The Langlands lemma and the Betti numbers of stacks of $G$--bundles on a curve

TL;DR

This paper demonstrates how the Langlands lemma can be applied to invert recursion relations for Poincaré series of semi-stable G-bundles on a curve, connecting Eisenstein series theory with geometric stack invariants.

Contribution

It introduces a novel application of the Langlands lemma to compute Betti numbers of stacks of G-bundles, extending previous recursive methods.

Findings

Inversion of recursion relations for Poincaré series using Langlands lemma.

Explicit computation of Betti numbers for stacks of G-bundles.

Bridging Eisenstein series theory with geometric invariants of moduli stacks.

Abstract

In this note we show that the Langlands lemma from the theory of Eisenstein series can be used to invert the recursion relation for the Poincar\'e series of the open substack of semi-stable $G$-bundles which was established by Atiyah/Bott and Harder/Narasimhan.