The canonical compactification of a finite group of Lie type

Putcha, Mohan S.; Renner, Lex E.

doi:10.1090/S0002-9947-1993-1091231-X

The canonical compactification of a finite group of Lie type
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by Mohan S. Putcha and Lex E. Renner PDF

Trans. Amer. Math. Soc. 337 (1993), 305-319 Request permission

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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