The Picky Conjecture for groups of Lie type

TL;DR

This paper proves the Picky Conjecture for all quasi-simple groups of Lie type at non-defining primes, advancing the understanding of character correspondences and classifying special elements within these groups.

Contribution

It establishes the Picky Conjecture for a broad class of groups and explores stronger versions under natural assumptions, also classifying key elements.

Findings

Proved the Picky Conjecture for all quasi-simple groups of Lie type at non-defining primes.

Classified picky 2- and 3-elements in these groups.

Demonstrated conditions under which stronger character value preservation holds.

Abstract

Recently, Moret\'o and Rizo proposed a conjecture, known as the Picky Conjecture, proposing new character correspondences extending the McKay Conjecture. We prove the Picky Conjecture for all quasi-simple groups of Lie type for non-defining primes. In favourable situations, we also obtain the stronger version postulating preservation of character values up to sign, and we show this stronger version holds in general when assuming certain natural properties of Lusztig's Jordan decomposition. Along the way, we complete the determination of semisimple picky elements in these groups by classifying picky 2- and 3-elements.