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Irreducible projective modules of the Hecke algebras of a finite Coxeter group. (English) Zbl 0689.20039
Let $$W$$ be a finite Coxeter group, $$t$$ an indeterminate, and $$q=t^ 2$$. Let $$H(W,q)$$ be the generic Hecke algebra over $${\mathbb C}(t)$$ associated with $$W$$. The author studies the irreducible projective modules of the specialized Hecke algebra $$H(W,\zeta^ 2)$$ where $$\zeta\in {\mathbb C}$$. It was shown by A. Gyoja [J. Algebra 86, 422–483 (1984; Zbl 0537.20020)] that any irreducible representation of $$H(W,q)$$ arises from a $$W$$-graph in the sense of B. Kazhdan and G. Lusztig [Invent. Math. 53, 165–184 (1979; Zbl 0499.20035)]. Let $$\lambda (W)$$ be a complete set of $$W$$-graphs corresponding to a set of representatives of the isomorphism classes of irreducible representations of $$H(W,q)$$. For $$\lambda\in \Lambda (W)$$ let $$\pi^ t_{\lambda}$$ be the corresponding representation of $$H(W,q)$$, and let $$d_{\lambda}(q)$$ be the generic degree polynomial associated with $$\pi^ t_{\lambda}$$. Let $$\pi^{\zeta}_{\lambda}$$ be the (not necessarily irreducible) representation of the $$\mathbb C$$-algebra $$H(W,\zeta^ 2)$$ obtained from $$\pi^ t_{\lambda}$$ by specializing $$t$$ to $$\zeta\in \mathbb C$$. The author shows (Theorem 3.1) that if $$\zeta$$ is not zero then a complete set of representatives for the isomorphism classes of irreducible projective $$H(W,\zeta^ 2)$$-modules is given by $$\pi^{\zeta}_{\lambda}$$, where $$\lambda$$ runs over elements of $$\Lambda (W)$$ such that $$\lim_{q\to \zeta^ 2}P_ W(q)/d_{\lambda}(q)$$ is not zero. (Here $$P_ W(q)$$ is the Poincaré polynomial of $$W$$.)
As a corollary he recovers a theorem of A. Gyoja and K. Uno [J. Math. Soc. Japan 41, 75–79 (1989; Zbl 0647.20038)] that $$H(W,\zeta^ 2)$$ is semisimple if and only if $$P_ W(\zeta^ 2)$$ is not zero. Finally, he also computes the irreducible projective representations of $$H(W,0)$$; these are one-dimensional.
Reviewer: B.Srinivasan

##### MSC:
 20G05 Representation theory for linear algebraic groups 20C08 Hecke algebras and their representations 20C25 Projective representations and multipliers 20C30 Representations of finite symmetric groups
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##### References:
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