I would like to understand in a more explicit way the Fell topology on unitary duals, that is to say the convergence of matrix coefficients of local representations.
If I consider a local representation $pi_lambda$ of a certain type (e.g. principal series for $GL(2)$ over a non-archimedean field), I would like to get an explicit dependence of the matrix coefficient $langle pi_lambda(g)v, wrangle_lambda$ in terms of the parameter $lambda$. This in particular involve explicit choices of a model for the representation and specific vectors: has this been done explicitly somewhere?
Ultimately, I want to see to what extent the Fell topology on representations corresponds to the usual topology on its (finite-dimensional) parameters.