Green functions for positive-depth Deligne--Lusztig induction

TL;DR

This paper provides an explicit geometric description of positive-depth Deligne--Lusztig induction for unramified elliptic pairs, linking it to $L$-packets and orbital integrals, with applications to the Springer hypothesis and character formulas.

Contribution

It introduces a simple characterization theorem for positive-depth Deligne--Lusztig induction and compares Green functions from algebraic and geometric perspectives, advancing understanding of supercuspidal representations.

Findings

Explicit description of positive-depth Deligne--Lusztig induction

Comparison of Green functions from different constructions

Proof of the positive-depth Springer hypothesis in the $0$-toral case

Abstract

Under a largeness assumption on the size of the residue field, we give an explicit description of the positive-depth Deligne--Lusztig induction of unramified elliptic pairs $(T,\theta)$. When $\theta$ is regular, we show that positive-depth Deligne--Lusztig induction gives a geometric realization of Kaletha's Howe-unramified regular $L$-packets. This is obtained as an immediate corollary of a very simple "litmus test" characterization theorem which we foresee will have interesting future applications to small-$p$ constructions. We next define and analyze Green functions of two different origins: Yu's construction (algebra) and positive-depth Deligne--Lusztig induction (geometry). Using this, we deduce a comparison result for arbitrary $\theta$ from the regular setting. As a further application of our comparison isomorphism, we prove the positive-depth Springer hypothesis in the $0$-toral setting and use it to give a geometric explanation for the appearance of orbital integrals in supercuspidal character formulae.