The bound $a_{J,v}$ of $J$-folded alcove path representations and Lusztig's $a$-function (Conjectures 6.14-6.16 of Guilhot-Little-Parkinson)

For an affine Weyl group with a positive weight function, Guilhot, Little and Parkinson constructed combinatorial matrix representations of the affine Hecke algebra from $J$-folded alcove paths, where $J$ is a subset of the simple generators. When such a representation is bounded, the maximum degree $a_{J,v}$ of its matrix entries and the set $\Gamma_{J,v}$ of elements attaining this bound ('the recognised cell') are conjectured to be governed by Kazhdan-Lusztig theory: $a_{J,v}$ should be given by an explicit degree formula involving Macdonald's c-function and Opdam's Plancherel density, should satisfy $a_{J,v}\le L(w_0)$ with equality only for $J$ empty, and the elements of $\Gamma_{J,v}$ lying in the non-extended affine Weyl group should lie in a single two-sided Kazhdan-Lusztig cell on which Lusztig's $a$-function is constantly $a_{J,v}$. These conjectures (5.14-5.16 in the first arXiv version; 6.14-6.16 in print) are verified for $J$ empty, for all affine types of rank at most 3 (finite root system of rank at most 2), and - by a 2024/2026 follow-up of Chapelier-Laget, Guilhot, Little and Parkinson - for all of affine type $\tilde A_n$, but remain open for the remaining affine types, that is, those whose finite root system has rank at least 3 and type other than $A_n$ (types $\tilde B_n$, $\tilde C_n$, $\widetilde{BC}_n$, $\tilde D_n$, $\tilde F_4$, $\tilde E_{6,7,8}$), with general $J$, general bounded parameter systems, and unequal parameters.

Background

Let $\Phi$ be a finite root system (possibly non-reduced, of type $BC_n$) with Weyl group $W_0$, let $W$ be the associated affine Weyl group with Coxeter generators $S=\{s_0,s_1,\dots,s_n\}$, and let $\widetilde{W}$ be the extended affine Weyl group, obtained from $W$ by adjoining the symmetries $\Sigma$ of the fundamental alcove ($\widetilde{W}=W\rtimes\Sigma$ with $\Sigma\simeq P/Q$, the weight lattice modulo the root lattice). The reflecting hyperplanes of $W$ decompose the underlying vector space into alcoves, and a (positively folded) alcove path is a sequence of alcoves obtained from a minimal gallery (a shortest sequence of pairwise adjacent alcoves) by 'folding' at certain walls so that every crossing is in a positive direction; such paths, introduced by Ram and Schwer, give combinatorial formulae for matrix coefficients of affine Hecke algebras. The affine Hecke algebra $\widetilde{H}(L)$ with (possibly unequal) parameters encoded by a positive weight function $L:W\to\mathbb{Z}_{\ge 0}$ (with $L(w)>0$ for $w\ne e$ and $L(uv)=L(u)+L(v)$ whenever $\ell(uv)=\ell(u)+\ell(v)$, the parameters being $q_i=q^{L(s_i)}$) is a deformation of the group algebra of $\widetilde{W}$ with quadratic relations $(T_{s_i}-q_i)(T_{s_i}+q_i^{-1})=0$ together with the braid relations of $W$. Guilhot, Little and Parkinson introduced the following refinement: for a subset $J$ of the simple generators one 'folds' the direction of a path only through walls in the parabolic subgroup $W_J$ (the subgroup of $W$ generated by $J$), producing $J$-folded alcove paths, and for each suitable tuple $v$ of parameters (a 'J-parameter system', specialising the indeterminates to powers of $q$ determined by $L$) these paths index a basis of a right $\widetilde{H}(L)$-module $M_{J,v}$, giving a matrix representation $(\pi_{J,v},M_{J,v},B_{J,v})$ whose entries are rational functions in $q$. The representation is bounded if the degrees of all matrix entries $[\pi_{J,v}(T_w)]_{b,b'}$, $b,b'\in B_{J,v}$, $w\in\widetilde{W}$, are bounded above (for a rational function $f(q)=q^N a'(q^{-1})/b'(q^{-1})$ with $a',b'$ polynomials with nonzero constant term, $\deg f=N$); the bound $a_{J,v}$ is the maximum of these degrees, and the recognised cell $\Gamma_{J,v}$ is the set of $w$ where the maximum is attained. On the other side, the Kazhdan-Lusztig basis of $\widetilde{H}(L)$ partitions the affine Weyl group $W$ into left, right, and two-sided cells, and Lusztig's $a$-function $a(w)$ is a cell-theoretic invariant defined from the degrees of the structure constants of that basis; it is constant on two-sided cells. The framework originated in work of Guilhot and Parkinson, who used explicitly constructed bounded 'cell representations' to prove Lusztig's conjectures P1-P15 - Lusztig's foundational conjectural axioms for the cell structure, the $a$-function, and the asymptotic algebra of Hecke algebras with unequal parameters - for the affine Weyl groups of the rank-2 root systems ($\tilde G_2$ and $\tilde C_2$) with arbitrary parameters. Macdonald's c-function $c(X)=\prod_{\alpha\in\Phi^+}\frac{1-q_{\alpha/2}^{-1}q_\alpha^{-2}X^{-\alpha^\vee}}{1-q_{\alpha/2}^{-1}X^{-\alpha^\vee}}$ is the normalising factor of the affine Hecke algebra's Plancherel measure, and Opdam's Plancherel theorem expresses the central decomposition of the Hecke algebra in terms of that measure; the conjectures below predict that the purely combinatorial statistic $a_{J,v}$ coincides with the degree of the Plancherel density $\psi_{J,v}(q^{2L(w_0)}c(X)c(X^{-1}))'$ of the corresponding character (the prime indicating that factors which vanish under the specialisation $\psi_{J,v}$ are omitted), where $w_0$ is the longest element of $W_0$ and $L(w_0)$ its length. The subtle point is that individual path contributions can exceed the matrix-entry bound: explicit examples in type $\tilde A_4$ show leading terms of different $J$-folded paths cancelling, so the bound is invisible at the level of single paths.

Problem Statement

Resolve, in full generality, the conjectural program of Guilhot-Little-Parkinson (arXiv:2212.10781; published as Algebras and Representation Theory 27 (2024), where it appears as Conjectures 6.14-6.16; numbered 5.14-5.16 in arXiv v1): for the affine Weyl group $W$ (inside the extended affine Weyl group $\widetilde{W}$) of a finite root system $\Phi$ (possibly non-reduced), equipped with an arbitrary positive weight function $L$, an arbitrary subset $J$ of the simple generators, and any $J$-parameter system $v$ for which the matrix representation $(\pi_{J,v},M_{J,v},B_{J,v})$ constructed from $J$-folded alcove paths is bounded, determine the bound $a_{J,v}=\max_{b,b'\in B_{J,v},\,w\in\widetilde{W}}\deg[\pi_{J,v}(T_w)]_{b,b'}$ and prove or disprove that: (i) $a_{J,v}\le L(w_0)$, where $w_0$ is the longest element of the finite Weyl group $W_0$, with equality if and only if $J=\emptyset$; (ii) $a_{J,v}$ is given by the explicit formula

$$a_{J,v}=L(w_0)-\tfrac{1}{2}\,\deg\prod_{\alpha\in\Phi}^{\prime}\frac{1-q_{\alpha/2}^{-1}v^{\alpha^\vee}}{1-q_{\alpha/2}^{-1}q_{\alpha}^{-2}v^{\alpha^\vee}},$$

where $\alpha_1,\dots,\alpha_n$ are the simple roots of $\Phi$ (with $\alpha_n$ short in the non-reduced case), $q_\alpha=q^{L(s_i)}$ for $\alpha\in W_0\alpha_i$ when $\Phi$ is reduced, while in the non-reduced case $q_\alpha=q^{L(s_i)}$ for $\alpha\in W_0\alpha_i$ with $i\ne n$, $q_\alpha=q^{L(s_n)-L(s_0)}$ for $\alpha\in W_0\alpha_n$, and $q_\alpha=q^{L(s_0)}$ for $\alpha$ long, and $\prod'$ omits any factors in the numerator or denominator that are identically $0$; the right-hand side is equivalently $\tfrac{1}{2}\deg\psi_{J,v}(q^{2L(w_0)}c(X)c(X^{-1}))'$, the degree of the Macdonald c-function/Opdam Plancherel density of the character $\chi_{J,v}$, the prime again indicating omission of factors vanishing under the specialisation $\psi_{J,v}$; and (iii) with $\Gamma_{J,v}$ the recognised cell, Lusztig's $a$-function satisfies $a(w)=a_{J,v}$ for all $w\in\Gamma_{J,v}\cap W$, and $\Gamma_{J,v}\cap W$ is contained in a single two-sided Kazhdan-Lusztig cell of the weighted Coxeter group $(W,L)$. The already-verified cases ($J=\emptyset$ in all types; all types of affine rank at most 3 with all parameters; affine type $\tilde A_n$ completely) may be assumed; the open target is the conjectures for the remaining affine types (finite root system of rank at least 3 and type other than $A_n$: $\tilde B_n$, $\tilde C_n$, $\widetilde{BC}_n$ with $n\ge 3$, $\tilde D_n$, $\tilde F_4$, $\tilde E_6$, $\tilde E_7$, $\tilde E_8$), all subsets $J$, all bounded $J$-parameter systems, and arbitrary unequal parameters.

References

1. Jérémie Guilhot, Eloise Little, James Parkinson, "On J-folded Alcove Paths and Combinatorial Representations of Affine Hecke Algebras", Algebras and Representation Theory 27(6), 2131-2185 (2024). DOI: 10.1007/s10468-024-10293-7, https://doi.org/10.1007/s10468-024-10293-7 (arXiv:2212.10781, https://arxiv.org/abs/2212.10781)
2. Nathan Chapelier-Laget, Jérémie Guilhot, Eloise Little, James Parkinson, "The asymptotic Plancherel formula and Lusztig's asymptotic algebra for $\tilde{\mathsf{A}}_n$", arXiv:2406.07004 (v4, 22 March 2026), to appear in Selecta Mathematica. https://arxiv.org/abs/2406.07004
3. J. Guilhot, E. Little, J. Parkinson, "Bounded weight functions on regular languages and groups", arXiv:2510.17061 (2025). https://arxiv.org/abs/2510.17061
4. Jérémie Guilhot, James Parkinson, "A proof of Lusztig's conjectures for affine type G2 with arbitrary parameters", Proceedings of the London Mathematical Society 118(5), 1188-1244 (2018/2019). DOI: 10.1112/plms.12211, https://doi.org/10.1112/plms.12211
5. Jérémie Guilhot, James Parkinson, "Balanced representations, the asymptotic Plancherel formula, and Lusztig's conjectures for $\tilde C_2$", Algebraic Combinatorics 2(5), 969-1031 (2019). DOI: 10.5802/alco.75, https://doi.org/10.5802/alco.75
6. George Lusztig, "Hecke Algebras with Unequal Parameters", CRM Monograph Series 18, American Mathematical Society (2003). DOI: 10.1090/crmm/018, https://doi.org/10.1090/crmm/018
7. Eric M. Opdam, "On the spectral decomposition of affine Hecke algebras", Journal of the Institute of Mathematics of Jussieu 3(3), 531-648 (2004). DOI: 10.1017/s1474748004000155, https://doi.org/10.1017/s1474748004000155
8. I. G. Macdonald, "Affine Hecke Algebras and Orthogonal Polynomials", Cambridge University Press, Cambridge Tracts in Mathematics 157 (2003). DOI: 10.1017/cbo9780511542824, https://doi.org/10.1017/cbo9780511542824