Unitriangular Shape of Decomposition Matrices of Unipotent Blocks

Abstract

We show that the decomposition matrix of unipotent $\ell$-blocks of a finite reductive group $\mathbf{G}({\mathbb{F}}_q)$ has a unitriangular shape, assuming $q$ is a power of a good prime and $\ell$ is very good for $\mathbf{G}$. This was conjectured by Geck in 1990 as part of his PhD thesis. We establish this result by constructing projective modules using a modification of generalised Gelfand--Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity one. This establishes a 30 year old conjecture of Kawanaka.

v1: 71 pages. v2: 73 pages. Added an acknowledgment, some references, and Lemma 10.2. Minor linguistic changes throughout. v3: 74 pages. Post referee. Added Lem 6.9. Improved the assumtpions in Thm A. Updated the proof in Section 11. To appear Ann. of Math

References in corpus (6)

• Nilpotent orbits over ground fields of good characteristic
• On Unipotent Supports of Reductive Groups with a Disconnected Centre
• Restriction of a character sheaf to conjugacy classes
• Quasi-semisimple elements
• Optimal SL(2)-homomorphisms
• Lusztig's conjecture for finite special linear groups

Cited by in corpus (5)

• Galois Automorphisms and Classical Groups
• Jordan decomposition for weights and the blockwise Alperin weight conjecture
• Decomposition numbers for the principal ${\Phi }_{2n}$-block of ${\mathrm{Sp}}_{4n}(q)$ and ${\mathrm{SO}}_{4n+1}(q)$
• Decomposition numbers for unipotent blocks with small ${\mathfrak{sl}}_2$-weight in finite classical groups
• Unitriangular basic sets, Brauer characters and coprime actions