On the unramified Eisenstein spectrum

TL;DR

This paper determines the spectrum of unramified Eisenstein series for split reductive groups over global fields, using advanced geometric and stack-theoretic methods involving Springer stacks and cobordisms.

Contribution

It introduces a novel geometric approach to analyze the unramified Eisenstein spectrum via Springer stacks and cobordism theory.

Findings

Explicit description of the unramified Eisenstein spectrum.

Connection between Eisenstein series and Springer stack decompositions.

New geometric tools for automorphic spectrum analysis.

Abstract

For a split reductive group ${}^L G$ over a global field, we determine the spectrum of the spherical Hecke algebra coming from the unramified Eisenstein series for the minimal parabolic ${}^L B$. This is done using a certain decomposition of the Springer stack $T^*(B \backslash G/ B)$ for the Langlands dual group in the additive group of cobordisms of cohomologically proper derived quotient stacks.