Hecke algebras of classical type and their representation type

Abstract

Let $W$ be a finite Weyl group of classical type which may not be irreducible, $F$ an algebraically closed field, $q$ an invertible element of $F$. We denote by $\mathcal H_W(q)$ the associated Hecke algebra. If $q=1$ then it is $FW$ and we know the representation type. Thus, we assume that $q\ne 1$. Let $P_W(x)$ be the Poincare polynomial of $W$. It is well-known that $\mathcal H_W(q)$ is semisimple if and only if $x-q$ does not divide $P_W(x)$. We show that the similar results hold for finiteness, tameness and wildness. In other words, the Poincare polynomial governs the representation type of $\mathcal H_W(q)$ completely. Note that the finiteness result was already given in the author's previous papers, some of which were written with Andrew Mathas. The proof uses the Fock space theory, which was developed for proving the LLT conjecture (see AMS Univ. Lec. Ser. 26), the Specht module theory, which was developed by Dipper, James and Murphy in this case, and results from the theory of finite dimensional algebras.