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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The canonical compactification of a finite group of Lie type


Authors: Mohan S. Putcha and Lex E. Renner
Journal: Trans. Amer. Math. Soc. 337 (1993), 305-319
MSC: Primary 20M30; Secondary 20G05
MathSciNet review: 1091231
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Abstract: Let $ G$ be a finite group of Lie type. We construct a finite monoid $ \mathcal{M}$ having $ G$ as the group of units. $ \mathcal{M}$ has properties analogous to the canonical compactification of a reductive group. The complex representation theory of $ \mathcal{M}$ yields Harish-Chandra's philosophy of cuspidal representations of $ G$. The main purpose of this paper is to determine the irreducible modular representations of $ \mathcal{M}$. We then show that all the irreducible modular representations of $ G$ come (via the 1942 work of Clifford) from the one-dimensional representations of the maximal subgroups of $ \mathcal{M}$. This yields a semigroup approach to the modular representation theory of $ G$, via the full rank factorizations of the 'sandwich matrices' of $ \mathcal{M}$. We then determine the irreducible modular representations of any finite monoid of Lie type.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1091231-X
PII: S 0002-9947(1993)1091231-X
Article copyright: © Copyright 1993 American Mathematical Society