Period Domains over Finite and p -adic Fields

by Jean-François Dat , Sascha Orlik , Michael Rapoport

• ISBN13: 9780521197694
• ISBN10: 0521197694
• Edition: Revised
• Format: Hardcover
• Copyright: 2010-08-16
• Publisher: Cambridge University Press

Summary

This book is, on the one hand, a pedagogical introduction to the formalism of slopes, of semi-stability and of related concepts in the simplest possible context. It is therefore accessible to any graduate student with a basic knowledge in algebraic geometry and algebraic groups. On the other hand, the book also provides a thorough introduction to the basics of period domains, as they appear in the geometric approach to local Langlands correspondences and in the recent conjectural p-adic local Langlands program. The authors provide numerous worked examples and establish many connections to topics in the general area of algebraic groups over finite and local fields. In addition, the end of each section includes remarks on open questions, historical context and references to the literature.

Table of Contents

Preface	p. vii
Introduction	p. xi
Period Domains for GLn over Finite Fields
Filtered Vector Spaces	p. 3
Filtrations	p. 3
The tensor product theorem	p. 13
The Harder-Narasimhan filtration	p. 23
Period Domains for GLn	p. 31
Definition and examples	p. 31
The relation to GIT	p. 35
The Harder-Narasimhan stratification	p. 41
Period domains over F1 and this cells	p. 51
Cohomology of Period Domains for GLn	p. 65
The Langlands Lemma	p. 66
The generalized Steinberg representations	p. 72
The Euler-Poincaré characteristic	p. 81
Period Domains for Reductive Groups over Finite Fields
Interlude on the Tannakian Formalism	p. 105
Tensor categories	p. 105
Gradings and filtrations	p. 115
Filtrations on Repk(G)	p. 125
Slopes	p. 126
Semi-stability	p. 128
The Harder-Narasimhan filtration	p. 132
Period Domains for Reductive Groups	p. 137
Definitions	p. 137
The relation to GIT	p. 144
The Harder-Narasimhan stratification	p. 149
Cohomology of Period Domains for Reductive Groups	p. 161
The Langlands Lemma and generalized Steinberg representations	p. 161
The Euler-Poincaré characteristic	p. 170
Period Domains over p-Adic Fields
Period Domains over p-Adic Fields	p. 189
Filtered isocrystals	p. 189
Period domains for GLn	p. 195
The Harder-Narasimhan stratification	p. 205
The relation to GIT	p. 213
Isocrystals with coefficients	p. 215
Period Domains for p-Adic Reductive Groups	p. 221
C-isocrystals and group schemes over Isoc(L)	p. 221
Filtrations on ¿G	p. 239
Automorphism groups and decency	p. 251
Structure of X*(G)Q/G for a reductive a.g.s	p. 259
Period domains	p. 271
The Harder-Narasimhan stratification	p. 277
Relation to GIT	p. 292
G-isocrystals with coefficients	p. 293
Cohomology of Period Domains over p-Adic Fields	p. 303
Generalized Steinberg representations	p. 303
Compactly supported l-adic cohomology of strata	p. 306
The Euler-Poincaré characteristic in the basic case	p. 310
Complements
Further Aspects of Period Domains	p. 319
The fundamental complex	p. 319
The relation to Deligne-Lusztig varieties	p. 325
The Drinfeld space for a p-adic field	p. 334
Local systems on p-adic period domains	p. 339
The cohomology complex of p-adic period domains	p. 348
References	p. 358
Index	p. 369
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