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Motivation: Let $G$ be an $\ell$-group (locally profinite group). A map $G\to \mathbb{C}$ is called smooth provided that it is continuous as a map $$G\to \mathbb{C}_{discrete}.$$This gives us the correct notion of smoothness for $\ell$-groups.

Question: Can we characterize smoothness topologically in other interesting cases, or is this just a coincidence?

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  • $\begingroup$ What you have written so far is a definition of smooth functions from $G$ to $\mathbb{C}$. I can easily believe that this is a useful definition for p-adic representation theory. But where is the characterization? In particular, do you have some a priori definition for smoothness in a topological group? $\endgroup$ Mar 2, 2010 at 20:52
  • $\begingroup$ Well, no, that's what I was trying to ask. Do you have a suggestion for me to rewrite the question better? $\endgroup$ Mar 2, 2010 at 21:22
  • $\begingroup$ @Marty: If you want to answer the question with that answer, I will accept it. $\endgroup$ Mar 2, 2010 at 21:48
  • $\begingroup$ @fpqc: Done. My comment is erased and my comment is the answer below. $\endgroup$
    – Marty
    Mar 2, 2010 at 21:58

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The question is a bit awkward, as Pete suggests. First, no need to take an $\ell$-group; an $\ell$-space is the right way to start. Second, you've stated the definition of the word "smooth" in this context. Definitions can't be "correct" -- but the word "smooth" is a good choice in this context, because of some parallels between harmonic analysis on p-adic groups and harmonic analysis on real Lie groups.

I think you should put this question aside for a bit.. look at Ralf Meyer's "Smooth Group Representations on Bornological Vector Spaces" to see an answer to something like your question.

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We won't be able to do it for varieties, at least, because a cusp is topologically the same as a smooth point (cuspidal cubic and $\mathbb{P}^1$ are homeomorphic, even over $\mathbb{C}$ in the analytic topology).

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