Generic Newton points and the Newton poset in Iwahori double cosets

Closure relations of Newton strata in Iwahori double cosets

We consider the Newton stratification on Iwahori double cosets for a connected reductive group. We prove the existence of Newton strata whose closures cannot be expressed as a union of strata, and show how this is implied by the existence of non-equidimensional affine Deligne-Lusztig varieties. We also give an explicit example for a group of type $A_4$.

Minimal Newton strata in Iwahori double cosets

The set of Newton strata in a given Iwahori double coset in the loop group of a reductive group G is indexed by a finite subset of the set B(G) of Frobenius-conjugacy classes. For unramified $G$ we show that it has a unique minimal element and determine this element. Under a regularity assumption we also compute the dimension of the corresponding Newton stratum. We derive corresponding results for affine Deligne-Lusztig varieties.

Levi子群中的Newton分层

Newton Strata in Levi Subgroups

本文研究了约化群环群中Iwahori双陪集的Newton分层，证明其与Levi子群中Iwahori双陪集的Newton分层关系比此前已知更为紧密。

Affine Deligne-Lusztig Varieties and Quantum Bruhat Graph

In this paper, we consider affine Deligne-Lusztig varieties $X_w(b)$ and their certain union $X(μ,b)$ inside the affine flag variety of a reductive group. Several important results in the study of affine Deligne-Lusztig varieties have been established under the so-called superregularity hypothesis. Such results include a description of generic Newton points in Iwahori double cosets of loop groups, covering relation in associated Iwahori-Weyl group and dimension formula for $X(μ,b)$. We show that one can considerably weaken the superregularity hypothesis and sometimes completely eliminate it, t…

Double affine Hecke algebras and 2-dimensional local fields

We give an interpretation of the double affine Hecke algebra of Cherednik as the (suitably regularized) algebra of double cosets of a group G by a subgroup J, extending the well known interpretations of finite and affine Hecke algebras. In this interpretation, G consists of K-points of a split reductive group where K is a 2-dimensional local field such as Q_p((t)) or F_q((t_1))((t_2)), and J is a certain analog of the Iwahori subgroup.

Chevalley多面体与Newton-Okounkov体

Chevalley Polytopes and Newton-Okounkov Bodies

构造了齐性空间射影嵌入中的Chevalley多面体，证明其在minuscule情形下为Newton-Okounkov体且Plücker坐标构成Khovanskii基，并提出一般情形的猜想与分解性质。