Character stabilizer limits relative to a normal nilpotent subgroup

Abstract

In his recent paper [ 11, Dade established the existence of some (rather unexpected) invariants of certain irreducible characters of finite groups. The purpose of the present paper is to state and to provide a relatively easy proof for a somewhat weakened and less general form of Dade’s result. In fact, Dade, in his introduction to [ 11, explicitly encouraged me to publish this simplified approach. To explain and motivate our theorem, we need to discuss “stabilizer limits.” Let x E Irr(G). (We work only with characters over the complex numbers. Dade, on the other hand, considers other characteristic zero fields too.) If Ma G and 8 is an irreducible constituent of xM, let T= Z&O), the inertia group. Then there is a unique character VE Irr(TI 0) such that qG =x. We call q the Clzfford correspondent of x with respect to 0. (Note that by taking M = 1, we see that x is one of its own Clifford correspondents.) We can repeat this process and consider Clifford correspondents (in T) for the Clifford correspondent q E Irr( T) of x. A character $ arising via any number of such iterations, we call a compound ClifSord correspondent (CCC) of x and we denote the set of these objects by CCC(x). Note that if Ic/ ECCC(X), then Ic/“=x. Consider “minimal” element of CCC(x). These are the compound Clifford correspondents $ of x which themselves have no proper Clifford correspondents. (In other words, they are quasiprimitive.) Following T. Berger and Dade, we call these characters the stabilizer limits of x. What do the different stabilizer limits for a given x E Irr(G) have in common? In general, the answer is “not much”.