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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)
CHEVIE
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##### References:
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