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• W1553792534 abstract "This note is an expository account of the theory of staggered sheaves, based on a series of lectures given by the author at RIMS (Kyoto) in October 2008. Perverse sheaves have become a tool of great importance in representation the- ory, largely because of the remarkable way in which they provide a link between geometry and algebra. A single perverse sheaf contains information reminiscent of classical algebraic topology; indeed, the prototypical example of a perverse sheaf comes from the Goresky-MacPherson theory of intersection homology. But the category of perverse sheaves behaves like the module categories typically seen in representation theory, and some of the most important theorems here are Ext- vanishing and complete reducibility criteria. Staggered sheaves, the subject of the present note, were introduced by the author in (A1) and subsequently studied in a series of papers (AT1, AT2, A2) by the author and D. Treumann. The category of staggered sheaves also enjoys a long list of re- markable algebraic properties, resembling the most important properties of perverse sheaves. Most notably, the category of staggered sheaves is quasi-hereditary and exhibits purity and decomposition phenomena. But whereas perverse sheaves are built out of local systems, staggered sheaves are built out of vector bundles, and they live in the derived category of coherent sheaves. This note is a self-contained exposition of the main results on staggered sheaves. It is meant to be accessible to anyone familiar with the basic theory of algebraic groups and a passing acquaintance with derived categories. Some familiarity with perverse sheaves may be helpful for motivation, but the relevant facts will be re- viewed in Section 2. Most proofs will be omitted. No new results appear here. Notation and definitions occupy Section 3, and the main theorems are listed in Sections 4-6. An example on P 1 is worked out in Section 7, and some open questions are posed in Section 8. Appendix A surveys the theory of baric structures, required for the proofs of the main theorems, and Appendix B describes differences in terminology, notation, and conventions among the various papers. Acknowledgements. This paper is based on a series of lectures given by the author at RIMS (Kyoto) in October 2008. I am deeply grateful to Syu Kato, Susumu Ariki, and Kyo Nishiyama for welcoming me to Kyoto and for their warm and generous hospitality during my stay there. The work described here was carried out with support from NSF grant DMS-0500873. 1. Prologue: Staggered Representations In this section, we carry out a construction that is parallel to, and exhibits the main features of, that of staggered sheaves. Sheaves do not explicitly appear in this section, but the ideas in this section will be revisited later. 1" @default.
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• W1553792534 date "2008-01-01" @default.
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• W1553792534 title "INTRODUCTION TO STAGGERED SHEAVES" @default.
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