The Langlands formula and perverse sheaves

TL;DR

This paper constructs a categorical framework linking perverse sheaves on Weyl group quotients to automorphic form operations, providing a geometric interpretation of the Langlands formula for Eisenstein series.

Contribution

It introduces a new category that models perverse sheaves and automorphic operations, revealing a geometric perspective on the Langlands formula.

Findings

Categorical model for perverse sheaves on Weyl group quotients

Identification of Langlands constant term formula with categorical identities

Connection between invariant categories and spectra of invariant algebras

Abstract

For a complex reductive Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$ and Weyl group $W$ we consider the category $\text{Perv}(W \backslash \mathfrak{h})$ of perverse sheaves on $W \backslash \mathfrak{h}$ smooth w.r.t. the natural stratification. We construct a category $\boldsymbol{\mathcal{C}}$ such that $\text{Perv}(W\backslash \mathfrak{h})$ is identified with the category of functors from $\boldsymbol{\mathcal{C}}$ to vector spaces. Objects of $\boldsymbol{\mathcal{C}}$ are labelled by standard parabolic subalgebras in $\mathfrak{g}$. 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 $\boldsymbol{\mathcal{C}}$. We define $\boldsymbol{\mathcal{C}}$ as the category of $W$-invariants (in an appropriate sense) in the category $Q$ describing perverse sheaves on $\mathfrak{h}$ smooth w.r.t. the root arrangement. This matches, in an interesting way, the definition of $W \backslash \mathfrak{h}$ itself as the spectrum of the algebra of $W$-invariants.