Simple endotrivial modules for quasi-simple groups

Abstract

We investigate simple endotrivial modules of finite quasi-simple groups and classify them in several important cases. This is motivated by a recent result of Robinson [41] showing that simple endotrivial modules of most groups come from quasi-simple groups.

A Murnaghan–Nakayama rule for values of unipotent characters in classical groups

“…Together with our previous work [10,11] with C. Lassueur and E. Schulte our results allow us to guarantee the existence of zeroes of characters of quasi-simple groups on -singular elements once the -rank is at least 3, as claimed in Theorem 1 which we restate: (1), (2) and (3) can arise. For the remaining groups of Lie type first note that the Steinberg character vanishes on the product of any -element with a unipotent element in its centraliser, so it can be discarded from our discussion.…”

“…(c) The results of [11], [10] and of the present paper show that even for -rank 2 there exist only relatively few irreducible characters of quasi-simple groups not vanishing on some -singular element. Nevertheless we refrain from attempting to give an explicit list in that case.…”

“…Since the -rank of G is at least 3, it must be of type F 4 , E 6 , 2 E 6 , E 7 or E 8 . For these it is shown in the proof of [11,Thm. 6.11] that their irreducible characters χ vanish on some -singular element unless either χ is unipotent or |(q 2 + 1) in G = E 8 (q) and χ lies in the Lusztig series of an isolated element with centraliser D 8 (q).…”

“…Our approach relies on the fact, proven in [11,Thm. 1.3] that any endotrivial module is liftable to a characteristic 0 representation, which can then be studied by ordinary character theory.…”

“…We will use the fact shown in [11,Thm. 1.3] that any endotrivial module is liftable to a characteristic 0 representation.…”