On endomorphism algebras arising from multiparameter Hecke algebras
Let \(\{G(q)\}\) be a family of finite groups of Lie type having \((W,S)\) as irreducible Coxeter system and let \(H\) be the Hecke algebra corresponding to \(G(q)\) over \(A=\mathbb{Z}[q^{1/2},q^{-1/2}]\); the ring of Laurent polynomials in the indeterminate \(q^{1/2}\). For any \(I\subseteq S\), let \(x_I=\sum_{w\in W_I} T_\omega\), where \(W_I\) is the standard parabolic subgroup of \(W\) generated by \(I\). Let \({\mathcal A}=\text{End}_H(\oplus_I x_iH)\). When \(G(q)=\text{GL}(n,q)\) then \({\mathcal A}\) is known as a \(q\)-Schur algebra. In this paper the author considers the case when \(G=U_{2n}(q)\) or \(U_{2n+1}(q)\) and studies the representation theory of \({\mathcal A}\) in those cases by using Kazhdan-Lusztig theory. The main result (Theorem 2.7) is that the Kazhdan-Lusztig basis of \({\mathcal A}\) is a standard basis (see 2.1 for definition). He also describes the simple modules and standard modules in the category of left \({\mathcal A}\)-modules.
scientific article; zbMATH DE number 1076277
scientific article