Supersingular representations of $p$-adic reductive groups

Abstract: The local Langlands conjectures predict that (packets of) irreducible complex representations of $p$-adic reductive groups (such as $\mathrm{GL}_n(\mathbb{Q}_p)$, $\mathrm{GSp}_{2n}(\mathbb{Q}_p)$, etc.) should be parametrized by certain representations of the Weil-Deligne group.  A special role in this hypothetical correspondence is held by the supercuspidal representations, which generically are expected to correspond to irreducible objects on the Galois side, and which serve as building blocks for all irreducible representations.  Motivated by recent advances in the mod-$p$ local Langlands program (i.e., with mod-$p$ coefficients instead of complex coefficients), I will give an overview of what is known about supersingular representations of $p$-adic reductive groups, which are the "mod-$p$ coefficients" analogs of supercuspidal representations.  This is joint work with Florian Herzig and Marie-France Vigneras.

number theory

Audience: researchers in the topic

Harvard number theory seminar