$\DeclareMathOperator\Kern{Kern}\DeclareMathOperator\GL{GL}\DeclareMathOperator\C{\mathbb{C}}\DeclareMathOperator\Z{\mathbb{Z}}\DeclareMathOperator\diag{diag}\DeclareMathOperator\Ind{Ind}$$\DeclareMathOperator\Kern{Kern}\DeclareMathOperator\GL{GL}\DeclareMathOperator\C{\mathbb{C}}\DeclareMathOperator\Z{\mathbb{Z}}\DeclareMathOperator\diag{diag}\DeclareMathOperator\Ind{Ind}\newcommand\iddots{\mathinner{
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}}$Setting:
$F$ : non-archimedean local field (of char. $0$)
$F$ : non-archimedean local field (of char. $0$)$(\pi) = \mathfrak{p} \subset \mathcal{O} \subset F$ : uniformizer, max.ideal and ring of integers respectively
$(\pi) = \mathfrak{p} \subset \mathcal{O} \subset F$ : uniformizer, max.ideal and ring of integers respectively For $e = (e_1,\ldots,e_n) \in \Z^{n}$ write
$$
\pi^e := \begin{pmatrix} \pi^{e_1} & & \\ & \ddots & \\ & & \pi^{e_n} \end{pmatrix} \in \GL_n(F)
$$
For $e = (e_1,\ldots,e_n) \in \Z^{n}$ write
$$
\pi^e := \begin{pmatrix} \pi^{e_1} & & \\ & \ddots & \\ & & \pi^{e_n} \end{pmatrix} \in \GL_n(F)
$$$\kappa(F)$ : the residue field of $F$
$\kappa(F)$ : the residue field of $F$$B_n$ : standard Borel of upper-triangular matrices
$B_n \subset \GL_n$ : standard Borel of upper-triangular matrices$U_n \subset B_n$ : maximal unipotent
$U_n \subset B_n$ : maximal unipotent$\psi \colon F \to \mathbb{C}^{\times}$ additive character with $\mathcal{O} \subset \Kern(\psi)$, but $\pi^{-1}\mathcal{O} \not\subset \Kern(\psi)$.
$T_n \subset \GL_n$ : diagonal max. torus$\psi$ induces naturally a character of $U_n(F)$ (also denoted by $\psi)$ via
$$
\psi((u_{ij})) = \psi(u_{12} + u_{23} + \ldots + u_{n-1,n}).
$$
$W_n$ : Weyl group, in this case we drop the modulo condition (this does not affect my needs) and work only with permutation matrices$\delta_n \colon B_n(F) \to \C$ : (the inverse of) the modular character of $B_n(F)$
$w_n \in W_n$ : long Weyl-element 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 $I(\tau)$.
$\psi \colon F \to \mathbb{C}^{\times}$ additive character with $\mathcal{O} \subset \Kern(\psi)$, but $\pi^{-1}\mathcal{O} \not\subset \Kern(\psi)$. 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.
$\psi$ induces naturally a character of $U_n(F)$ (also denoted by $\psi)$ via
$$
\psi((u_{ij})) = \psi(u_{12} + u_{23} + \ldots + u_{n-1,n}).
$$$\mathcal{W}(ug) = \psi(u) \cdot \mathcal{W}(g)$ for all $(u,g) \in U_n(F) \times \GL_n(F)$,
$\delta_n \colon B_n(F) \to \C$ : (the inverse of) the modular character of $B_n(F)$$\mathcal{W}$ is locally constant, i.e. there exists an open-compact $K \subset \GL_n(\mathcal{O})$, s.t. $\mathcal{W}(gk) = \mathcal{W}(g)$ for all $(g,k) \in \GL_n(F) \times K$. I will call such a Whittaker function $K$-spherical.
Let me now fix a nice (explained later in Iwahori-sphericity) character $\tau \colon B_n(F) \to \mathbb{C}^{\times}$ and denote by $I(\tau) := \Ind_{B_n(F)}^{\GL_n(F)}(\tau)$ (smooth induction) the corresponding principal series representation. Then the theory tells us that we have a unique embedding of $\GL_n(F)$-representations
$$
I(\tau) \hookrightarrow \Ind^{\GL_n(F)}_{U_n(F)}(\psi).
$$
The image $\mathfrak{W}(\tau,\psi)$ is called the Whittaker model of $I(\tau)$.- A Whittaker function is an element $\mathcal{W} \in \mathfrak{W}(\psi) := \Ind^{\GL_n(F)}_{U_n(F)}(\psi)$; thus it is a function $\mathcal{W} \colon \GL_n(F) \to \C$, s.t.
- $\mathcal{W}(ug) = \psi(u) \cdot \mathcal{W}(g)$ for all $(u,g) \in U_n(F) \times \GL_n(F)$,
- $\mathcal{W}$ is locally constant, i.e. there exists an open-compact $K \subset \GL_n(\mathcal{O})$, s.t. $\mathcal{W}(gk) = \mathcal{W}(g)$ for all $(g,k) \in \GL_n(F) \times K$. I will call such a Whittaker function $K$-spherical.
The Iwahori group $J_n$ is the preimage of $B_n(\kappa(F))$ under the canonical projection $\GL_n(\mathcal{O}) \to \GL_n(\kappa(F))$. In other words
$$
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}.
$$
NowDue to the more refined Bruhat-Iwasawa decomposition $\GL_n(F) = \bigsqcup_{w \in W_n} B_n(F) w J_n$, we can take the standard basis on $I(\tau)^J_n$ given by
$
\varphi_{w}(bw'j) = (\delta^{1/2}_n \otimes \tau)(b)
$$
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.).
Moreover, the dimension of the space
If $\mathfrak{W}(\tau,\psi)^{J_n}$$\tau$ is bounded byassumed $n!$regular (i.e. $\tau^{w} \neq \tau$ for any $w \in W_n), then ${\mathcal{W}{w}}{w \in W_n}$ is indeed a basis of $\mathfrak{W}(\tau, which is the order of the Weyl\psi)^{J_n}$. Every $\mathcal{W}{w_n w}$ is 'easy' to compute at the point $\pi^e w$ (assuming $e$ almost-$w$-dominant), giving
$$
\mathcal{W}{w_n w}(\pi^e w) = q^{-groupl(w)} \cdot $W_n$, and indeed by putting some restrictions on the 'niceness'(\delta^{1/2}_n \otimes \tau^{w_n})(\pi^e).
$$
The computation of the other arguments $\tau$, we can have equality, so let's say there$\mathcal{W}_{w_n w}(\pi^e w')$ is a basis $\{\mathcal{W}_{v}\}$ indexed by $v \in W_n$rather tedious. Bump and co. derive a recursive expression (in simple reflections) for the various $\mathcal{W}_{v}(\pi^e w)$$\mathcal{W}_{w}(\pi^e w')$ and mention their connection to specialization to non-symmetric MacDonald polynomials.
Do the various Iwahori-sphericalexpressions $\mathcal{W}_{v}$ also have$\mathcal{W}_{w}(\pi^e w')$ possess a representation-theoretic description as characters of some representations? (I am particularly interested in the case where $v = w_0$$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.
