A proof of Lusztig's conjectures for affine type G2 with arbitrary parameters
A proof of Lusztig's conjectures for affine type G2 with arbitrary parameters
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DOI：
10.1112/plms.12211
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发表时间：
2017-11
影响因子：
1.8
通讯作者：
J. Guilhot;J. Parkinson
中科院分区：
文献类型：
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作者：
J. Guilhot;J. Parkinson
对于所有参数选择，我们证明了 G∼2 型仿射 Weyl 群的 Lusztig 猜想 P1 – P15。我们计算 Lusztig 函数的方法基于 Hecke 代数的“细胞表示的平衡系统”的概念。我们证明，对于任意 Coxeter 类型，细胞表示的平衡系统的存在足以计算 a 函数，并且我们针对任意参数显式地构造了 G∼2 类型的这样一个系统。然后，我们研究 Kazhdan-Lusztig 细胞与 G∼2 型中的 Plancherel 定理之间的联系，使我们能够证明 P1 并确定 Duflo 对合集。从这里开始，剩下的猜想的证明就非常自然了，本质上是来自 G2 和 A1 类型的 Weyl 特征的组合，以及有限单元的一些显式计算。
We prove Lusztig's conjectures P1 – P15 for the affine Weyl group of type G∼2 for all choices of parameters. Our approach to compute Lusztig's a ‐function is based on the notion of a ‘balanced system of cell representations’ for the Hecke algebra. We show that for arbitrary Coxeter type the existence of balanced system of cell representations is sufficient to compute the a ‐function and we explicitly construct such a system in type G∼2 for arbitrary parameters. We then investigate the connection between Kazhdan–Lusztig cells and the Plancherel theorem in type G∼2 , allowing us to prove P1 and determine the set of Duflo involutions. From there, the proof of the remaining conjectures follows very naturally, essentially from the combinatorics of Weyl characters of types G2 and A1 , along with some explicit computations for the finite cells.