Smooth representations and Hecke modules in characteristic p
Smooth representations and Hecke modules in characteristic p
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特征 p 中的平滑表示和 Hecke 模
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发表时间：
2015
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通讯作者：
P. Schneider
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作者：
P. Schneider
设G是一个p-adic李群，I ∈ G是一个紧的开子群，它是一个挠自由的pro-p-群.在特征为p的系数域k上，我们引入了一个微分分次Hecke代数，并证明了G在k-向量空间中的光滑表示的导出范畴自然等价于这个Hecke DGA上的微分分次模的导出范畴. 1背景与动机设G是d维p-adic李群，k是任意域。我们用Modk（G）表示k-向量空间中的光滑G-表示范畴。它显然有任意的直和。我们固定一个紧开子群I ∈ G.在Modk（G）中，我们有表示indI（1）：= G/I上具有有限支撑的所有k值函数，其中G通过左平移作用。indI（1）由单个元素生成，是平凡陪集的特征函数，是Modk（G）中的紧致对象。它生成Modk（G）中所有表示V的全子范畴Modk（G），这些表示由它们的I-固定向量VI生成。一般来说，Modk（G）不是交换范畴。定义I的Hecke代数为自同态环HI：= EndModk（G）（ind G I（1））op。设Mod（HI）表示左单位HI -模范畴.有一对伴随函子H：Modk（G）−→ Mod（HI）V 7−→ V I = HomModk（G）（ind G I（1），V），和T0：Mod（HI）−→ Modk（G）<$Modk（G）M 7−→ indI（1）<$HI M。
Let G be a p-adic Lie group and I ⊆ G be a compact open subgroup which is a torsionfree pro-p-group. Working over a coefficient field k of characteristic p we introduce a differential graded Hecke algebra for the pair (G, I) and show that the derived category of smooth representations of G in k-vector spaces is naturally equivalent to the derived category of differential graded modules over this Hecke DGA. 1 Background and motivation Let G be a d-dimensional p-adic Lie group, and let k be any field. We denote by Modk(G) the category of smooth G-representations in k-vector spaces. It obviously has arbitrary direct sums. We fix a compact open subgroup I ⊆ G. In Modk(G) we then have the representation indI (1) := all k-valued functions with finite support on G/I with G acting by left translations. Being generated by a single element, which is the characteristic function of the trivial coset, indI (1) is a compact object in Modk(G). It generates the full subcategory Modk(G) of all representations V in Modk(G) which are generated by their I-fixed vectors V I . In general Modk(G) is not an abelian category. The Hecke algebra of I by definition is the endomorphism ring HI := EndModk(G)(ind G I (1)) op . We let Mod(HI) denote the category of left unital HI -modules. There is the pair of adjoint functors H : Modk(G) −→ Mod(HI) V 7−→ V I = HomModk(G)(ind G I (1), V ) , and T0 : Mod(HI) −→ Modk(G) ⊆ Modk(G) M 7−→ indI (1)⊗HI M .