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.
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Notes
The dual word arises here due to our definition of the twist map. This choice is motivated by later computations related to the Iwasawa decomposition.
This is simply to align with the conventions of [10].
Here we ignore the dependence on \(\underline{i}\).
This distinction is similar to the Class I, Class II, and totally resonant cases arising in the analysis of Fourier coefficeints of metaplectic Eisenstein series of type B in [16].
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Acknowledgements
I want to thank Solomon Friedberg for introducing me to questions which led directly to this project, as well as for many helpful conversations. I also wish to thank both Ben Brubaker and Dan Bump for helpful conversations on crystal graphs, Whittaker functions, and other topics on multiple occasions. Finally, I wish to thank the anonymous referee who explained to me the notion of (upper) seminormality and suggesting the correct statement of Theorem 6.8.
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Appendix A: Verification of Conjecture 4.6 for \(A_3\)
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:
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
Following the construction of \(g_i(t_\alpha ,w_\alpha )\) in Sect. 4.2, we have \(g_2(t_\alpha ,w_\alpha ) = t_6\),
On the other hand, if we apply the algorithm (3.3), we find that \(h_2=g_2\), \(h_3=g_3\), but
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
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
where I(a, b) is defined in Sect. 5 and
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
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
which is also true for (25). This exhausts the cases, proving the proposition. \(\square \)
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Leslie, S. Resonant Mirković–Vilonen polytopes and formulas for highest-weight characters. Sel. Math. New Ser. 25, 41 (2019). https://doi.org/10.1007/s00029-019-0486-7
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DOI: https://doi.org/10.1007/s00029-019-0486-7
Keywords
- p-adic reductive groups
- Spherical Whittaker functions
- Total positivity
- Crystal bases
- Mirković–Vilonen polytopes
- Resonance
- Tokuyama’s formula