Unipotent variety in the group compactification

The unipotent variety of a reductive algebraic group $G$ plays an important role in the representation theory. In this paper, we will consider the closure \bar{\Cal U} of the unipotent variety in the De Concini-Procesi compactification $\bar{G}$ of a connected simple algebraic group $G$. We will prove that \bar{\Cal U}-\Cal U is a union of some $G$-stable pieces introduced by Lusztig in \cite{L4}. This was first conjectured by Lusztig. We will also give an explicit description of \bar{\Cal U}. It turns out that similar results hold for the closure of any Steinberg fiber in $\bar{G}$.Comment: 21 pages. Final version. To appear in Adv. in Mat

Normality and Cohen-Macaulayness of local models of Shimura varieties

We prove that in the unramified case, local models of Shimura varieties with Iwahori level structure are normal and Cohen Macaulay.Comment: 13 pages, final versio

Some results on affine Deligne-Lusztig varieties

The study of affine Deligne-Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne-Lusztig varieties are purely Lie-theoretic in nature. This survey deals with recent progress on several important problems on affine Deligne-Lusztig varieties. The emphasis is on the Lie-theoretic aspect, while some connections and applications to arithmetic geometry will also be mentioned.Comment: 2018 ICM report, reference update

Total positivity in the De Concini-Procesi compactification

We study the nonnegative part \bar{G_{>0}} of the De Concini-Procesi compactification of a reductive algebraic group G, as defined by Lusztig. Using positivity properties of the canonical basis and parametrization of flag varieties, we will give an explicit description of \bar{G_{>0}}. This answers the question of Lusztig in [L4]. We will also prove that \bar{G_{>0}} has a cell decomposition which was conjectured by Lusztig.Comment: 23 pages. Some corrections. References update

Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties

In this paper, we prove a conjecture of Kottwitz and Rapoport on a union of (generalized) affine Deligne-Lusztig varieties $X(\mu, b)_J$ for any tamely ramified group $G$ and its parahoric subgroup $P_J$. We show that $X(\mu, b)_J \neq \emptyset$ if and only if the group-theoretic version of Mazur's inequality is satisfied. In the process, we obtain a generalization of Grothendieck's conjecture on the closure relation of \s-conjugacy classes of a twisted loop group.Comment: 19 page

Minimal length elements in some double cosets of Coxeter groups

We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig's $G$-stable pieces and the generalization of $G$-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in \cite{GKP}.Comment: 35 page