Steinberg characters for Chevalley groups over finite local rings. (English) Zbl 1124.20029
Let \(G\) be a Chevalley group, possibly ‘extended’ with an enlarged torus as in the case of \(\text{GL}_n\). Let \(R\) be a finite local ring whose residue field has very good characteristic for \(G\). In the case of \(\text{GL}_n\), both P. Lees and G. Hill have introduced a Steinberg character of \(G(R)\). The author generalizes it and shows how it arises as the character of a representation in the top homology of a simplicial complex.
Unlike the case where \(R\) is a field, originally considered by Steinberg, the character need no longer be irreducible and the issue is to find its decomposition. The constituents are described in terms of Gelfand-Graev characters. If \(G\) is a fully extended Chevalley group, then the Steinberg character is irreducible and corresponds with a linear character of a Hecke algebra. This linear character is related to a sign character.
Unlike the case where \(R\) is a field, originally considered by Steinberg, the character need no longer be irreducible and the issue is to find its decomposition. The constituents are described in terms of Gelfand-Graev characters. If \(G\) is a fully extended Chevalley group, then the Steinberg character is irreducible and corresponds with a linear character of a Hecke algebra. This linear character is related to a sign character.
Reviewer: Wilberd van der Kallen (Utrecht)
MSC:
| 20G05 | Representation theory for linear algebraic groups |
| 20G35 | Linear algebraic groups over adèles and other rings and schemes |
| 20C08 | Hecke algebras and their representations |
| 20E42 | Groups with a \(BN\)-pair; buildings |
| 20J05 | Homological methods in group theory |
Keywords:
Chevalley groups; finite local rings; Steinberg characters; combinatorial buildings; Hecke algebras; Gelfand-Graev characters; irreducible constituentsSoftware:
GAPReferences:
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