Pro-p-Iwahori Hecke ring and supersingular -representations
Pro-p-Iwahori Hecke ring and supersingular -representations
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DOI：
10.1007/s00208-004-0592-4
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发表时间：
2005-03
影响因子：
1.4
通讯作者：
M. Vigneras
中科院分区：
文献类型：
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作者：
M. Vigneras
本文的动机是寻找一个朗兰兹对应模。我们证明了支持P-岩堀海克环 具有q个元素的有限剩余域Fq的局部域Ff上的分裂可约ep-adic群G，允许Iwahori-Matsumoto表示和BernsteinZ-基，并且我们确定了它的中心。我们证明戒指 是在其中心上作为一个模块生成的。文[11]仅对Iwahori Hecke环证明了这些结果。设p是整除q的素数，k是特征p的代数闭域。一个来自 tokwhich是“as null as possible”将被称为null。简单 中心特征标为空的模称为超奇异模。当G =GL（n）时，我们证明了每个简单的 含仿射子环特征标的维数模 是超奇异的，使用Haines的最小表达式推广到 当Hecke代数侧的一个单化子PF的作用和Galois侧的一个Frobenius FRF的行列式的作用固定时，n维Weil群WF的不可约k-表示的个数等于n维Weil群WF的不可约k-表示的个数，即在Fq [X]中的degreen的酉不可约多项式的个数Nn（q）.我们知道，当n =2 [10]和n =3时，通过显式计算，匡威命题成立（Rachel Ollivier）。
The motivation of this paper is the search for a Langlands correspondence modulop. We show that the pro-p-Iwahori Hecke ring of a split reductivep-adic groupGover a local fieldFof finite residue fieldFqwithqelements, admits an Iwahori-Matsumoto presentation and a BernsteinZ-basis, and we determine its centre. We prove that the ring is finitely generated as a module over its centre. These results are proved in [11] only for the Iwahori Hecke ring. Letpbe the prime number dividingqand letkbe an algebraically closed field of characteristicp. A character from the centre of tokwhich is “as null as possible” will be called null. The simple -modules with a null central character are called supersingular. WhenG=GL(n), we show that each simple -module of dimensionncontaining a character of the affine subring is supersingular, using the minimal expressions of Haines generalized to , and that the number of such modules is equal to the number of irreduciblek-representations of the Weil groupWFof dimensionn(when the action of an uniformizerpFin the Hecke algebra side and of the determinant of a Frobenius FrFin the Galois side are fixed), i.e. the numberNn(q) of unitary irreducible polynomials inFq[X] of degreen. One knows that the converse is true by explicit computations whenn=2 [10], and whenn=3 (Rachel Ollivier).