2024/12/02 by Mikhail Kapranov, Vadim Schechtman, Kapranov, Mikhail +5 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Geometry and complex manifolds #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2412.01638
For a complex reductive Lie algebra \mathfrakg with Cartan subalgebra \mathfrakh and Weyl group W we consider the category Perv(W \backslash \mathfrakh) of perverse sheaves on W \backslash \mathfrakh smooth w.r.t. the natural stratification. We construct a category \boldsymbolC such that Perv(W\backslash \mathfrakh) is identified with the category of functors from \boldsymbolC to vector spaces. Objects of \boldsymbolC are labelled by standard parabolic subalgebras in \mathfrakg. It has morphisms analogous to the operations of parabolic induction (Eisenstein series) and restriction (constant term) of automorphic forms. In particular, the Langlands formula for the constant term of an Eisenstein series has a counterpart in the form of an identity in \boldsymbolC. We define \boldsymbolC as the category of W-invariants (in an appropriate sense) in the category Q describing perverse sheaves on \mathfrakh smooth w.r.t. the root arrangement. This matches, in an interesting way, the definition of W \backslash \mathfrakh itself as the spectrum of the algebra of W-invariants.