Structure of Iwahori-Hecke algebras of type \(B\) and \(D\). Generic isomorphisms (Q2109060)

Let \(S_{n}\) be the symmetric group of degree \(n\) and \(C_{2}\) the cyclic group of order 2. The Weyl group \(WB_{n}\) of type B is (as abstract group) the wreath product \(WB_{n} \simeq C_{2} \wr S_{n} \simeq (C_{2})^{n} \rtimes S_{n}\). The Weyl group \(WD_{n}\) of type D can obtained from the previous one and, as an abstract group, it is isomorphic to \((C_{2})^{n-1} \rtimes S_{n}\). In a previous work [J. Algebra 581, 278--302 (2021; Zbl 07354445)] the author gave a explicit algebraic construction of a generic isomorphism between the Iwahori-Hecke algebra of type A and the group algebra of the symmetric group \(S_{n}\). In the paper under review he generalizes these results by constructing an isomorphism between the Hecke algebra \(\mathcal{H}(WB_{n})(p,q)\) and the group algebra of \(WB_{n}\). The isomorphism thus constructed is compatible with Hoefsmit's model of irreducible representations of these two algebras. Moreover, the restriction of this isomorphism to a subalgebra of \(\mathcal{H}(WB_{n})(p,q)\), isomorphic to the Hecke algebra of type D, allows the author to establish an analogous isomorphism between the Hecke algebra of type D and the group algebra of the Weyl group \(WD_{n}\).