Orbit method for $p$-Sylow subgroups of finite classical groups

Abstract

For the $p$-Sylow subgroups $U$ of the finite classical groups of untwisted Lie type, $p$ an odd prime, we construct a monomial $\mathbb C U$-module $M$ which is isomorphic to the regular representation of $\mathbb C G$ by a modification of Kirillov's orbit method called monomial linearisation. We classify a certain subclass of orbits of the $U$-action on the monomial basis of $M$ consisting of so called staircase orbits and show, that every orbit module in $M$ is isomorphic to a staircase one. Finally we decompose the Andr\'e-Neto supercharacters of $U$ into a sum of $U$-characters afforded by staircase orbit modules contained in $M$.