# Explicit description for action of Weyl element in Whittaker model for GL2

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.

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 $$\int_{\widehat{F^{\times}}} f(\chi) \, d\chi = \sum_{m = 0}^{\infty} \sum_{\substack{\beta \in \mathfrak{X} \\ c(\beta) = m}} \int_{0}^{1} f\left(\beta |\cdot|^{\frac{2\pi it}{\log q}}\right) \, dt.$$

• Thank you for the stellar explanation and references, this is exactly what I was looking for! Commented Oct 27, 2023 at 20:00
• @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... Commented Oct 27, 2023 at 20:09