The scalar product formula for parahoric Deligne--Lusztig induction

TL;DR

This paper proves that the scalar product formula holds for parahoric Deligne--Lusztig induction in a broad class of cases, confirming a fundamental conjecture in the representation theory of p-adic groups.

Contribution

It establishes the scalar product formula for all Howe-factorizable split-generic pairs, including all characters when T is elliptic and p is not a torsion prime.

Findings

Scalar product formula verified for all Howe-factorizable split-generic pairs

Confirmed the conjecture for characters when T is elliptic and p is not a torsion prime

Advances understanding of positive-depth representations of p-adic groups

Abstract

Parahoric Deligne--Lusztig induction gives rise to positive-depth representations of parahoric subgroups of $p$-adic groups. The most fundamental basic question about parahoric Deligne--Lusztig induction is whether it satisfies the scalar product formula. We resolve this conjecture for all Howe-factorizable split-generic pairs $(T,\theta)$ -- in particular, for all characters $\theta$ when $T$ is elliptic and $p$ is not a torsion prime for the root system of the $p$-adic group.