Period Domains over Finite and p-adic Fields (Cambridge Tracts in Mathematics, Series Number 183)

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. Contents......Page 7 Preface......Page 9 Introduction......Page 13 Part 1 Period Domains for GLn over Finite Fields......Page 25 1 Filtrations......Page 27 2 The tensor product theorem......Page 37 3 The Harder–Narasimhan filtration......Page 47 1 Definition and examples......Page 55 2 The relation to GIT......Page 59 3 The Harder–Narasimhan stratification......Page 65 4 Period domains over F1 and thin cells......Page 75 III. Cohomology of Period Domains for GLn......Page 89 1 The Langlands Lemma......Page 90 2 The generalized Steinberg representations......Page 96 3 The Euler–Poincar characteristic......Page 105 Part 2 Period Domains for Reductive Groups over Finite Fields......Page 127 1 Tensor categories......Page 129 2 Gradings and filtrations......Page 139 V. Filtrations on Repk(G)......Page 149 1 Slopes......Page 150 2 Semi-stability......Page 152 3 The Harder–Narasimhan filtration......Page 156 1 Definitions......Page 161 2 The relation to GIT......Page 168 3 The Harder–Narasimhan stratification......Page 173 1 The Langlands Lemma and generalized Steinberg representations......Page 185 2 The Euler–Poincar characteristic......Page 194 Part 3 Period Domains over p-adic Fields......Page 211 1 Filtered isocrystals......Page 213 2 Period domains for GLn......Page 219 3 The Harder–Narasimhan stratification......Page 229 4 The relation to GIT......Page 237 5 Isocrystals with coefficients......Page 239 1 C-isocrystals and group schemes over Isoc(L)......Page 245 2 Filtrations on ωG......Page 263 3 Automorphism groups and decency......Page 275 4 Structure of X*(G)Q/G for a reductive a.g.s.......Page 283 5 Period domains......Page 295 6 The Harder–Narasimhan stratification......Page 301 7 Relation to GIT......Page 316 8 G-isocrystals with coefficients......Page 317 1 Generalized Steinberg representations......Page 327 2 Compactly supported l-adic cohomology of strata......Page 330 3 The Euler–Poincar characteristic in the basic case......Page 334 Part 4 Complements......Page 341 1 The fundamental complex......Page 343 2 The relation to Deligne–Lusztig varieties......Page 349 3 The Drinfeld space for a p-adic field......Page 358 4 Local systems on p-adic period domains......Page 363 5 The cohomology complex of p-adic period domains......Page 372 References......Page 382 Index......Page 393 This Systematic Exposition Of The Basics Of Period Domains Is A Pedagogical Introduction Accessible To Any Graduate Student With A Basic Knowledge In Algebraic Geometry And Algebraic Groups. The Authors Provide Numerous Worked Examples, Remarks On Open Questions, Historical Context And References To The Literature. Pt. 1. Period Domains For Gln Over Finite Fields -- Pt. 2. Period Domains For Reductive Groups Over Finite Fields -- Pt. 3. Period Domains Over P-adic Fields. Jean-françois Dat, Sascha Orlik, Michael Rapoport. Includes Bibliographical References (p. 358-368) And Index.