Path models and combinatorial representations of affine Hecke algebras

Abstract

In this thesis we explore particular combinatorial representations of affine Hecke algebras with general parameters. We introduce the combinatorial model of J-folded alcove paths and show that the matrix entries of our representations are described by these alcove paths. We categorise which of the representations are bounded and give conjectures connecting the representations to Kazhdan-Lusztig theory and Opdam’s Plancherel Theorem. Upon specialising to type affine An, we show that the combinatorial representations form a balanced system of cell representations. For this type, we prove an asymptotic version of Opdam’s Plancherel Theorem and develop a Janalogue to the classical Satake isomorphism. Using this asymptotic Plancherel Theorem we construct a new explicit description of Lusztig’s asymptotic algebra in type affine An in terms of a ring of specialised matrices formed from the constructed representations.