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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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