For $\mu,\nu\in \mathbb{C}^2$ we denote $I(\mu,\nu)$ to be the principal series of $\mathrm{GL}_2(\mathbb{Q}_p)$ induced from $|.|^\mu\otimes |.|^\nu$. For $s=\mu-\nu$ one defines the standard intertwining operator $$M(s):I(\mu,\nu)\to I(\nu,\mu),\quad\Re(s)>0$$ by $$f\mapsto \{g\mapsto \int_N f(wng) dn\}.$$ One can analytically continue $M(s)$ by introducing suitable normalization, e.g. $$M^*(s) := \frac{1-p^{-s}}{1-p^{-1}}M(s),$$ which is holomorphic everywhere. One can also check that the operator vanishes whenever $\mu-\nu=-1$. One can also compute that, for an absolute constant $C$ (depending on $p$) $$M^*(-s)M^*(s)=C(1-p^{\mu-\nu-1})(1-p^{\nu-\mu-1}).$$

On the other hand if $I(\mu,\nu)$ is tempered i.e. $\mu,\nu \in i\mathbb{R}$ we can compute the $L^2$ norms of the vectors in the Whittaker model of $I(\mu,\nu)$, where the inner product is taken in the corresponding Kirillov model. I checked that the squared $L^2$ norms of the spherical vectors $W$ in the Whittaker model with $W(1)=1$ have $(1-p^{\mu-\nu-1})(1-p^{\nu-\mu-1})$ in the denominators. For the spherical vector $W$ in the Whittaker model of $I(\mu,\nu)$ with $W(1)=1$,

the function $$(\mu,\nu)\mapsto \|W\|^2_{\mathcal{K}}$$ has poles whenever the product of normalized intertwining operators $M^*(-s)M^*(s)$ has a zero.

My questions are

(1) Does $L^2$ norm (with respect to the Kirillov model inner product) of any Whittaker function $W$ with $W(1)=1$ diverge as $\mu-\nu=\pm 1$, i.e. does the quoted section remain true even for non-spherical vectors?

(2) Is the fact, that zeros of product of normalized intertwining operators and poles of squared $L^2$ norm of a Whittaker vector match, just a coincidence or is there a good reason behind it?


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