Combinatorics of Iwahori Whittaker Functions

Consider a split reductive group over a non-archimedean local field. The principal series representation is a representation parabolically induced from a character of the maximal torus. We consider Whittaker functions on the group which are certain matrix coefficients on the principal series representation corresponding to Whittaker functionals and spherical/Iwahori vectors. See Appendix A for details.

The non-achimedean Whittaker functions were introduced by Jacquet in [Jac67] and were given in the form of a $p$-adic Jacquet integral over the maximal unipotent subgroup. There is a rich theory of combinatorial evaluation of the non-achimedean Whittaker functions explored in [McN11, BBF11b, BBBG19, BBBG20] and related special functions in [Tok88, Oka90, HK07] and others.

The spherical Whittaker function $W_{0}$ is the first and simplest example. It was computed explicitly by T. Shintani in [Shi76] for Lie groups of type A. Later Casselman and Shalika in [CS80] gave a formula for all Lie types as a product of the deformed Weyl denominator times the character of the dual Langlands group.

The spherical Whittaker functions $W_{0}^{n}$ of the metaplectic $n$-covers for Lie groups of type A is the next example. In [McN11], McNamara gave the expression of $W_{0}^{n}$ in terms of Lusztig data and showed that this data is equivalent to Mirković-Vilonen cycles of the affine grassmanian. In [BBF11b], Brubaker, Bump, and Friedberg also expressed $W_{0}^{n}$ (as the Whittaker coefficient of Borel Eisenstein series) in terms of crystal graphs (and in terms of strict Gelfand-Tsetlin patterns).

The Iwahori Whittaker functions $W_{w}$ in type A are refinements of the spherical Whittaker functions that comes from the Iwahori decomposition instead of the Iwasawa decomposition. In [BBBG19], a recursive evaluation of the functions $W_{w}$ was given using the intertwining operators. By connecting the intertwiners with Yang-Baxter equations, functions $W_{w}$ were expressed in terms of colored lattice models. Moreover, In [BBBG20], the calculation was extended for metaplectic $n$-covers of Lie groups of type A in terms of supersymmetric lattice models using similar methods. At the present, it is the most general result which specializes to all functions mentioned above.

Thanks to the Casselman-Shalika formula (the CS-formula) for the spherical Whittaker function $W_{0}$, one can study the combinatorics by analyzing the explicit expression given in terms of the special functions. In this approach, one does not need any representation theory to obtain the combinatorial results. These special functions occur independently in the literature with no relation to the Whittaker functions.

Tokuyama in [Tok88] found an exact expression for the generating function of strict Gelfand-Tsetlin patterns which incidentally matched the CS-formula. In [Oka90], the CS-formula was expressed in terms of partially strict shifted plane partitions. In [HK07], it was computed combinatorically by applying jeu de taquin to the primed tableaux. In [BBF11a], Brubaker, Bump, and Friedberg evaluated the CS-formula as the partition function of a free-fermionic solvable lattice model which gave a new combinatorial interpretation using Yang-Baxter equation, a tool from statistical mechanics. A posteriori, all these methods give a combinatorial evaluation of the spherical Whittaker function $W_{0}$.

Some progress was made for other types other Lie types as well. In [HK02], a variation of the CS-formula was given combinatorically by Hamel and King. Moreover, they introduced lattice models with U-turns. In [Iva12], Ivanov used lattice models with U-turns and introduced Yang-Baxter equation that allowed to get the CS-formula for the symplectic group as the partition function of his model. Later, in [Gra17] Gray extended lattice models with U-turns and conjectured that the partition function is the spherical Whittaker function for the metaplectic covers of the symplectic group. In [FZ16], a partial evaluation of the CS-formula for type B was found by Friedberg and Zhang. Also, in [DeF21] DeFranco gave the CS-formula for type $G_{2}$ by brute force. A non-finite variant of such computations was given for the metaplectic cover of $G_{2}$ by Leslie in [Les19].

The main goal of this paper is to extend McNamara’s approach from the spherical Whittaker function to the Iwahori Whittaker functions. An advantage of McNamara’s approach is the description of the spherical Whittaker function as a sum over combinatorial data. Each term in the sum corresponds to a Mirković-Vilonen cycle. The combinatorial sum is obtained directly, without using intertwining operators, recursive relations, explicit expressions, or multiple Dirichlet series. This geometric approach has the following steps.

1. (1)

   Write explicit Iwasawa decomposition for the maximal unipotent subgroup $U$ that corresponds to Mirković-Vilonen cycles in the affine grassmanian. It gives the decomposition $U=\bigsqcup_{m}C_{m}^{\textbf{i}}$ into cells;

2. (2)

   The spherical Whittaker function $W_{0}^{n}$ is defined by a $p$-adic integral over $U$. The explicit decomposition for $U$ gives a combinatorial sum $\sum_{m}\int_{C_{m}^{\textbf{i}}}$;

3. (3)

   Evaluate each individual integral $\int_{C_{m}^{\textbf{i}}}$ explicitly in appropriate coordinates to obtain combinatorial weights for each term;

4. (4)

   Describe the decomposition and weights combinatorically, for example, in terms of Gelfand-Tsetlin patterns.

In Section 4 of [McN11], McNamara found an explicit Iwasawa decomposition for the maximal unipotent subgroup $U$ into disjoint cells $C_{m}^{\textbf{i}}$. In Section 2, we extend his decomposition and produce explicit Iwahori decomposition for $U$ into cells $S_{m,\sigma}^{\textbf{i}}$ that are the refinement of McNamara’s decomposition.

In Section 8 of [McN11], McNamara computes the values of the spherical Whittaker functions using the explicit decomposition. In Section 4 we extend the calculation to the case of the Iwahori Whittaker functions using cells $S_{m,\sigma}^{\textbf{i}}$.

There Iwahori case has several difficulties compared to the spherical case from McNamara’s work. First, the space of Iwahori Whittaker functions is not one dimensional which results in computing each Iwahori basis element separately. Second, the Iwahori Whittaker functions are defined by the affine Weyl group, not just by the diagonal which results in two parameters for the group element to consider. Third, we introduce new combinatorial data that we call colored data. We give equivalent definitions of colored Lusztig data, colored Gelfand-Tsetlin patterns, and colored lattice models.

The lattice model we introduce is dual to the one from [BBBG19, BBBG20]. This gives a geometric meaning to the admissible states of the lattice model as they are in bijection with the cells $S_{m,\sigma}^{\textbf{i}}$ and the weights are the $p$-adic integrals over the corresponding cells. We remark that we don’t use the intertwining operators or Yang-Baxter equation in our approach. Instead, we give the combinatorial evaluation directly from the definition of the Iwahori Whittaker functions by the Jacquet integral.

Let us be more precise now.

Let $G$ be a split reductive group over a non-achimedean local field and let $\widetilde{G}$ be an $n$-fold metaplectic cover of $G$. Let $\widetilde{T}$ be a fixed split maximal torus of $\widetilde{G}$, $\widetilde{B}$ the Borel subgroup, $U^{\pm}$ be the maximal unipotent subgroup and its opposite, $J$ the opposite Iwahori subgroup, and $W$ the Weyl group. See Section A.1 for details. In Section 2, we give the explicit Iwahori decomposition for $U$ that depends on a long word decomposition i:

We use this explicit decomposition to write $U$ as a union of cells which are related to the generalized Mirković-Vilonen cycles in the affine flag varieties. In this paper, it will be important that we can describe each cell in appropriate coordinates to compute integrals explicitly.

Similar to Section 7 of [McN11], we can show that cells $S_{m,\sigma}^{\textbf{i}}$ are in bijection with the generalized Mirković-Vilonen cycles in the affine flag variety which gives a parametrization of the generalized Mirković-Vilonen cycles. Alternative parametrization in terms of refined alcove paths was explored in [PRS09] by Parkinson, Ram, and Schwer.

Let now specialize to $G=\operatorname{GL}_{r+1}$. In Section A.4 we define Iwahori Whittaker functions $W_{w}\colon\widetilde{G}\to\mathbb{C}$ parametrized by Weyl group elements $w\in W$. In short, these are the matrix coefficients on the unramified genuine principal series representation of $\widetilde{G}$ corresponding to the averaged Whittaker functional and an Iwahori fixed vector. In Section 4, we evaluate all values of $W_{w}$ in terms of colored Lusztig data, combinatorial objects we introduce in Section 3.

Compare it with Theorem 8.6 from [McN11]:

As usual, one can use the combinatorial formulas to get the branching rules, specializations, and asymptotics for the Iwahori Whittaker functions. We do not do it here.

In Section 3, we define colored Lusztig data, colored Gelfand-Tsetlin patterns, and supersymmetric lattice models. We show that all of them are equivalent. Thus, Theorem 4.6 can be written over any of these equivalent data. In examples in Section 5 we primarily use colored GT-patterns.

This result connects our approach to colored lattice models and supersymmetric lattice models dual to ones introduced in [BBBG19] and [BBBG20]. We note that the alternative choice of the long word decomposition yields the exact lattice models. By combining two models, one can obtain the standard results like the Cauchy identities and duality for the supersymmetric ice.

We finally remark that our approach can be adapted in other settings. Instead of the general linear group, one can choose any other split reductive group and its covers. Instead of the delta word decomposition, one can pick the gamma word decomposition to obtain combinatorics matching colored lattice models from [BBBG19, BBBG20]. The main difficulty lies in the explicit expression of the character $\psi_{\lambda}$ in coordinates which is the reason why we need to pick a “good” word decomposition.

We use notation $\mathbb{N}=\mathbb{Z}_{\geq 0}$.

Acknowledgements. We thank Dan Bump for being a great advisor and his support throughout the project. We thank Alexei Borodin and Ben Brubaker for their helpful comments on the presentation of the paper. We thank Claire Frechette for the discussion about the general metaplectic covers. Thank you!

In Section 4 of [McN11], McNamara gives an explicit Iwasawa decomposition for the opposite maximal unipotent subgroup of a metaplectic cover of a split reductive group. We extend his results to get an explicit Iwahori decomposition. We loosely follow the exposition of Section 4 of [McN11].

Let $G$ be a split reductive group over a non-archumedian local field $F$. Let $\widetilde{G}$ be an $n$-fold metaplectic cover of $G$. We use notation from Section A.1, A.4: ring of integers $\mathcal{O}_{F}$, maximal ideal $\mathfrak{p}$, torus $\widetilde{T}$, Borel subgroup $\widetilde{B}$, maximal unipotent subgroup $U^{+}$ and its opposite $U=U^{-}$, and the opposite Iwahori subgroup $J$. And from Section A.5: positive roots $\Phi^{+}$, coroots $\Phi^{\vee}$, the Weyl group $W$, simple reflections $s_{i}$, and generators $e_{\alpha}$ and $h_{\alpha}$.

Let $\textbf{i}=(i_{1},\dots,i_{N})$ be a $N$-tuple of indices such that $w_{0}=s_{i_{1}}s_{i_{2}}\dots s_{i_{N}}$ is a reduced decomposition of the long word $w_{0}\in W$. The choice of i defines a total ordering $<_{\textbf{i}}$ on the set of positive roots given by

where $\gamma_{j}=s_{i_{N}}s_{i_{N-1}}\dots s_{i_{j+1}}\alpha_{i_{j}}$. The proof can be found in any Lie theory text, for example [Bou68][Ch VI, §6].

For each $k=0,1,2,\dots,N$, let $G_{k}$ denote the set of elements $g\in\widetilde{G}$ which can be expressed in the form

Note that $G_{0}=U^{+}$. It slightly differs from $G_{k}$ from Section 4 of [McN11] as we don’t include the torus element in our definition.

Before we give an explicit Iwahori decomposition, we need a technical result which is an extension of Algorithm 4.4 of [McN11] to the Iwahori subgroup.

We can also reformulate the result in terms of simple reflections. We favor this reformulation because it is easier to give a combinatorial description in terms of simple reflections.

Now we extract the combinatorial essence from the results above. We want to describe all possible sequences of $\sigma_{1},\sigma_{2},\dots,\sigma_{N+1}$ that can occur in the explicit Iwahori decomposition. We drop variables $y_{k}$ and keep only their valuations $m_{k}=\max(\operatorname{val}(y_{k}^{-1}),0)$, in other words, if $y_{k}\not\in\mathcal{O}$, we take the valuation of its inverse, and if $y_{k}\in\mathcal{O}$, we set $m_{k}$ to be zero.

We call $w^{\prime}$ the input, and $w$ the output of the coloring.

Informally, start with $w^{\prime}\in W$ and then read $(m_{1},m_{2},\dots,m_{N})$ from right to left. At each entry, update the current element. If $m_{k}>0$, skip it. If $m_{k}=0$, multiply by $s_{i_{k}}$ if applying the inverse of the current element to $\alpha_{i_{k}}$ lies in $\Phi^{+}$. Otherwise, either multiply or skip it. Then $\Sigma(m,w,w^{\prime})$ is the set of such sequences that end with $w\in W$. Note that $\Sigma(m,w,w^{\prime})$ may be empty.

Write $y_{k}=\varpi^{-m_{k}}u_{k}$, $u_{k}\in\mathcal{O}_{F}\times$ for $y_{k}\not\in\mathcal{O}_{F}$ and $y_{k}=u_{k}$, $u_{k}\in\mathcal{O}_{F}$ for $y_{k}\in\mathcal{O}_{F}$. From the proof of Theorem 2.9, we find a cell $S_{m,\sigma}$ is a set of elements in $U$ with $u_{k}\in D_{k}$, where the domains $D_{k}=D_{k}(m,\sigma)$ are defined by

Now we give the relation to the cells introduced by McNamara in [McN11].

We have the following relation of our cells with McNamara’s cells.

In the case of the general linear group and a good choice of a long word decomposition, it is possible to write down colorings from Section 2 combinatorically which will be useful when computing Iwahori Whittaker functions in Section 4. In that section we will need weights and contributions of the colored data. For coherency, we define them in this section next to the corresponding data. The statistics $r_{ij}$ and $s_{ij}$ and the Gauss sums $g(r_{ij},s_{ij})$ are defined in Section 4.

Let $G=\operatorname{GL}_{r+1}$. Realize the root system as $\Phi=\{(i,j)\in\mathbb{Z}^{2}\mid 1\leq i<j\leq r+1\}$ and the positive roots as $\Phi^{+}=\{(i,j)\in\mathbb{Z}^{2}\mid 1\leq i<j\leq r+1\}$. Denote the order of $\Phi^{+}$ by $N=r(r+1)/2$. We identify weights $\Lambda=X_{*}(T)\cong\mathbb{Z}^{r+1}$ and use the standard elementary basis $e_{i}$, so that simple roots $\alpha_{i}=e_{i}-e_{i+1}$. The fundamental weights $\varpi_{i}$ are defined by

We write a weight $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r+1})$ uniquely as $\lambda=\sum_{i}\Lambda_{i}\varpi_{i}$ with $\Lambda_{i}=\lambda_{i}-\lambda_{i+1}$.

Let $\textbf{i}=\Delta$ be the delta word, where

The choice of the delta word induces a total ordering $<_{\Delta}$ on the positive roots given by $(i,j)<_{\Delta}(i^{\prime},j^{\prime})$ if $j<j^{\prime}$ or if $j=j^{\prime}$ and $i<i^{\prime}$. I.e.,

We choose the delta word because it allows us to use the induction argument. To emphasize it, we write data labeled by positive roots in the form of a table, reading the data from the end and writing in the table from right to left, from top to bottom. For example,

We identify $W\cong S_{r+1}$. For the delta word, we interpret colorings in terms of actions on the set $\{1,2,\dots,r+1\}$; in this context we refer to numbers $1,2,\dots,r+1$ as colors.