Isometries parfaites entre blocs de groupes linéaires ou unitaires. (Perfect isometries between blocks of linear or unitary groups)
The purpose of this paper is to prove the following theorem ``Let G and \(G'\) be general linear groups, or unitary groups, on the same finite field. Let \(\ell\) be a prime different from the characteristic of the basic field, and b and \(b'\) be \(\ell\)-blocks of G and \(G'\) respectively. If the weights of b and \(b'\) are equal, there exists a perfect isometry between b and \(b'.''\) The weight of an \(\ell\)-block of a general linear or unitary finite group is a family of integers and was introduced by \textit{J. B. Olsson} and \textit{G. O. Michler} [Math. Z. 184, 203-233 (1983; Zbl 0505.20009)]. A perfect isometry between two blocks - in fact between their Grothendieck groups on the complex field - consists of a bijection between the sets of irreducible ordinary characters and a sign-valued function with some good arithmetical properties; this notion was introduced by M. Broué in 1988. The existence of a perfect isometry between b and \(b'\) implies that corresponding ordinary characters have equal heights, that the blocks have the same number of irreducible modular characters, and that the algebras of b and \(b'\) on a suitable complete valued ring have isomorphic centers. The method of proof is similar to the one already used by the same author for the blocks of equal weights of symmetric groups. So it confirms ``the particularly close fit of the Deligne-Luztig theory, the Brauer theory, and the combinatorial theory underlying the character theory of the symmetric group'' [\textit{P. Fong} and \textit{B. Srinivasan}, Invent. Math. 69, 109-153 (1982; Zbl 0507.20007)]. By a theorem of \textit{M. Broué} and \textit{J. Michel} [J. Reine Angew. Math. 395, 56-67 (1989; Zbl 0654.20048)] the proof reduces immediately to the case where each of the blocks contains a unipotent character. Then the weight is an integer and the block is defined by a partition. The sets of conjugacy classes of Brauer's elements are in natural bijection. Elementary facts on partitions, hooks and cores are used to define the required bijection and sign-valued function. This is done for every pair of corresponding classes of Brauer's elements. It is shown, using properties of Lusztig's map \(R_ L^ G\), where L is a regular subgroup of G, and an induction argument, that these isometries preserve ``projectivity'' of generalized characters; this last fact implies that the isometry between b and \(b'\) is perfect.
scientific article; zbMATH DE number 4136220