Kavli Affiliate: Mikhail Kapranov
| Summary:
For a complex reductive Lie algebra $mathfrakg$ with Cartan subalgebra $mathfrakh$ and Weyl group $W$ we consider the category $textPerv(W backslash mathfrakh)$ of perverse sheaves on $W backslash mathfrakh$ smooth w.r.t. the natural stratification. We construct a category $boldsymbolmathcalC$ such that $textPerv(Wbackslash mathfrakh)$ is identified with the category of functors from $boldsymbolmathcalC$ to vector spaces. Objects of $boldsymbolmathcalC$ 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 $boldsymbolmathcalC$. We define $boldsymbolmathcalC$ 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.
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