Meromorphic continuation of local zeta integrals

Question

Let $f$ be a Maass cusp form for $\text{SL}_2(\mathbb{Z})$ on the upper half plane. Let $\varphi_0$ be its lift to an automorphic form on $G = \text{PGL}_2(\mathbb{R})$ and let $\pi = \pi_{f} =\langle \varphi_0 \rangle$ denote the cyclic subrepresentation of $L^2(\Gamma \backslash G)$ (where $\Gamma = \text{PGL}_2(\mathbb{Z})$) generated by $\varphi_0$. To each smooth vector $\varphi \in \pi$ we can associate its Whittaker function

$$W_{\varphi}(g) = \int_{\mathbb{Z} \backslash \mathbb{R}}{\varphi(n(x)g) e(-x)}dx\qquad n(x) = \begin{pmatrix} 1 & x\\ 0 & 1 \end{pmatrix} $$ and the Mellin-transform thereof, known as the local zeta integral attached to $\varphi$ : $$ Z(W_{\varphi}, s) = \int_{\mathbb{R}^{\times}}{W_{\varphi}(a(y)) |y|^{s-1/2} d^{\times}y} \qquad a(y) = \begin{pmatrix} y & 0\\ 0 & 1 \end{pmatrix} $$

It is known (in my case, from Proposition 1.5 in Cogdell Piatetski-Shapiro "The arithmetic and spectral analysis of Kloosterman sums") that the assignment $\varphi \mapsto Z(W_{\varphi},s)$ is continuous for the smooth topology on $\pi^{\infty}$ and even uniformly so for $s$ varying in a compact subset of some right half-plane.

Does this uniform continuity remain true for the meromorphic continuation of the local local zeta integral? In particular, I'd like to know whether this holds for compact subsets of $\Re(s) = 1/2$.

The local functional equation gives uniform continuity for $s$ varying in compact subsets of $\{|\Re(s)-1/2| > A\} - \{\text{poles}\}$ for some $A > 0$ (I think $A = 5/2$ is valid).

Answer 1

As we are considering continuity of the local zeta integral it is sufficient to pose the problem locally. Thus we think the Archimedean local component of $\pi$ is the principal series $\pi_\mu$ where $\mu\in \mathbb{C}$ such that the Laplacian eigenvalue of $f$ is $1/4-\mu^2$.

Let $\{\mathcal{D}_i\}$ be the set of invariant differential operators of degree $\le 2$. We define a Sobolev norm (semi-norm) on $\pi_\mu$ by $$S(v):=\sum_{i}\|\mathcal{D}_iv\|_\pi.$$ Thus to prove the uniform continuity it is enough to show the following: For every compact set $K\subseteq \{s\mid\Re(s)> 0\}$, $$|Z(W_{v_1},s)-Z(W_{v_2},s)|\ll_K S(v_1-v_2),$$ whenever $s\in K$. Using linearity of the zeta integral it is enough to show that $$Z(W_v,s)\ll_K S(v).$$ But the LHS is $$\int_{\mathbb{R}^\times}W_v(a(y))|y|^{s-1/2}d^\times y\ll_KS(v)\int_{\mathbb{R}^\times}\min(|y|^{1/2-\eta},|y|^{-2})|y|^{\Re(s)-1/2}\frac{dy}{|y|}.$$ We have used Proposition 3.2.3 of Michel-Venkatesh in a slightly modified form (their seminorm is only on the Kirillov model of $\pi_\mu$). If $\pi_\mu$ is tempered, i.e. $\mu$ is purely imaginary then $\eta$ can be replaced by any $\epsilon > 0$ (with $\ll_K$ replaced by $\ll_{K,\epsilon}$). As $\Re(s)> 0$, the last integral is $\ll_K 1$ choosing $\epsilon$ small enough.