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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 $\langle \pi_\lambda(g)v, w\rangle_\lambda$ 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.

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  • $\begingroup$ A lot has been known for a long time for unramified representations of p-adic groups (representations with a non-zero fixed vector under a good maximal compact subgroup). These representations belong to the principal series. Cf. the works of Macdonald, Casselman ... $\endgroup$
    – Paul Broussous
    Commented Oct 18, 2020 at 11:15
  • $\begingroup$ @PaulBroussous Thanks! My knowledge in representation theory is limited and this is why I am asking for some guidance. I have the feeling that a lot is known and for sure what I want is scattered somewhere. For instance, I bet I just aim at exemplifying Tadic's paper on the Fell topology to the case of GL(2). However, even after having read some of Casselman's introductory notes and book, and just went through MacDonald, I'm still uneasy. I know standard realizations of principal series, definition of matrix coefficients, but what about explicit choice of inner product, test vectors, etc.? $\endgroup$
    – Desiderius Severus
    Commented Oct 18, 2020 at 13:19
  • $\begingroup$ These matrix coefficients computations and convergence issues may be written somewhere, and I may be underlooking many of the papers I read. However, I have the feeling that computations are either done explicitly with other aims in mind (viz. L-functions or analytic issues, e.g. Ralf Schmidt's notes) or stay in a very general framework concerning matrix coefficients (viz. orthogonality or traces formulas, e.g. Peter-Weyl or Bekka-de la Harpe ; or more general groups, e.g. Tadic) I want to understand precisely how to cope with matrix coefficients, if this can help clarifying the OP. $\endgroup$
    – Desiderius Severus
    Commented Oct 18, 2020 at 13:30
  • $\begingroup$ You can compute the matrix coefficients, at least in the unramified case as Paul says, if you choose v, w to be (normalized) Whittaker newvectors. E.g. see Godement's notes on JL. Is this what you're looking for? I'm not sure I completely understand the question. $\endgroup$
    – Kimball
    Commented Oct 18, 2020 at 14:41

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