Tits groups of Iwahori-Weyl groups and presentations of Hecke algebras (Q6655956)

Let \(\mathcal{H}_n=\mathcal{H}(G(F),I_n)\) be the \(\mathbb{Z}\)-algebra of the compactly supported, \(I_n\)-biinvariant functions on \(G(F)\). Here \(G(F)\) is connected reductive group over a non-archimedean local field \(F\) and for \(n\in \mathbb{N}\) and one denotes as \(I_n\) be the \(n\)-th congruence subgroup of a an Iwahori subgroup \(I\subset G(F)\).\N\NThe first main result of the paper is the generalization of the Iwahori-Matsumoto presentation to \(\mathcal{H}_n\): the generators are the characteristic functions on the \(I_n\)-double cosets on \(G(F)\) and the multiplications of the characteristic functions are given via the conditions on the length function of \(W\), where \(W\) is a quasi-Coxeter group, namely, it is a semidirect product of an affine Weyl group with a group of length-zero elements.\N\NThe second result is a construction of a generalizaion of the Howe-Tits presentation for \(\mathcal{H}_n\).