On the blocks of modular Hecke algebras (Q1210086)

Let \(G = (G,B,N,R,U)\) be a finite group with a split \(BN\)-pair of characteristic \(p\). Let \(k\) be a field of characteristic \(p\) and \((W,R)\) be the Coxeter system of \(G\). It is known from \textit{M. Cabanes} [J. Fac. Sci., Univ. Tokyo, Sect. I A 36, 347-362 (1989; Zbl 0727.20006)] that the \(p\)-blocks of the Hecke algebra \(E = \text{End}_{kG}(k^ G_ U)\) are indexed by the \(W\)-orbits of \(\widehat H\); the set of multiplicative characters of \(H = B \cap N\). If \(\chi \in \widehat H\), write \(\mathbb{B}_ \chi\) for the corresponding \(p\)-block of \(E\). In the paper under review a basis and a presentation for \(\mathbb{B}_ \chi\) are given by which the central elements of \(\mathbb{B}_ \chi\) are (partially) described. It turns out that \(\dim_ k\mathbb{B}_ \chi = | W||(\chi)|\), where \((\chi)\) is the \(W\)-orbit of \(\chi\) in \(\widehat H\). The case when \(\chi\) is regular is fully studied. In this case the irreducible characters as well as the projective indecomposable modules are determined. It turns out that such blocks have singular Cartan matrices.