Assume $\pi$ is an unramified representation of $GL\_n(F)$, where $F$ is a p-adic field. And $\phi$ is an unramified vector for $\pi$. Assume $W\_{\phi}$ is a Whittaker function associated to $\phi$. Then whether $|W\_{\phi}(1)|^2=\frac{1}{L(1,\pi,Ad)}$?


I will appreaciate if you can show me a detailed calculation of the equality in an specific example. Thank you all the time.



Corollary for Casselman-Shalika formula
The answer depends on your normalisation of $W$. Perhaps the most standard normalisation of $W$ is such that $W(1\_n) = 1$, where $1\_n$ is the $n \times n$ identity matrix.


Let $\pi$ be a unitary spherical representation of $\mathrm{GL}\_n(F)$ with Satake parameters $\alpha\_1,\ldots,\alpha\_n$. Let $\widetilde{W}$ be the spherical vector of the contragredient representation $\widetilde{\pi}$. Then it is true that $$\langle W, \widetilde{W} \rangle = \int\_{N\_{n - 1}(F) \backslash \mathrm{GL}\_{n - 1}(F)} W\begin{pmatrix}g&0\\ 0&1\end{pmatrix} \widetilde{W}\begin{pmatrix}g&0\\ 0&1\end{pmatrix} \, dg$$
is equal to
\[\frac{L(1,\pi \otimes \widetilde{\pi})}{\zeta\_F(n)} = (1 - q^{-n}) \prod\_{j = 1}^{n} \prod\_{k = 1}^{n} \frac{1}{1 - \alpha\_j \alpha\_k^{-1} q^{-1}}.\]
Here $q$ is the cardinality of the residue field of $F$.


In fact, a more general statement is true, where $\pi$ is allowed to be ramified and $W$ is the newform. A proof appears in Section 7 of "[Large sieve inequalities for $\mathrm{GL}(n)$-forms in the conductor aspect](https://doi.org/10.1016/j.aim.2005.11.001)" by Akshay Venkatesh.


There is another nice method to prove this, which is to use the method of Michitaka Miyauchi in "[Whittaker functions associated to newforms for $\mathrm{GL}(n)$ over $p$-adic fields](https://doi.org/10.2969/jmsj/06610017)", using Theorem 4.1, which is the Shintani-Casselman-Shalika formula for newforms, together with Cauchy-Schur identities, since the integral over $N\_{n - 1}(F) \backslash \mathrm{GL}\_{n - 1}(F)$ reduces to an integral over
\[\{\mathrm{diag}(\varpi^{f\_1},\ldots,\varpi^{f\_{n - 1}},1) : f\_1 \geq \cdots \geq f\_{n - 1} \geq 0\},\]
where $\varpi$ is a uniformiser for $F$.


Remarkably, such a result is also true, once suitably modified, for archimedean $F \in \{\mathbb{R},\mathbb{C}\}$, even if $\pi$ is ramified. For $\mathrm{GL}\_2$, this is quite standard; for $\mathrm{GL}\_n$ and $\pi$ unramified, this is due to [Stade](https://doi.org/10.1007/BF02784531); for $\mathrm{GL}\_n$ and $\pi$ ramified, this is [Theorem 5.10 of my paper with Yeongseong Jo](https://arxiv.org/abs/2112.06860) based on [recent work of mine](https://arxiv.org/abs/2008.12406).)


All this is the adèlic version of the more classical statement that
\[\frac{|a\_f(1)|^2}{\langle f,f\rangle} = \frac{1}{2\Lambda(1, \operatorname{ad} f)}\]
for an even Maass form $f$ on $\mathrm{SL}\_2(\mathbb{Z}) \backslash \mathbb{H}$ with first Fourier coefficient $a\_f(1)$.


