2
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ 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
    Feb 13, 2017 at 22:32
  • 1
    $\begingroup$ @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
    Feb 13, 2017 at 22:42
  • 1
    $\begingroup$ 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
    Feb 14, 2017 at 3:39

1 Answer 1

1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.