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Let $F$ be a non-archimedean local field and let $\pi =\mathscr{B}(\chi, \chi^{-1})$ be a principal series representation of $\mathrm{PGL}_2(F)$ induced from a character $\chi$ of $F^\times$. Let $w = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.$ View $\mathscr{B}(\chi, \chi^{-1})$ in its Whittaker model $\mathcal{W}(\pi, \psi)$ for a fixed additive character $\psi$ on $F$. Let $W\begin{pmatrix} y & \\ & 1 \end{pmatrix}$ be a vector in the Whittaker model. Then I want an explicit description of $$(\pi(w)W)\begin{pmatrix} y & \\ & 1 \end{pmatrix},$$ in either the Whittaker or Kirillov model of $\pi$. From trying to find an answer to this question I have learned that I may need to utilize a sort of $\mathrm{GL}_2 \times \mathrm{GL}_1$ local functional equation which I think would take the shape $$\gamma(\pi \otimes \chi, 1/2, \psi)\int_{F^\times} W(t)\chi(t)\,d^\times t = \int_{F^\times} (\pi(w)W)(t)\omega^{-1}(t)\chi^{-1}(t)\,d^\times t,$$ now viewing $W$ in its Kirillov model with $\omega$ the central character. Assuming for the moment the central character is trivial, Mellin inversion should yield $$(\pi(w)W)(y) = \int_{\chi} \chi(y)\gamma(\pi \otimes \chi, 1/2, \psi)\left(\int_{F^\times} W(t)\chi(t)\,d^\times t\right)\,d\chi.$$ This leads me to ask two questions:

  1. How to explicitly integrate over characters $\chi$? What does this integration mean?
  2. What is $\gamma(\pi \otimes \chi, 1/2, \psi)$ explicitly and how can I calculate it in a way that is amenable to then integrating with it?

I would appreciate either direct answers or hints or being pointed to a reference that would answer these questions.

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Let $F$ be a local field, let $\omega_1,\omega_2$ be characters of $F^{\times}$, and let $\pi = \omega_1 \boxplus \omega_2$ be the principal series representation obtained via normalised parabolic induction from these two characters. Let $\psi$ be an additive character of $F$ (which I will assume to be unramified) and let $\mathcal{W}(\pi,\psi)$ denote the Whittaker model of $\pi$. When $F$ is nonarchimedean, I'll write $\mathcal{O}$ for the ring of integers of $F$, $\varpi$ for a uniformiser, and $\mathfrak{p} = \varpi \mathcal{O}$.

The $\mathrm{GL}_2 \times \mathrm{GL}_1$ local functional equation states that for any $W \in \mathcal{W}(\pi,\psi)$ and any character $\chi$ of $F^{\times}$, $$\int_{F^{\times}} W\left(\begin{pmatrix} y & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\right) \omega_{\pi}^{-1} \chi^{-1}(y) |y|^{\frac{1}{2} - s} \, d^{\times}y = \gamma(s,\pi \otimes \chi,\psi) \int_{F^{\times}} W\begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} \chi(a) |a|^{s - \frac{1}{2}} \, d^{\times}a.$$ Here $\omega_{\pi} = \omega_1 \omega_2$ is the central character of $\pi$.

Via the Mellin inversion formula, it follows that $$W\left(\begin{pmatrix} y & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\right) = \int_{\widehat{F^{\times}}} \gamma\left(\frac{1}{2},\pi \otimes \chi,\psi\right) \omega_{\pi}\chi(y) \int_{F^{\times}} W\begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} \chi(a) \, d^{\times}a \, d\chi.$$ See Section 3.2.2 of Michel and Venkatesh's paper "The Subconvexity Problem for $\mathrm{GL}_2$" for this approach.

Alternatively, one can interchange the order of integration and evaluate the integral over $\chi$, yielding $$W\left(\begin{pmatrix} y & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\right) = \omega_{\pi}(y) \int_{F^{\times}} W\begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} j_{\pi,\psi}(ay) \, d^{\times}a.$$ Here $j_{\pi,\psi}$ is the Bessel distribution given by the conditionally convergent integral $$j_{\pi,\psi}(y) = \zeta_F(1)^{-2} \omega_1\omega_2(-1) \omega_2^{-1}(y) |y|^{\frac{1}{2}} \int_{F^{\times}} \omega_1\omega_2^{-1}(a) \psi(ay + a^{-1}) \, d^{\times}a.$$ When $F$ is archimedean, this is a classical $J$- or $K$-Bessel function. This approach is sketched in Section 1.11 of Godement's book "Notes on Jacquet-Langlands' Theory".

Let's return to the former identity. First let me describe the gamma factor. This is $$\gamma\left(\frac{1}{2},\pi \otimes \chi,\psi\right) = \varepsilon\left(\frac{1}{2},\pi \otimes \chi,\psi\right) \frac{L\left(\frac{1}{2},\widetilde{\pi} \otimes \chi^{-1}\right)}{L\left(\frac{1}{2},\pi \otimes \chi\right)}.$$ Since $\pi = \omega_1 \boxplus \omega_2$, we can write this further as $$\varepsilon\left(\frac{1}{2},\omega_1\chi,\psi\right) \varepsilon\left(\frac{1}{2},\omega_2\chi,\psi\right) \frac{L\left(\frac{1}{2}, \omega_1^{-1} \chi^{-1}\right) L\left(\frac{1}{2}, \omega_2^{-1} \chi^{-1}\right)}{L\left(\frac{1}{2},\omega_1 \chi\right) L\left(\frac{1}{2},\omega_2 \chi\right)}.$$ When $F$ is nonarchimedean, these $L$-functions are just the usual local Euler factors (i.e. $1/P(q^{-1/2})$, where $q$ is the order of the residue field of $F$ and $P$ is a polynomial of degree at most $1$ with constant term $1$). The epsilon factor is essentially a Gauss sum, since $\omega_i \chi$ is basically a Dirichlet character; see Theorem 2.3.8 of Goldfeld and Hundley's book "Automorphic Representations and $L$-Functions for the General Linear Group" for an explicit evaluation of this.

What about the integral over $\chi$? When $F$ is archimedean, this is almost the usual Mellin transform, since every unitary character of $\mathbb{R}^{\times}$ can be written uniquely in the form $\chi(y) = \operatorname{sgn}(y)^{\kappa} |y|^{it}$ for some $\kappa \in \{0,1\}$ and $t \in \mathbb{R}$: we then have that $$\int_{\widehat{\mathbb{R}^{\times}}} f(\chi) \, d\chi = \frac{1}{2} \sum_{\kappa \in \{0,1\}} \frac{1}{2\pi} \int_{-\infty}^{\infty} f\left(\operatorname{sgn}^{\kappa} |\cdot|^{it}\right) \, dt.$$ When $F$ is nonarchimedean, we may write each unitary character $\chi$ of $F^{\times}$ uniquely in the form $\chi(y) = \beta(y) |y|^{\frac{2\pi it}{\log q}}$. Here $t \in [0,1]$, while $\beta$ is a character of $F^{\times}$ that is trivially on $\varpi^{\mathbb{Z}}$. The character $\beta$ descends to a character of $\mathcal{O}^{\times}/(1 + \mathfrak{p}^m) \cong (\mathcal{O}/\mathfrak{p}^m \mathcal{O})^{\times}$ for some nonnegative integer $m$, which we denote by $c(\beta)$ and call the conductor exponent of $\beta$. (For $F = \mathbb{Q}_p$, this means that you should think of $\beta$ as being a primitive Dirichlet character of conductor $p^m$.) Writing $\mathfrak{X}$ for the set of all such characters $\beta$, we have that $$\int_{\widehat{F^{\times}}} f(\chi) \, d\chi = \sum_{m = 0}^{\infty} \sum_{\substack{\beta \in \mathfrak{X} \\ c(\beta) = m}} \int_{0}^{1} f\left(\beta |\cdot|^{\frac{2\pi it}{\log q}}\right) \, dt.$$

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    $\begingroup$ Thank you for the stellar explanation and references, this is exactly what I was looking for! $\endgroup$ Commented Oct 27, 2023 at 20:00
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    $\begingroup$ @StephCurry No worries. By the way, one can explicitly write down a formula for the Bessel distribution $j_{\pi,\psi}$; it is basically a Kloosterman sum. But I can't find a good reference for this... $\endgroup$ Commented Oct 27, 2023 at 20:09

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