$\DeclareMathOperator\Kern{Kern}\DeclareMathOperator\GL{GL}\DeclareMathOperator\C{\mathbb{C}}\DeclareMathOperator\Z{\mathbb{Z}}\DeclareMathOperator\diag{diag}\DeclareMathOperator\Ind{Ind}\newcommand\iddots{\mathinner{ \kern1mu\raise1pt{.} \kern2mu\raise4pt{.} \kern2mu\raise7pt{\Rule{0pt}{7pt}{0pt}.} \kern1mu }}$Setting:

Following players:

• $F$ : non-archimedean local field (of char. $0$)
• $(\pi) = \mathfrak{p} \subset \mathcal{O} \subset F$ : uniformizer, max.ideal and ring of integers respectively
• For $e = (e_1,\ldots,e_n) \in \Z^{n}$ write $$ \pi^e := \begin{pmatrix} \pi^{e_1} & & \\ & \ddots & \\ & & \pi^{e_n} \end{pmatrix} \in \GL_n(F) $$
• $\kappa(F)$ : the residue field of $F$
• $B_n \subset \GL_n$ : standard Borel of upper-triangular matrices
• $U_n \subset B_n$ : maximal unipotent
• $T_n \subset \GL_n$ : diagonal max. torus
• $W_n$ : Weyl group, in this case we drop the modulo condition (this does not affect my needs) and work only with permutation matrices
• $w_n \in W_n$ : long Weyl-element
• $\psi \colon F \to \mathbb{C}^{\times}$ additive character with $\mathcal{O} \subset \Kern(\psi)$, but $\pi^{-1}\mathcal{O} \not\subset \Kern(\psi)$.
• $\psi$ induces naturally a character of $U_n(F)$ (also denoted by $\psi)$ via $$ \psi((u_{ij})) = \psi(u_{12} + u_{23} + \ldots + u_{n-1,n}). $$
• $\delta_n \colon B_n(F) \to \C$ : (the inverse of) the modular character of $B_n(F)$
• Let me now fix a nice (explained later in Iwahori-sphericity) character $\tau \colon B_n(F) \to \mathbb{C}^{\times}$ and denote by $I(\tau) := \Ind_{B_n(F)}^{\GL_n(F)}(\tau)$ (smooth induction) the corresponding principal series representation. Then the theory tells us that we have a unique embedding of $\GL_n(F)$-representations $$ I(\tau) \hookrightarrow \Ind^{\GL_n(F)}_{U_n(F)}(\psi). $$ The image $\mathfrak{W}(\tau,\psi)$ is called the Whittaker model of $I(\tau)$.
• A Whittaker function is an element $\mathcal{W} \in \mathfrak{W}(\psi) := \Ind^{\GL_n(F)}_{U_n(F)}(\psi)$; thus it is a function $\mathcal{W} \colon \GL_n(F) \to \C$, s.t.
• $\mathcal{W}(ug) = \psi(u) \cdot \mathcal{W}(g)$ for all $(u,g) \in U_n(F) \times \GL_n(F)$,
• $\mathcal{W}$ is locally constant, i.e. there exists an open-compact $K \subset \GL_n(\mathcal{O})$, s.t. $\mathcal{W}(gk) = \mathcal{W}(g)$ for all $(g,k) \in \GL_n(F) \times K$. I will call such a Whittaker function $K$-spherical.

Question (First attempt):

I am interested in a representation-theoretic description of these Whittaker functions.

History & Motivation:

As far as I know, it was Langlands who first conjectured some connection between the Whittaker functions and the representations of the Langlands Dual group (don't have really a reference for this), which is in this case just $^{L}\GL_n(F) = \GL_n(\mathbb{\C})$.

$K = \GL_n(\mathcal{O})$-sphericity:

The theory tells us, that the space $\mathfrak{W}(\tau,\psi)^{\GL_n(\mathcal{O})}$ is one-dimensional. It was Shintani in 1976, who first obtained an explicit expression of these. It turns out that indeed, there is such a connection as predicted by Langlands:

Let us suppose $\mathcal{W}$ is such a function. Then the Iwasawa decomposition of $\GL_n(F)$ tells us that $\mathcal{W}$ is already uniquelly determined by the values $\mathcal{W}(\pi^e)$ for $(e \in \Z^n)$. Moreover the properties 1. and 2. listed above force $\mathcal{W}(\pi^e)$ to vanish unless $e$ is dominant. Let us thus suppose that $e$ is dominant, i.e. $e_1 \geq e_2 \geq \ldots \geq e_n$. Shintani discovered that in this case

$$ \mathcal{W}(\pi^{e}) = \delta_{n}^{1/2}(\pi^{e}) \cdot \chi_{e}(A_{\tau}), $$ where $\chi_{e}$ is the character of the irreducible $\GL_n(\C)$-representation with highest vector $e \in \Z^n$ (i.e. the Schur polynomial) and $A_{\tau}$ is the Satake-parameter of $\tau$.

$K = J_n$: Iwahori-sphericity:

The Iwahori group $J_n$ is the preimage of $B_n(\kappa(F))$ under the canonical projection $\GL_n(\mathcal{O}) \to \GL_n(\kappa(F))$. In other words $$ J_n = \begin{pmatrix} \mathcal{O}^{\times} & \mathcal{O} & \ldots & \mathcal{O} \\ \mathfrak{p} & \mathcal{O}^{\times} & \ddots & \vdots \\ \vdots & \ddots & \ddots & \mathcal{O} \\ \mathfrak{p} & \ldots & \mathfrak{p} & \mathcal{O}^{\times} \end{pmatrix}. $$ Due to the more refined Bruhat-Iwasawa decomposition $\GL_n(F) = \bigsqcup_{w \in W_n} B_n(F) w J_n$, we can take the standard basis on $I(\tau)^J_n$ given by $ \varphi_{w}(bw'j) = (\delta^{1/2}_n \otimes \tau)(b) $$ if $w = w'$ and $0$ otherwise. The *Whittaker-transform* $\mathfrak{W} \colon I(\tau)^{J_n} \stackrel{\sim}{\to} \mathfrak{W}(\tau, \psi)^{J_n}$ is then explicitely given by $$ \mathcal{W}_v(g) := \mathfrak{W}(\varphi_{v})(g) := \int_{U_n(F)} \varphi_{v}(w_n u g) \overlin{\psi}(u) du $$ whenever convergent. If one assumes $\tau$ sufficiently nice, something like $|\tau(\alpha^{\vee}(\pi))| < 1$ for any simple cocharacter $\alpha^{\vee}$, then the expression should converge everywhere.

Now any such Iwahori-spherical function $\mathcal{W}$ is completely determined by its values at $\mathcal{W}(\pi^e w)$ for $e \in \Z_n$ and $w \in W_n$. As before, $\mathcal{W}(\pi^e w) = 0$ unless $e$ is almost $w$-dominant (Def.3.4 in Colored Vertex Models and Iwahori Whittaker Functions of Bump and co.).

If $\tau$ is assumed regular (i.e. $\tau^{w} \neq \tau$ for any $w \in W_n), then ${\mathcal{W}{w}}{w \in W_n}$ is indeed a basis of $\mathfrak{W}(\tau, \psi)^{J_n}$. Every $\mathcal{W}{w_n w}$ is 'easy' to compute at the point $\pi^e w$ (assuming $e$ almost-$w$-dominant), giving $$ \mathcal{W}{w_n w}(\pi^e w) = q^{-l(w)} \cdot (\delta^{1/2}_n \otimes \tau^{w_n})(\pi^e). $$

The computation of the other arguments $\mathcal{W}_{w_n w}(\pi^e w')$ is rather tedious. Bump and co. derive a recursive expression (in simple reflections) for the various $\mathcal{W}_{w}(\pi^e w')$ and mention their connection to specialization to non-symmetric MacDonald polynomials.

I think have read somehwere of non-symmetric MacDonald polynomials in connection with Demazure characters. I do not really know what these are. But the question that naturally rises, in analogy with the $\GL_n(\mathcal{O})$-case:

Question (Second attempt):

Do the various expressions $\mathcal{W}_{w}(\pi^e w')$ possess a representation-theoretic description? (I am particularly interested in the case where $w = w_0$ is the long Weyl-element). I have an infinite sum over such Whittaker functions in the context of local zeta-integrals. I would like to evaluate the sum, but I think the right approach would be some type Cauchy-identity, which is why I hope to have such description.