Irreducible characters of finite simple groups constant at the $p$-singular elements

Abstract

In representation theory of finite groups an important role is played by irreducible characters of $p$-defect 0, for a prime $p$ dividing the group order. These are exactly those vanishing at the $p$-singular elements. In this paper we generalize this notion investigating the irreducible characters that are constant at the $p$-singular elements. We determine all such characters of non-zero defect for alternating, symmetric and sporadic simple groups.

We also classify the irreducible characters of quasi-simple groups of Lie type that are constant at the non-identity unipotent elements.In particular, we show that for groups of BN-pair rank greater than 2 the Steinberg and the trivial characters are the only characters in question. Additionally, we determine all irreducible characters whose degrees differ by 1 from the degree of the Steinberg character.