Program:

• Sunday: Arrival
• Monday: State of the art before localization
  • Semisimple complex Lie algebras, root systems, Weyl groups, Borel subalgebras, universal enveloping algebras, Harish-Chandra isomorphism, finite dimensional representations [10].
  • Representations of compact groups. Irreducible implies finite dimensional. Continuous representations of semisimple Lie groups in Banach spaces. Unitarity, irreducibility and admissibility. Harish-Chandra modules [12], [13], [20].
  • Classification of simple Harish-Chandra modules following Langlands. Discrete series, tempered representations [13], [20].
  • Harish-Chandra modules for complex semisimple Lie groups and highest-weight modules [8].
• Tuesday: D-modules and localization
  • D-modules on smooth varieties. Inverse and direct images. Coherence, characteristic variety, Bernstein's theorem on the dimension of the characteristic variety [2], [7].
  • Holonomic modules. Preservation of holonomicity under inverse and direct images. Duality. Classification of irreducible holonomic modules [2], [7].
  • Panorama: D-modules with regular singularities, Riemann-Hilbert correspondence, perverse sheaves, intersection cohomology, decomposition theorem [2], [5], [7].
  • The localization theorem. Twisted sheaves of differential operators. Borel-Weil theorem [3], [6], [14], [15].
• Wednesday: Applications to Harish-Chandra modules
  • Harish-Chandra sheaves. Localization of Harish-Chandra modules. Harish-Chandra sheaves are holonomic. Classification of irreducible Harish-Chandra sheaves. Standard Harish-Chandra sheaves Example: Localization of category of highest weight modules [14], [15], [16].
  • Geometric classification of Harish-Chandra modules. K-orbits in flag varieties. Cohomology of standard Harish-Chandra sheaves Connection of geometric classification with the Langlands classification. Examples: SL(2,R), SL(2,C) and SU(2,1) [14], [15], [16].
• Thursday: Detailed study of the category of highest weight modules
  • The Kazhdan-Lusztig conjectures and their proof [18].
  • The Jantzen conjectures and their proof [4].
  • The selfduality theorem and its proof [17].
  • Various interpretations KL-polynomials and the formalism of Koszul duality [9].
• Friday: Things to dream about
  • Vogan's character duality [1]
  • The p-adic case [19].