Explicit Plancherel theorems for \({\mathcal H}(q_1,q_2)\) and \(\text{SL}_2(F)\)
For real numbers \(q_1, q_2\) with \(q_1 \geq q_2 \geq 1\), the authors introduce an affine Hecke algebra \(\mathcal H = \mathcal H(q_1, q_2)\) in two parameters, and its completion \(A = A(q_1, q_2)\) with respect to a norm on \(\mathcal H\). They determine explicitly the topological structure of the full unitary dual \(\hat{A}\) of the enveloping \(C^*\)-algebra \(C^*(A)\) and determine an explicit Plancherel formula for the reduced \(C^*\)-algebra \(C^*_r(A)\) (a quotient of \(C^*(A)\)). Then, as an application, they determine an explicit Plancherel formula for \(G = \mathbb S\mathbb L_2(F)\), where \(F\) is a complete nonarchimedean local field of arbitrary residual characteristic \(p\). Explicit Plancherel formulas for affine Hecke algebras and more general algebras have already been known and also for \(\mathbb S\mathbb L_2(F)\) in the case \(p \neq 2\), but the authors say in the introduction that they use a deformation argument, completely avoiding residue calculus and that their approach to the Plancherel formula for \(\mathbb S\mathbb L_2(F)\) is very different from previous methods and provides a conceptional route to that formula.
scientific article; zbMATH DE number 5543155