Resonant Mirković–Vilonen polytopes and formulas for highest-weight characters

Abstract

Formulas for the product of an irreducible character $\chi _\lambda$ of a complex Lie group and a deformation of the Weyl denominator as a sum over the crystal ${\mathcal {B}}(\lambda +\rho )$ go back to Tokuyama. We study the geometry underlying such formulas using the expansion of spherical Whittaker functions of p-adic groups as a sum over the canonical basis ${\mathcal {B}}(-\infty )$, which we show may be understood as arising from tropicalization of certain toric charts that appear in the theory of total positivity and cluster algebras. We use this to express the terms of the expansion in terms of the corresponding Mirković–Vilonen polytope. In this non-archimedean setting, we identify resonance as the appropriate analogue of total positivity, and introduce resonant Mirković–Vilonen polytopes as the corresponding geometric context. Focusing on the exceptional group $G_2$, we show that these polytopes carry new crystal graph structures which we use to compute a new Tokuyama-type formula as a sum over ${\mathcal {B}}(\lambda +\rho )$ plus a geometric error term coming from finitely many crystals of resonant polytopes.

Appendix A: Verification of Conjecture 4.6 for \(A_3\)

In this appendix, we verify Conjecture 4.6 in the simplest case. Let $G={{\,\mathrm{SL}\,}}(4)$, and consider the reduced expression $\underline{i}=(3,2,1,2,3,2)$. We enumerate the three simple roots in the standard way. The induced ordering on the positive roots $\{\gamma _k\}_{k=1}^6$ is given by:

$$\begin{aligned} \alpha _1<\alpha _1+\alpha _2<\alpha _1+\alpha _2+\alpha _3<\alpha _3<\alpha _2+\alpha _3<\alpha _2. \end{aligned}$$

Given our choice of $\underline{i}$, there is a unique $\underline{i}$-trail from $\Lambda _2\longrightarrow w_0s_2\Lambda _2$, two from $\Lambda _3\longrightarrow w_0s_3\Lambda _3$, and four $\underline{i}$-trails from $\Lambda _1\longrightarrow w_0s_1\Lambda _1$. This may be checked by applying [10, Proposition 9.2], as $\Lambda _k$ is always minuscule in type A. Then by Proposition 4.3, we find that if $u=z_{\underline{i}}(b_\bullet )$, then ${\mathfrak {s}}_2(u)=1/b_6=X_6$, ${\mathfrak {s}}_3(u)=1/b_5+b_6/b_4b_5=X_5+X_4$, and finally

$$\begin{aligned} {\mathfrak {s}}_1(u) = \frac{1}{b_3}+\frac{b_4}{b_2b_3}+\frac{b_5b_6}{b_1b_2b_3}+\frac{b_6}{b_2b_3}=X_3+X_2+X_1+\frac{X_2X_4}{X_5}. \end{aligned}$$

Following the construction of $g_i(t_\alpha ,w_\alpha )$ in Sect. 4.2, we have $g_2(t_\alpha ,w_\alpha ) = t_6$,

$$\begin{aligned} g_2(t_\alpha ,w_\alpha ) = t_5+\frac{t_4w_5}{w_6}, \text { and }g_1(t_\alpha ,w_\alpha )=t_3+\frac{t_2w_3}{w_4}+\frac{t_1w_2w_3}{w_5w_6}+\frac{t_2t_4w_3}{w_4w_6}. \end{aligned}$$

On the other hand, if we apply the algorithm (3.3), we find that $h_2=g_2$, $h_3=g_3$, but

$$\begin{aligned} h_1(t_\alpha ,w_\alpha )=g_1(t_\alpha ,w_\alpha )+\frac{t_3w_4}{w_6}\left( 1-\frac{t_4}{w_4}\right) . \end{aligned}$$

As before, we denote the bounding data $s_k={{\,\mathrm{val}\,}}(\varpi ^{\lambda _{i^*}}X_k)$.

Proposition A.1

Let $\underline{i}=(3,2,1,2,3,2)$. For any $\underline{i}$-Lusztig datum $\mathbf{m}$ and dominant coweight $\lambda$, we have the equality

$$\begin{aligned} I_\lambda (\mathbf{m}) = \int _{C^{\underline{i}}(\mathbf{m})}f(u)\psi \left( \sum _{i\in I}\varpi ^{\lambda _{i^*}}g_i(t_\alpha ,w_\alpha )\right) du. \end{aligned}$$

(25)

That is, Conjecture 4.6 holds in this case.

Up to passing to dual long words $\underline{i}\mapsto \underline{i}^*$, this is the only long word of type $A_3$ for which the conjecture is not immediate. The analysis below relies in a precise fashion on the various piecewise linear relations between the bounding data.

Proof

For simplicity, we assume that $n=1$, though the statement holds in general but with more tedious notation. Note that this is obvious if $t_4=w_4$, so we assume that $m_4=0$. We may express $I_\lambda (\mathbf{m})$ as

$$\begin{aligned} \prod _{\alpha \in \Delta ^+}(q^{-1}x_\alpha )^{m_\alpha }I(s_1,m_1)I(s_6,m_6)J_{2,3,4,5}, \end{aligned}$$

where I(a, b) is defined in Sect. 5 and

$$\begin{aligned} J_{2,3,4,5}= & {} \displaystyle \int \int \int \int \psi \left( \varpi ^{\lambda _2}\left( t_5+\frac{t_4w_5}{w_6}\right) \right. \\&\left. +\varpi ^{\lambda _3}\left( {t_2w_3}+\frac{t_2t_4w_3}{w_6}+t_3\left( 1+\frac{1-t_4}{w_6}\right) \right) \right) dt, \end{aligned}$$

and the precise domain of integration depends on the circling pattern of $\mathbf{m}$. Note that if $m_6>0$, then $1+(1-t_4)/w_6\in {\mathcal {O}}^\times$, so a simple change of variables reduces this to

$$\begin{aligned} \displaystyle \int \int \int \int \psi \left( \varpi ^{\lambda _2}\left( t_5+\frac{t_4w_5}{w_6}\right) +\varpi ^{\lambda _3}\left( {t_2w_3}+\frac{t_2t_4w_3}{w_6}+t_3\right) \right) dt, \end{aligned}$$

which matches (25). If $m_6=0$, the same argument works unless $\lambda _3-m_3\le -1$ and ${{\,\mathrm{val}\,}}(2-t_4)>0$. However in this case, we have the inner integration

$$\begin{aligned} \int \psi \left( \varpi ^{\lambda _3}t_2w_3\left( 1+t_4\right) \right) dt_2=0, \end{aligned}$$

which is also true for (25). This exhausts the cases, proving the proposition. $\square$