An explicit decomposition of higher Deligne-Lsuztig representations

TL;DR

This paper provides an explicit decomposition of elliptic higher Deligne-Lusztig representations into irreducible components, revealing their structure via characters and connecting to supercuspidal representations.

Contribution

It shows that the geometric analog of the Weil-Heisenberg representation differs from the original by a character and identifies this character explicitly under mild conditions.

Findings

The character difference is explicitly identified as the quadratic character by Fintzen-Kaletha-Spice.

Each unramified Yu type appears in the cohomology of higher Deligne-Lusztig varieties.

Unramified Kaletha's regular supercuspidal representations are induced from higher Deligne-Lusztig representations.

Abstract

In a previous paper, the second named author obtains a decomposition of an elliptic higher Deligne-Lusztig representation into irreducible summands, which are built in the same way as Yu types using a geometric analog $\kappa'$ of the Weil-Heisenberg representation $\kappa$. In this note, we show that $\kappa'$ and $\kappa$ differs by a character $\chi$. Moreover, under a mild condition on the cardinality $q$ of the residue field (for instance $q > 3$), we show that $\chi$ equals the quadratic character constructed by Fintzen-Kaletha-Spice, which gives an explicit irreducible decomposition result on elliptic higher Deligne-Lusztig representations. As an application, we deduce (under the mild condition on $q$) that each unramified Yu type appears in the cohomology of higher Deligne-Lusztig varieties, and each unramified Kaletha's regular supercuspidal representation is the compact induction of a specified higher Deligne-Lusztig representation up to a sign.