\(R\)-matrix arising from affine Hecke algebras and its application to Macdonald's difference operators
A construction of \(R\)-matrices in the sense of Cherednik is presented for an arbitrary affine root system. It means that for any root \(\alpha\) an operator \(R_ \alpha\) is constructed acting on a \(| W|\)- dimensional space \(V\) over a certain field of fractions of the group algebra of a lattice \(L\), including the root lattice and being contained in the weight lattice; \(W\) is the Weyl group of the underlying finite- dimensional Lie algebra. The operators \(R_ \alpha\), \(R_ \beta\) satisfy the analog of the Yang-Baxter equation defined by the structure of the corresponding rank two root system. The construction is based on the two actions of a Hecke algebra on \(V\). A \(q\)-analog of the Knizhnik- Zamolodchikov equations is written down by means of the corresponding \(R\). Cherednik's equations are their specializations. It is shown that the solutions are eigenvectors of Macdonald's difference operators.
scientific article; zbMATH DE number 694967
scientific article