) Hecke eigenvalues of \( G^{\sigma} \) to mod-\( l \) Hecke eigenvalues of \( G \), where \( G^{\sigma} \) is the connected component of the fixed points of \( \sigma \). They also made some conjectures for representation theory of \( p \)-adic groups and these conjectures predict that the (local) functoriality lifting is compatible with Tate cohomology for the action of \( \langle \sigma \rangle \).

In this talk, we discuss this conjecture in the setting of cyclic base change for the general linear group \( G = \mathrm{GL}_n \). Say \( F \) is a finite extension of \( \mathbb{Q}_p \) and let \( E \) be a finite Galois extension of \( F \) with degree of extension \( l \), where \( l \) and \( p \) are distinct primes. Let \( \pi \) be an irreducible integral \( l \)-adic representation of \( \mathrm{GL}_n(F) \), and \( \Pi \) be an irreducible integral \( l \)-adic representation of \( \mathrm{GL}_n(E) \) obtained as base change lift of \( \pi \). Then there are two representations of \( \mathrm{GL}_n(F) \), defined over \( \overline{\mathbb{F}}_l \), namely the mod-\( l \) reduction \( r_l(\pi) \) and the Tate cohomology group \( \widehat{H}^i(\sigma, \Pi) \), \( i \in \{0, 1\} \), for the action of \( \langle \sigma \rangle \) on \( \Pi \). Treumann–Venkatesh’s conjecture relates these two representations. We discuss this in the case where \( \pi \) and \( \Pi \) are both generic, and \( l \) does not divide the order of \( \mathrm{GL}_{n-1}(\mathbb{F}_q) \) for \( n \geq 3 \), where \( q \) is the cardinality of the residue field of \( F \).

[TV16] David Treumann and Akshay Venkatesh, Functoriality, Smith theory, and the Brauer homomorphism, Ann. of Math. (2) 183 (2016), no. 1, 177–228. MR 3432583