Whittaker at Archimedean Test vector

Question

Let $\pi$ be a generic irreducible Casselman Wallach representation of $GL_n(\mathbb{R})$. Let $\phi$ be a test vector for the archimedean local L function $L(s,\pi)$. Is there an explicit expression for the Whittaker function at $\phi$ on the torus inside $GL_n(\mathbb{R})$. Any help is appreciated.

Answer 1

The short answer is no: there is no simple closed form for such a Whittaker function.

The long answer is as follows. When $F$ is nonarchimedean and $\pi$ is a generic irreducible admissible smooth representation of $\mathrm{GL}_n(F)$ with Whittaker model $\mathcal{W}(\pi,\psi)$, the local $L$-function is represented by the $\mathrm{GL}_n$ Godement-Jacquet zeta integral $$\int_{\mathrm{GL}_n(F)} \langle \pi(g) \cdot W, \widetilde{W}\rangle \Phi(g) \left|\det g\right|^{s + \frac{n - 1}{2}} \, dg,$$ where $W \in \mathcal{W}(\pi,\psi)$ and $\widetilde{W} \in \mathcal{W}(\widetilde{\pi},\overline{\psi})$.

I showed in a short paper (https://doi.org/10.1112/blms.12401) that if one takes $W$ and $\widetilde{W}$ to be Whittaker newforms and chooses $\Phi$ appropriately, then this zeta integral is exactly equal to $L(s,\pi)$. If $\pi$ is spherical (unramified), then the Whittaker newform is just the spherical vector, and a famous formula of Casselman-Shalika-Shintani gives an explicit expression for $W$ evaluated on the diagonal torus $\mathrm{A}_n(F)$ in $\mathrm{GL}_n(F)$. Work of Matringe shows that the Casselman-Shalika-Shintani formula also extends in a natural way to Whittaker newforms. So this gives a nice satisfactory answer to your question when $F$ is nonarchimedean.

When $F$ is archimedean, on the other hand, and $\pi$ is a generic irreducible Casselman-Wallach representation of $\mathrm{GL}_n(F)$, then there is no such satisfactory answer. Even if $\pi$ is spherical, the spherical Whittaker function $W \in \mathcal{W}(\pi,\psi)$ does not have a nice closed form when restricted to the diagonal torus $\mathrm{A}_n(F)$, unlike the Casselman-Shalika-Shintani formula. (Roughly speaking, this is because $W$ is just some crazy special function; already for $n = 2$, this is a Bessel function.)

Nonetheless, I showed in a recent paper (https://arxiv.org/abs/2008.12406) that there is an archimedean version of the Whittaker newform, and if one chooses $W$ and $\widetilde{W}$ to be Whittaker newforms and chooses $\Phi$ appropriately, then the $\mathrm{GL}_n$ Godement-Jacquet zeta integral is exactly equal to $L(s,\pi)$.

The most "natural" formulae for the Whittaker newform on the diagonal torus $\mathrm{A}_n(F)$ is a recursive formula that gives an expression for $W(\mathrm{diag}(a_1,\ldots,a_n))$ as an integral involving a Whittaker function for $\mathrm{GL}_{n - 1}(F)$. I call this a propagation formula in my paper. I write down such a formula explicitly when $\pi$ is spherical in Lemma 7.2 of this paper: https://arxiv.org/abs/2112.06860. A similar formula holds in the nonspherical case.