$\DeclareMathOperator\Kern{Kern}\DeclareMathOperator\GL{GL}\DeclareMathOperator\C{\mathbb{C}}\DeclareMathOperator\Z{\mathbb{Z}}\DeclareMathOperator\diag{diag}\DeclareMathOperator\Ind{Ind}\newcommand\iddots{\mathinner{
\kern1mu\raise1pt{.}
\kern2mu\raise4pt{.}
\kern2mu\raise7pt{\Rule{0pt}{7pt}{0pt}.}
\kern1mu
}}$**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 $F$ with $q$ elements.
- $B_n \subset \GL_n$ : 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
$$
\mathcal{W}_v(g) := \mathfrak{W}(\varphi_{v})(g) := \int_{U_n(F)} \varphi_{v}(w_n u g) \overline{\psi}(u) du
$$
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 $\mathcal{W}_{w_nw}(\pi^e w')$ and mention their connection to specializations of non-symmetric MacDonald polynomials. It goes very roughly like this: if $w = s_k \cdot \ldots \cdot s_1 w'$ are simple reflections moving from $w$ to $w'$, then (up to $q$-powers), $$ \mathcal{W}_{w_nw}(\pi^e w') = (\delta_k \circ \ldots \circ \delta_1)(\tau(\pi^e)). $$ The $\delta_i$ are by Bump and co. called Demazure-Whittaker-operators, see formula (11) in his paper.

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 $w = w_n$ 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 of **Cauchy-identity**, which is why I hope to have such a description of the Whittaker functions.