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Edit due to the comment.

Consider $G=GL(2)$ over a local field $F$. The Fell topology on the unitary dual of $G(F)$ is seperable.

Given a sequence of irreducible unitary representations $(\pi_n)$ of $G(F)$ and a unitary representation $\pi$, are the following statements equivalent?

  1. $\pi_n$ converges to $\pi$ in the Fell topology, i.e., for any matrix coefficient $f$ of $\pi$, there exists a sequence of matrix coefficients $f_n$ of $\pi_n$ with $f_n \rightarrow f$ locally uniformly.

  2. The tuple $[ L(s, \pi_n, \psi), \epsilon(s, \pi_n, \psi)]$ converge to $[ L(s, \pi, \psi), \epsilon(s, \pi, \psi)]$ for all $s \in \mathbb{C}$ and some/each fixed additive character $\psi$ of $F$.

  3. The character distributions of $\pi_n$ converge to that of $\pi$ as distributions (functionals on $C_c^\infty(G(F))$.

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Btw, it holds for GL(1) as well;) – Marc Palm Jul 4 '13 at 13:07
@PM, the Fell topology is not Hausdorff in general (it can be shown that restricted to the principal series rep. it is indeed T2), so one have to be a bit careful with the convergence notions. Do you know for example if it works for sequence of principal series rep. which converge to limits of discrete series (I'm thinking about $SL_2(\mathbb{R})$ here)? – Asaf Jul 5 '13 at 10:42
For one limit of discrete series representation, only half of the matrix coefficient can be approximated by matrix coefficients from irreducible principal series, or not? – Marc Palm Jul 5 '13 at 16:05
Ah okay, I see that the limits of the L-functions therefore coincide (almost by definition) and so do the $\epsilon$-factors. – Marc Palm Aug 14 '13 at 12:31

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