On representations of twisted group rings

Abstract

We generalize certain parts of the theory of group rings to the twisted case. Let G be a finite group acting (possibly trivially) on a field L of characteristic coprime to the order of the kernel of this operation. Let K in L be the fixed field of this operation, let S be a discrete valuation ring with field of fractions K, maximal ideal generated by pi and integral closure T in L. We compute the colength of the twisted group ring T G in a maximal order in L G. Moreover, if S/pi S is finite, we compute the S/pi S- dimension of the center of T G/Jac(T G). If this quotient is split semisimple, this yields a formula for the number of simple T G-modules, generalizing Brauer's formula.