In representation theory of finite groups an important role is played by irreducible characters of -defect 0, for a prime dividing the group order. These are exactly those vanishing at the -singular elements. In this paper we generalize this notion investigating the irreducible characters that are constant at the -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.
Cite this article
Marco A. Pellegrini, Alexandre Zalesski, Irreducible characters of finite simple groups constant at the -singular elements. Rend. Sem. Mat. Univ. Padova 136 (2016), pp. 35–50DOI 10.4171/RSMUP/136-4