In the $GL_2(\mathfrak{R})$ case, assume that $\pi$ is an irreducible unitary representation and $W_{\pi}(g)$ is the Whittaker functional associates with $\pi$. Then there is a Bessel function $j_{\pi}(y)$ which behaves like a integral kernel such that $$W(w \left(\begin{array}{ccc}y & 0 \\0 & 1\end{array}\right) ) = \int_{\mathfrak{R^{\times}}}j(y)W( \left(\begin{array}{ccc}y & 0 \\0 & 1\end{array}\right) )dy.$$ where $w$ is the longest Weyl element and $j_{\pi}$ depends on the representation $\pi$.

Now my question is whether there is a $GL_3$ version corresponds to the longest Weyl element. More specific, I am expecting a formula of the form $$W(w \left(\begin{array}{ccc}h & 0 \\0 & 1\end{array}\right) ) = \int_{N_2\GL_2}j(h)W( \left(\begin{array}{ccc}h & 0 \\0 & 1\end{array}\right) )dh.$$

It seems like a Bessel distribution is kind of related to this. I know the results from Shalika and many others such that $<\lambda, \pi^{*}(g)\lambda'>$ defines a Bessel function. But it is still not clear to me whether we can compute Wittaker model at $w$ through those Bessel functions.