Inducing \(W\)-graphs.
Let \(W=(W,S)\) be a Coxeter group with \(S\) the set of simple reflections. Let \(\mathcal H\) be the corresponding Hecke algebra over the ring \(\mathcal A=\mathbb{Z}[q^{-1/2},q^{1/2}]\). \textit{D. Kazhdan} and \textit{G. Lusztig} defined a \(W\)-graph [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)] which provides a representation of the algebra \(\mathcal H\). In the paper under review, the authors show that if \(W_J\) is a standard parabolic subgroup of \(W\) and \(V\) is a module for the corresponding Hecke algebra \({\mathcal H}_J\), then a \(W_J\)-graph structure for \(V\) gives rise to a \(W\)-graph structure for the induced module \({\mathcal H}\otimes_{{\mathcal H}_J}V\). In the case that \(J=\emptyset\) and \(V\) has dimension 1, the construction given in the paper coincides with that by Kazhdan and Lusztig for the regular representation of \(\mathcal H\). For an arbitrary \(J\) and one-dimensional \(V\), the paper recovers the constructions by \textit{M. Couillens} [J. Algebra 213, No. 2, 687-720 (1999; Zbl 0981.20031)] and \textit{V. Deodhar} [J. Algebra 190, No. 1, 214-225 (1997; Zbl 0889.20021)].
scientific article; zbMATH DE number 1965497