Smooth modular representation theory of p-adic reductive groups

朗兰兹纲领是现代数学中最有影响力的驱动力之一。我们的研究项目是关于年轻的本地Langlands计划在特征p的变种。在第一个资助期间，我们已经展示了如何模型范畴的理论可以卓有成效地应用到研究的光滑表示的p-adic约化群的特征p。特别是，我们推广了一个重要的结果Cabanes关于有限约化群的p-adic设置。此外，我们可以展示如何从布鲁哈特-提茨建筑上的适当系数系统过渡到等变层的对偶理论。这是将Schneider和Stuhler的理论推广到mod-p设置的又一步。在第二个资助期内，我们打算进一步发展我们的方法，并将其应用到与当地朗兰兹程序的特征p.特别是，这涉及Gorenstein派生类别，计算派生生成同伦类别的Hecke模块和有限性的研究广义（phi，伽玛）-模块通过紧化的Bruhat-山雀建筑物。

The Langlands program is one of the most influential driving forces in modern mathematics. Our research project is concerned with the young variant of the local Langlands program in characteristic p. During the first funding period we have shown how the theory of model categories can be applied fruitfully to the study of smooth representations of p-adic reductive groups in characteristic p. In particular, we generalized an important result of Cabanes concerning finite reductive groups to the p-adic setting. Moreover, we could show how to pass from suitable coefficient systems on the Bruhat-Tits building to a dual theory of equivariant sheaves. This is another step towards a generalization of the theory of Schneider and Stuhler to the mod-p setting. During the second funding period we intend to develop our approaches further and apply them to questions related to the local Langlands program in characteristic p. In particular, this concerns Gorenstein derived categories, the computation of derived generators in homotopy categories of Hecke modules and the study of finiteness properties of generalized (phi,Gamma)-modules via compactifications of Bruhat-Tits buildings.