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Given a $p$-adic reductive group $G$ with Grothendieck group $R(G)$ and $f$ an element of the Hecke Algebra $H(G)$ we can consider the function $x: R(G) \to \mathbb{C}$ given by $\pi \mapsto trace \pi(f)$ (for $f \in H(G)$). It is stated

(condition (i) in 1.2 here, for instance: https://publications.ias.edu/sites/default/files/Number55.pdf)

that $x$ restricted to the Bernstein variety of $G$ is a regular function. This should be easy, but I would appreciate an explanation of why this is true.

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  • $\begingroup$ It's way less interesting than you think: the analytic structure on the Bernstein spectrum comes from twisting by unramified characters. If $f$ is the characteristic function of a coset of a compact, open subgroup, then $\operatorname{tr} (\pi \otimes \chi)(f)$ equals $\chi(g)\operatorname{tr} \pi(f)$ for any $g$ in the support of $f$. The general result follows by writing an arbitrary $f$ as a combination of such characteristic functions. $\endgroup$
    – LSpice
    Commented Feb 13, 2017 at 22:32
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    $\begingroup$ @LSpice Right, but I guess in general, one is considering something like $\mathrm{tr} I_P^G( \pi \otimes \chi)(f)$ and it wasn't clear to me how to deal with this induction. $\endgroup$
    – Alexander
    Commented Feb 13, 2017 at 22:42
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    $\begingroup$ Oh, right. I'm a supercuspidals guy, and I just plumb forgot about parabolic induction. I think that Theorem 2, p. 233, of van Dijk's "Computation of certain induced characters of $\mathfrak p$-adic groups" will do it, though it may be overkill. $\endgroup$
    – LSpice
    Commented Feb 14, 2017 at 3:39

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As in my comment, Theorem 2 of van Dijk - Computation of certain induced characters of $p$-adic groups (MR) says, with the notation of that result, that \begin{align*} \Theta_{\mathrm{ind}_P^G(\rho\chi)}(f) & = \int_M \chi(m)\int_K \int_N f(k m n k^{-1})\delta_P(m)^{1/2}\theta_\rho(m)\mathrm dn\,\mathrm dk\,\mathrm dm \\ & = \sum_{m \in M/K \cap M} \chi(m)\operatorname{meas}(K \cap M)\int_K \int_N f(k m n k^{-1})\delta_P(m)^{1/2}\theta_\rho(m)\mathrm dn\,\mathrm dk. \end{align*} (Actually the stated integral is in a different order, but each integral is compactly supported, so the interchange is OK.) The sum is finite because $f$ is compactly supported.

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