仿射Deligne-Lusztig簇的维数：基于标号折叠凹室行走和根算子的新方法
Dimensions of affine Deligne-Lusztig varieties: a new approach via labeled folded alcove walks and root operators
摘要：
设G是域F=k（（t））上的约化群，其中k是有限域的代数闭包，W是G的（扩展）仿射外尔群. Rapoport引入了相关的仿射Deligne-Lusztig簇$X_x(b)$，它们由G（F）中的元素b和W中的元素x索引。关于变种$X_x(b)$的基本问题在很大程度上仍然是开放的，包括它们何时是非空的，如果非空，它们的维数。我们使用几何群论和表示论启发的技巧，在b是纯平移的情况下解决这些问题，从而证明了Görtz、Haines、Kottwitz和Reuman猜想9.5.1的一个尖锐版本。我们的方法是建设性的和无类型的，揭示了新的光的情况下，b是基本的现有结果的原因，并揭示了新的模式。由于我们只在G（F）的建筑物的标准公寓中工作，因此我们的结果在p-adic上下文中也成立，在那里我们定义了p-adic Deligne-Lusztig集的维数。我们提出了两个直接后果，我们的主要结果，类多项式的仿射Hecke代数和仿射反射长度。
英文摘要：
Let G be a reductive group over the field F=k((t)), where k is an algebraic closure of a finite field, and let W be the (extended) affine Weyl group of G. The associated affine Deligne-Lusztig varieties $X_x(b)$, which are indexed by elements b in G(F) and x in W, were introduced by Rapoport. Basic questions about the varieties $X_x(b)$ which have remained largely open include when they are nonempty, and if nonempty, their dimension. We use techniques inspired by geometric group theory and representation theory to address these questions in the case that b is a pure translation, and so prove much of a sharpened version of Conjecture 9.5.1 of Görtz, Haines, Kottwitz, and Reuman. Our approach is constructive and type-free, sheds new light on the reasons for existing results in the case that b is basic, and reveals new patterns. Since we work only in the standard apartment of the building for G(F), our results also hold in the p-adic context, where we formulate a definition of the dimension of a p-adic Deligne-Lusztig set. We present two immediate consequences of our main results, to class polynomials of affine Hecke algebras and to affine reflection length.
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