# On $p$-adic Iwahori-spherical Whittaker functions

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

• 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 Apr 15, 2023 at 16:32
• 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 $S_n$.
– ArB
Apr 16, 2023 at 9:42
• 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. Apr 17, 2023 at 13:01
• 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. Apr 17, 2023 at 14:21

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.
• 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 $$\mathcal{W}(\pi^e w) = (\delta^{1/2}_2 \otimes \tau^{w})(\pi^{e}) \cdot \left(\frac{1 - q^{-1} + q^{-1}\left(\frac{\alpha}{\beta}\right)^{e_2-e_1 + 2} - \left(\frac{\alpha}{\beta}\right)^{e_2-e_1 + 1}}{1 - \frac{\alpha}{\beta}}\right).$$ Apr 15, 2023 at 18:09