$\DeclareMathOperator\Kern{Kern}\DeclareMathOperator\GL{GL}\DeclareMathOperator\C{\mathbb{C}}\DeclareMathOperator\Z{\mathbb{Z}}\DeclareMathOperator\diag{diag}\DeclareMathOperator\Ind{Ind}$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 $F$

• $B_n$ : standard Borel of upper-triangular matrices

• $U_n \subset B_n$ : maximal unipotent

• $\psi \colon F \to \mathbb{C}^{\times}$ 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 $$ J_n = \begin{pmatrix} \mathcal{O}^{\times} & \mathcal{O} & \ldots & \mathcal{O} \\ \mathfrak{p} & \mathcal{O}^{\times} & \ddots & \vdots \\ \vdots & \ddots & \ddots & \mathcal{O} \\ \mathfrak{p} & \ldots & \mathfrak{p} & \mathcal{O}^{\times} \end{pmatrix}. $$ 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.). Moreover, the dimension of the space $\mathfrak{W}(\tau,\psi)^{J_n}$ is bounded by $n!$, which is the order of the Weyl-group $W_n$, and indeed by putting some restrictions on the 'niceness' of $\tau$, we can have equality, so let's say there is a basis $\{\mathcal{W}_{v}\}$ indexed by $v \in W_n$. Bump and co. derive a recursive expression (in simple reflections) for the various $\mathcal{W}_{v}(\pi^e w)$ 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 Iwahori-spherical $\mathcal{W}_{v}$ also have a description as characters of some representations? (I am particularly interested in the case where $v = w_0$ is the long Weyl-element)