The irreducible Brauer characters of the finite special linear groups in non-describing characteristics
Let \(\widetilde G:=\text{GL}_n(q)\) and \(G:=\text{SL}_n(q)\). In the present paper the author studies the irreducible Brauer characters for \(G\) and primes \(\ell\) not dividing \(q\). They are described in terms of a basic set of ordinary characters of \(G\) which form a basis for the Abelian group of Brauer characters of \(G\). Their decomposition matrices are closely connected to the corresponding \(\ell\)-decomposition matrices for \(\widetilde G\) and the decomposition matrices of `extended \(q\)-Schur algebras' \({\mathcal S}({\mathcal H})\). These are defined in section two of the paper as endomorphism rings of certain modules of extended Hecke algebras of type \(A\). In particular, it is proved that these later decomposition matrices are lower unitriangular and even square lower unitriangular, if \({\mathcal S}({\mathcal H})\) is an extension of a \(q\)-Schur algebra which is coprime to \(\ell\). It is known that ordinary \(q\)-Schur algebras determine the decomposition matrix of \(\widetilde G\). The main theme of this article is to show how the extended \(q\)-Schur algebras \({\mathcal S}({\mathcal H})\) do the same for \(G\). Major tools in the \(\ell\)-modular representation theory of \(G\) and \(\widetilde G\) are the Gelfand-Graev characters, which are induced from non-degenerate characters of the unipotent (\(\ell'\)-)group \(U\) and therefore provide useful projective lattices. All needed results about those and their connections to cuspidal Brauer characters are collected in Section 3. As a major tool to link decomposition numbers with Harish-Chandra functors and series, the theory of `projective restriction systems' developed jointly with R. Dipper is introduced in Section 4. This theory can be formulated for groups with split BN-pair and thus it can be generalized to other finite groups of Lie type. It relies on a number of assumptions (e.g. purity of certain sublattices) which are later verified to hold for the useful submodules of the Gelfand-Graev lattices in Section 5. In Section 6 all these results are used to show that the decomposition matrix of the characters in the Lusztig series \({\mathcal E}(G,s)\) can be completely derived from the decomposition matrices of a certain extended \(q\)-Schur algebra \({\mathcal S}({\mathcal H})\) defined with respect to a minimal Levi subgroup of \(\widetilde G\) which contains the semisimple \(\ell'\)-element \(s\). Using some technical results of Section 7, this result is extended in Section 8 to certain Lusztig series parametrized by \(\ell\)-singular elements of the dual group \(G^*\). In Section 9 a basic set of characters of \(G\) is described, which is a subset of those series considered earlier and thus provides a complete description of the \(\ell\)-decomposition matrix of \(G\) in terms of \(q\)-Schur algebras of type \(A\).
scientific article; zbMATH DE number 1295452
scientific article