The canonical compactification of a finite group of Lie type. (English) Zbl 0787.20039
For a finite group \(G\) of Lie type, a finite monoid \(M\) having \(G\) as the group of units is constructed. The complex representations of \(M\) yield Harish-Chandra philosophy of cuspidal representations of \(G\), and have been used by the first author and the reviewer to prove the complete reducibility of all finite monoids of Lie type [Int. J. Algebra Comput. 1, No. 1, 33-47 (1991; Zbl 0752.20034) and Trans. Am. Math. Soc. 323, No. 2, 563-581 (1991; Zbl 0745.20057)]. The authors determine all modular representations of \(M\). They come from the one-dimensional representations of the maximal subgroups of \(M\) and restrict to irreducible representations of \(G\). This gives a new approach to the modular representation theory of finite groups of Lie type – via so called sandwich matrices of the monoid \(M\), defined entirely in terms of \(G\).
Reviewer: J.Okniński (Warszawa)
MSC:
| 20M30 | Representation of semigroups; actions of semigroups on sets |
| 20G05 | Representation theory for linear algebraic groups |
| 20C33 | Representations of finite groups of Lie type |
| 20M20 | Semigroups of transformations, relations, partitions, etc. |
| 20G40 | Linear algebraic groups over finite fields |
Keywords:
finite monoid; group of units; complex representations; cuspidal representations; complete reducibility; finite monoids of Lie type; modular representations; one-dimensional representations; maximal subgroups; irreducible representations; finite groups of Lie type; sandwich matricesReferences:
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