165 research outputs found
Unipotent variety in the group compactification
The unipotent variety of a reductive algebraic group 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 of a connected simple algebraic group . We will
prove that \bar{\Cal U}-\Cal U is a union of some -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 .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 for any tamely
ramified group and its parahoric subgroup . We show that 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 -stable pieces and the generalization of
-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
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