Deligne-Lusztig Constructions for Division Algebras and the Local Langlands Correspondence

TL;DR

This paper demonstrates that for the case n=2, a Deligne-Lusztig scheme provides a geometric realization of the local Langlands and Jacquet-Langlands correspondences for division algebras over local fields.

Contribution

It proves that the $p$-adic Deligne-Lusztig scheme induces a correspondence matching the LLC and JLC for n=2, linking geometric constructions to representation theory.

Findings

The scheme induces a correspondence between characters of $L^ imes$ and representations of $D^ imes$.

The correspondence aligns with the local Langlands and Jacquet-Langlands correspondences.

The result confirms Lusztig's conjectural geometric approach for n=2.

Abstract

Let $K$ be a local non-Archimedean field of positive characteristic and let $L$ be the degree-$n$ unramified extension of $K$. Via the local Langlands and Jacquet-Langlands correspondences, to each sufficiently generic multiplicative character of $L$, one can associate an irreducible representation of the multiplicative group of the central division algebra $D$ of invariant $1/n$ over $K$. In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive $p$-adic groups analogous to Deligne-Lusztig theory for finite reductive groups. In this paper we prove that when $n=2$, the $p$-adic Deligne-Lusztig (ind-)scheme $X$ induces a correspondence between smooth one-dimensional representations of $L^\times$ and representations of $D^\times$ that matches the correspondence given by the LLC and JLC.