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Left cells and constructible representations. (English) Zbl 1124.20003
To a finite Coxeter group \(W\), we can associate a Hecke algebra \(\mathcal H\), with its coefficient ring involving either one parameter or \(\geqslant 2\) unequal parameters. In the one parameter case, G. Lusztig showed [in C. R. Acad. Sci., Paris, Sér. I 302, 5–8 (1986; Zbl 0615.20020)] that (i) the representations of \(\mathcal H\) afforded by the left cells of \(W\) (called the left cell representations) are precisely the constructible representations; and also showed [in Algebraic groups and related topics, Proc. Symp., Kyoto and Nagoya/Jap. 1983, Adv. Stud. Pure Math. 6, 255–287 (1985; Zbl 0569.20032); J. Algebra 109, 536–548 (1987; Zbl 0625.20032)] that (ii) \(W\), \(\mathcal H\) satisfy the properties (P1)-(P15). Later, G. Lusztig conjectured [in Hecke algebras with unequal parameters. CRM Monogr. Ser. 18, Providence, RI: AMS (2003; Zbl 1051.20003)] that both assertions (i)-(ii) should also hold in the \(\geqslant 2\) unequal parameters case.
The paper under review shows that in the \(\geqslant 2\) unequal parameters case, assertion (i) is a consequence of assertion (ii). The arguments make use of the idea of A. Gyoja [in Osaka J. Math. 33, No. 2, 307–341 (1996; Zbl 0872.20040)] and R. Rouquier [in C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 12, 1037–1042 (1999; Zbl 0947.20005)] that all the left cell representations of \(\mathcal H\) are projective in the modular representation theory. The arguments for \(W\) being of type \(B_m\) need a purely combinatorial identity about the constructible representations, the latter is proved by the fact that the constructible representations can be identified with the canonical basis of a certain irreducible representation of Lie algebra, while the arguments for \(W\) being of exceptional type rely on the properties (P1)-(P15) and some explicit computations on characters.

MSC:
20C08 Hecke algebras and their representations
20F55 Reflection and Coxeter groups (group-theoretic aspects)
Software:
CHEVIE
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References:
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