Certain group dynamical systems induced by Hecke algebras

Abstract

In this paper, we study dynamical systems induced by a certain group \(\mathfrak{T}_{N}^{K}\) embedded in the Hecke algebra \(\mathcal{H}(G_{p})\) induced by the generalized linear group \(G_{p} = GL_{2}(\mathbb{Q}_{p})\) over the \(p\)-adic number fields \(\mathbb{Q}_{p}\) for a fixed prime \(p\). We study fundamental properties of such dynamical systems and the corresponding crossed product algebras in terms of free probability on the Hecke algebra \(\mathcal{H}(G_{p})\).

Keywords

• free probability
• free probability, free moments
• free cumulants
• Hecke algebra
• normal Hecke subalgebra
• free probability spaces
• representations
• operators
• Hilbert spaces
• dynamical systems
• crossed product algebras