The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties

2007 ◽  
Vol 2007 (609) ◽  
Author(s):  
Ulrich Görtz ◽  
Thomas J Haines
Keyword(s):  
Shimura Varieties ◽  
Flag Varieties ◽  
Nearby Cycles ◽  
Affine Flag

scholarly journals Local models in the ramified case. III Unitary groups

2009 ◽  
Vol 8 (3) ◽  
pp. 507-564 ◽  
Author(s):  
G. Pappas ◽  
M. Rapoport
Keyword(s):  
Local Structure ◽  
Level Structure ◽  
Quadratic Extension ◽  
Unitary Groups ◽  
Shimura Varieties ◽  
Flag Varieties ◽  
Shimura Variety ◽  
Local Models ◽  
Affine Flag

AbstractWe continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primespat which the group defining the Shimura variety ramifies. We describe ‘good’p-adic integral models of these Shimura varieties and study their étale local structure. In the present paper we mainly concentrate on the case of unitary groups for a ramified quadratic extension. Some of our results are applications of the theory of twisted affine flag varieties that we developed in a previous paper.

scholarly journals Affine Deligne–Lusztig varieties in affine flag varieties

2010 ◽  
Vol 146 (5) ◽  
pp. 1339-1382 ◽  
Author(s):  
Ulrich Görtz ◽  
Thomas J. Haines ◽  
Robert E. Kottwitz ◽  
Daniel C. Reuman
Keyword(s):  
Level Structure ◽  
Flag Manifold ◽  
Shimura Varieties ◽  
Flag Varieties ◽  
Algorithmic Approach ◽  
Divisible Groups ◽  
Affine Flag

AbstractThis paper studies affine Deligne–Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, and extends previous conjectures concerning their dimensions. We generalize the superset method, an algorithmic approach to the questions of non-emptiness and dimension. Our non-emptiness results apply equally well to the p-adic context and therefore relate to moduli of p-divisible groups and Shimura varieties with Iwahori level structure.

scholarly journals Local models for Weil-restricted groups

2016 ◽  
Vol 152 (12) ◽  
pp. 2563-2601 ◽  
Author(s):  
Brandon Levin
Keyword(s):  
Hecke Algebra ◽  
Shimura Varieties ◽  
Reductive Groups ◽  
Local Models ◽  
Central Function ◽  
Nearby Cycles ◽  
Weil Restriction ◽  
Trace Of Frobenius ◽  
Invent Math

We extend the group-theoretic construction of local models of Pappas and Zhu [Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math.194(2013), 147–254] to the case of groups obtained by Weil restriction along a possibly wildly ramified extension. This completes the construction of local models for all reductive groups when$p\geqslant 5$. We show that the local models are normal with special fiber reduced and study the monodromy action on the sheaves of nearby cycles. As a consequence, we prove a conjecture of Kottwitz that the semi-simple trace of Frobenius gives a central function in the parahoric Hecke algebra. We also introduce a notion of splitting model and use this to study the inertial action in the case of an unramified group.

scholarly journals Alcove walks and nearby cycles on affine flag manifolds

2007 ◽  
Vol 26 (4) ◽  
pp. 415-430 ◽  
Author(s):  
Ulrich Görtz
Keyword(s):  
Flag Manifolds ◽  
Nearby Cycles ◽  
Affine Flag

Affine flag varieties

2020 ◽  
pp. 191-206
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Group Scheme ◽  
Flag Variety ◽  
Flag Varieties ◽  
Connected Component ◽  
Witt Vector ◽  
Classical Definition ◽  
Definition Of ◽  
Affine Flag Variety ◽  
Affine Flag

This chapter reviews affine flag varieties. It generalizes some of the previous results to the case where G over Zp is a parahoric group scheme. In fact, slightly more generally, it allows the case that the special fiber is not connected, with connected component of the identity G? being a parahoric group scheme. This case comes up naturally in the classical definition of Rapoport-Zink spaces. The chapter first discusses the Witt vector affine flag variety over Fp. This is an increasing union of perfections of quasiprojective varieties along closed immersions. In the case that G° is parahoric, one gets ind-properness.

scholarly journals Twisted loop groups and their affine flag varieties

2008 ◽  
Vol 219 (1) ◽  
pp. 118-198 ◽  
Author(s):  
G. Pappas ◽  
M. Rapoport
Keyword(s):  
Flag Varieties ◽  
Loop Groups ◽  
Affine Flag

scholarly journals Degenerate Affine Flag Varieties and Quiver Grassmannians

2020 ◽  
Author(s):  
Alexander Pütz
Keyword(s):  
Flag Varieties ◽  
Rational Singularities ◽  
Quiver Grassmannians ◽  
Finite Dimensional ◽  
Irreducible Components ◽  
Affine Flag

AbstractWe study finite dimensional approximations to degenerate versions of affine flag varieties using quiver Grassmannians for cyclic quivers. We prove that they admit cellular decompositions parametrized by affine Dellac configurations, and that their irreducible components are normal Cohen-Macaulay varieties with rational singularities.

scholarly journals Nearby cycles of Whittaker sheaves on Drinfeld’s compactification

2018 ◽  
Vol 154 (8) ◽  
pp. 1775-1800
Author(s):  
Justin Campbell
Keyword(s):  
Lie Algebra ◽  
Geometric Construction ◽  
Intersection Cohomology ◽  
Flag Varieties ◽  
Perverse Sheaves ◽  
Nilpotent Radical ◽  
Perverse Sheaf ◽  
Nearby Cycles

In this article we give a geometric construction of a tilting perverse sheaf on Drinfeld’s compactification, by applying the nearby cycles functor to a family of nondegenerate Whittaker sheaves. Its restrictions along the defect stratification are shown to be certain perverse sheaves attached to the nilpotent radical of the Langlands dual Lie algebra. We also describe the subquotients of the monodromy filtration using the Picard–Lefschetz oscillators introduced by Schieder. We give an argument that the subquotients are semisimple based on the action, constructed by Feigin, Finkelberg, Kuznetsov, and Mirković, of the Langlands dual Lie algebra on the global intersection cohomology of quasimaps into flag varieties.

Euler characteristic of certain affine flag varieties

1996 ◽  
Vol 1 (1-2) ◽  
pp. 35-39 ◽  
Author(s):  
C. Kenneth Fan
Keyword(s):  
Euler Characteristic ◽  
Flag Varieties ◽  
Affine Flag
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