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

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    $\begingroup$ If I remember correctly M. Reeder has some work in this direction. A good starting point to search the literature might be the paper eudml.org/doc/90194 $\endgroup$ Apr 15, 2023 at 16:32
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    $\begingroup$ I don't know if it's the same Whittaker function, but the symmetric Macdonald functions $P_{\lambda}(q,t)$ specialized at $t=0$ are sometimes referred to as $q$-Whittaker functions. These have representation theoretic meaning: they are characters of the local Weyl modules for the borel of current algebra. There is also a meaning in terms of $S_n$ representations, they are the Frobenius characteristics of some module for $S_n$. $\endgroup$
    – ArB
    Apr 16, 2023 at 9:42
  • $\begingroup$ Probably not, since my functions should be non-symmetric polynomials. Those $q$-Whittaker-functions may be Whittaker functions with some other invariance property, but I do not know it either. $\endgroup$ Apr 17, 2023 at 13:01
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    $\begingroup$ I think Demazure characters are just the familiar characters in characteristic $0$, and the only divergence comes in positive characteristic when representations of reductive algebraic groups, while still parameterised by dominant co-weights, don't have the characters you expect. $\endgroup$
    – LSpice
    Apr 17, 2023 at 14:21

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I'm afraid that this question has a disappointingly simple answer. Yes, the values of the Iwahori-spherical Whittaker functions have an interpretation as characters of representations; but they are one-dimensional representations!

To see this, it suffices to check out $n = 2$. Here the Satake parameter is an unordered pair of (nonzero) complex numbers $\alpha, \beta$. One checks that $\mathfrak{W}(\tau, \psi)^{J_2}$ is 2-dimensional with a natural basis $$\{ \mathcal{W}_\alpha, \mathcal{W}_\beta \} $$ and the values of these on the maximal torus are given (up to some powers of $q$ which I am ignoring) by $$\mathcal{W}_\alpha \left( \begin{pmatrix} \pi^r & 0 \\ 0 & \pi^s\end{pmatrix}\right) = \alpha^r \beta^s. $$

The picture for general $n$ is similar: there is a natural basis of the Whittaker functions indexed by the $n!$ Weyl-group permutations $w \cdot \tau$ of the inducing character $\tau$, and the Whittaker function associated to $w \cdot \tau$ just restricts to $w \cdot \tau$ on the torus.

So there is much less interesting structure in the values of the Iwahori-level Whittaker functions than there is in the spherical ones. The interesting combinatorics is in how to stick together Iwahori Whittaker functions to get spherical Whittaker functions.

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  • $\begingroup$ while it is true that (at least for the long Weyl-element), the (Iwahori-spherical) Whittaker function restricted to the diagonal torus has the form you stated, its values on $\pi^e \cdot w$ are not so easy to describe. In deed, if one sticks to $n=2$, then for example for $w = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ one gets $$ \mathcal{W}(\pi^e w) = (\delta^{1/2}_2 \otimes \tau^{w})(\pi^{e}) \cdot \left(\frac{1 - q^{-1} + q^{-1}\left(\frac{\alpha}{\beta}\right)^{e_2-e_1 + 2} - \left(\frac{\alpha}{\beta}\right)^{e_2-e_1 + 1}}{1 - \frac{\alpha}{\beta}}\right). $$ $\endgroup$ Apr 15, 2023 at 18:09
  • $\begingroup$ Bump and co. calls this operator in the recursion a modified Kazdhan-Lusztig operator. What I would be interested in is understand this transition by means of rep.theory. $\endgroup$ Apr 15, 2023 at 18:13
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    $\begingroup$ I see. So it looks like you were actually asking a different and more subtle question from the one I answered – apologies for jumping to conclusions! Do you want me to delete this answer? That might increase the chances of a real expert coming along and answering your actual question. $\endgroup$ Apr 15, 2023 at 23:14
  • $\begingroup$ I think it is better if you leave the answer. I now added a little more to my already overloaded content by trying to explain in more detail what I wish to obtain. $\endgroup$ Apr 17, 2023 at 12:51

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