WARTHOG 2024

 Coherent-constructible equivalences in local Geometric Langlands and Representation Theory

Workshop on Algebra and Representation Theory, Held on Oregonian Grounds

22-26 July, 2024
University of Oregon
Eugene, OR

The main themes of this workshop will be the following:

  • Constructible sheaves and perverse sheaves on complex algebraic varieties
  • Coherent sheaves on schemes
  • Representation theory

    Expected background and suggested reading

    The background material that participants should study in preparation is divided into four sections below. Each section consists of required and bonus topics, with references.

    We will assume that all participants are familiar, but not experts, with the required topics. For required topics, the workshop will recall the needed facts and give examples, but it will be hard to follow unless you have played with these concepts a bit before. It is more important to know the definitions and statements of theorems, and to get a feeling for examples, rather than knowing the details of the proofs.

    Prior exposure to the bonus topics will be helpful, but we'll cover the bonus topics as needed during the workshop. They are listed to give more advanced participants something to work on before the summer school.

    It looks like four topics with imposing names, but actually the required topics are quite manageable. Part I and half of III are in many standard graduate algebra classes. Part IV is very short.

    I. Homological algebra.

    Required topics:

  • the notions of category and functor
  • additive and abelian categories
  • adjoint functors
  • complexes in an additive category, cohomology of a complex of objects in an abelian category

    This material is covered in Sections 1, 3.1-3.2, and 4.1-4.5 of the notes [Sc] by P. Schapira. Another option is Sections 1.1-1.3 of the book [KS] by Kashiwara-Schapira, but this is probably more terse than [Sc].

    Bonus topics:

  • derived categories
  • derived functors

    Some of the main ideas in the workshop can most naturally be expressed only in the derived setting, but we'll explain how to work in this setting during the workshop. Chapter 1 of [KS] gives a self-contained exposition.

    II. Sheaf theory.

    Required topics:

  • presheaves and sheaves on topological spaces
  • tensor product and internal Hom for sheaves of k-modules
  • Push/pull functors associated with continuous maps of topological spaces

    These topics are covered in pp. 83-103 of [KS]. (You can ignore statements involving the functors (-)_Z and \Gamma_Z or involving flabby sheaves.)

    Bonus topics:

  • derived versions of the topics above
  • constructible and perverse sheaves on complex algebraic varieties

    For derived sheaf functors, see section 2.6 of [KS]. One possible reference for constructible and perverse sheaves is Chapters 1-3 of Pramod's book [Ac]. (However, this book assumes familiarity with derived categories throughout.)

    III. Algebraic geometry.

    Required topics:

  • The beginnings of algebraic geometry: Algebraic sets and their coordinate rings, morphisms between algebraic sets/varieties, the relationship to algebras and algebra morphisms. Projective varieties.
  • Complex algebraic varieties, and quasi-coherent sheaves on them
  • Locally-free sheaves; sheaf of sections of a vector bundle

    The beginnings of algebraic geometry are covered well in chapter 1 of [E]. The book [Ar] by Arapura gives a friendly introduction to the remaining topics: see especially sections 2.1-2.4 and 3.5.

    Bonus topics:

  • varieties and schemes over algebraically closed fields, and quasi-coherent sheaves in this generality
  • functor-of-points perspective on schemes

    Schemes and quasi-coherent sheaves are covered in II.2 and II.5 of Hartshorne [Ha].

    IV. Algebraic group theory.

    Required topics:

  • How to view GL_n and other matrix groups as affine algebraic varieties
  • definition of a representation of an algebraic group

    One reference that Pramod likes for these topics is the book [MT] by Malle and Testerman. See especially sections 1.1-1.2 and 5.1-5.2. This book emphasizes examples but omits many proofs.

    Bonus topics:

  • Hopf algebra / comodule language for algebraic groups and representations.
  • structure theory of reductive groups
  • weight spaces of representations; highest-weight classification of irreducible representations

    For the Hopf algebra perspective, see Chap. 3-4 of the book [Mi] by Milne. Structure theory and weight spaces are covered in Chap. 8, 9, and 15 of [MT]. For structure theory, see also the lectures [Ma] by Makisumi.

    References

    References for background reading:

  • [Ac] P. Achar, Perverse Sheaves and Applications to Representation Theory, Mathematical Surveys and Monographs 258, AMS.
  • [Ar] D. Arapura, Algebraic Geometry over the Complex Numbers
  • [E] D. Eisenbud, Commutative algebra.
  • [Ha] R. Hartshorne, Algebraic Geometry, Graduate texts in mathematics 52, Springer.
  • [KS] M. Kashiwara, P. Schapira, Sheaves on manifolds, Grundlehren der mathematischen Wissenschaften 292, Springer.
  • [Ma] S. Makisumi, Structure Theory of Reductive Groups through Examples , here
  • [MT] G. Malle and D. Testerman, Linear Algebraic Groups and Finite Groups of Lie Type
  • [Mi] Milne, Algebraic Groups - The Theory of Group Schemes of Finite Type over a Field , Cambridge University Press.
  • [Sc] P. Schapira, Categories and Homological Algebra, here

    References for content of workshop:

  • [AB] S. Arkhipov and R. Bezrukavnikov, Perverse sheaves on affine flags and Langlands dual group (with an appendix by R. Bezrukavnikov and I. Mirkovic), Israel J. Math. 170 (2009), 135-183.
  • [B1] R. Bezrukavnikov, Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves, Invent. Math. 166 (2006), 327-357.
  • [B2] R. Bezrukavnikov, On two geometric realizations of an affine Hecke algebra , Publ. Math. Inst. Hautes Etudes Sci. 123 (2016), 1-67.
  • [MV] I. Mirkovic and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. 166 (2007), 95-143.