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 CU -module M which is isomorphic to the regular representation of CU 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 André-Neto supercharacters of U into a sum of U -characters afforded by staircase orbit modules contained in M .

On coadjoint orbits for $p$-Sylow subgroups of finite classical groups

“…In this preliminary section we collect notation and some basic facts from [7]. Throughout U = U (B n ), U (C n ) or U (D n ) denotes the p-Sylow subgroup of the finite group of Lie type B n , C n and D n respectively, constructed in [7], where the describing characteristic of the group is the odd prime p.…”

“…More precisely for u ∈ U and (r, s) ∈ we have shown in [7] that u rs is determined by the entries on the positions which are to the left of (r, s) or to the left of or on (s, r), which we illustrate as follows:…”

“…In a previous paper [7] we exhibited a monomial linearisation for the p-Sylow subgroups U n of finite groups of Lie types B n , D n and C n , by constructing a monomial basis for the group algebra CU n . Thus we extended the construction of [8] to all classical groups of untwisted Lie type.…”

“…In this sequel to [7] we investigate even more specialized orbits, called main separated orbits and show, that every irreducible CU n -module occurs as irreducible constituent in one of those. Our goal for this paper is to prove, that the characters afforded by main separated orbit modules coincide or are orthogonal.…”

“…Section 2 is preliminary collecting the main definitions and results of [7]. In section 3 we define the notation of main separated orbits and show, that every irreducible CU n -module is constituent of a main separated orbit module.…”