Smooth modular representation theory of p-adic reductive groups

朗兰兹计划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特征p中局部朗兰兹规划的年轻变体。在第一个资助期内，我们已经展示了模型范畴理论如何有效地应用于特征p中p-进约群的光滑表示的研究。特别地，我们将Cabanes关于有限约化群的一个重要结果推广到p-进环境中。此外，我们还可以说明如何从Bruhat-Tits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环境的又一步。在第二个资助期，我们打算进一步发展我们的方法，并将它们应用于与特征p中的局部朗兰兹程序有关的问题，特别是涉及Gorenstein派生范畴，Hecke模的同伦范畴中导生元的计算，以及通过Bruhat-Tits建筑的紧化来研究广义(Phi，Gamma)-模的有限性质。

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.