On Brauer's Second Main Theorem and characters in blocks with nilpotent coefficient extensions

Abstract

1.1. In [1], J. Alperin and M. Broue introduced a local structure in any block of a finite group. With the structure M. Broue and L. Puig in [2] modified a Frobenius condition for finite p-nilpotent groups to introduce nilpotent blocks and showed how the characters of a nilpotent block are determined by the characters of its defect group. After, L. Puig came to the theory on pointed groups and extended Brauer’s Second Main Theorem, see [9]. Then in [11] Puig determined the precise structure of nilpotent blocks; with the structure theorem and Brauer’s Second Main Theorem in [9], he reproved in [11] the formula on characters in nilpotent blocks. These works are done by assuming that the ground-fields are large enough. Later, the first author considered the case of arbitrary ground-fields, and defined nilpotent blocks with the so-called local control condition (see (1.8.4) below), which is no longer equivalent to the Frobenius condition though the two conditions are equivalent for large enough ground-fields; and extended the structure theorem on nilpotent blocks in [4]. Inspired by a consideration of the Frobenius condition, Puig and the first author in [8] introduced the blocks with nilpotent coefficient extensions (see (1.8.5) below) and characterized the structure of such blocks precisely. The formula on characters in nilpotent blocks was extended to arbitrary ground-fields in [6], but the arguments were based on the formula of Puig because the author did not know what is the correct version of Brauer’s Second Main Theorem over small ground-fields. Here a suitable version we were looking for is presented, i.e., Theorem 1.6 below, which will be proved in Section 2. Applying it, we show a formula on characters in blocks