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

Gérard Laumon, Michael Rapoport|ArXiv.org|Mar 14, 1995

Advanced Algebra and Geometry3 citations

TL;DR

This paper applies the Langlands lemma from Eisenstein series theory to invert a recursion relation for the Poincaré series of semi-stable $G$-bundles on a curve, enabling explicit computation of the Betti numbers of the stack of $G$-bundles. The key contribution is a new method to determine the Betti numbers via inversion of the Atiyah-Bott and Harder-Narasimhan recursion using Langlands' lemma.

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 Poincaré series of the open substack of semi-stable $G$-bundles which was established by Atiyah/Bott and Harder/Narasimhan.

Motivation & Objective

• To resolve the recursion relation for the Poincaré series of semi-stable $G$-bundles on a curve.
• To provide a method for computing the Betti numbers of the stack of $G$-bundles on a curve.
• To apply the Langlands lemma from Eisenstein series theory to invert the Atiyah-Bott and Harder-Narasimhan recursion.
• To establish a connection between automorphic forms and the topology of moduli stacks of $G$-bundles.

Proposed method

• Utilizes the Langlands lemma, a result from the theory of Eisenstein series, to analyze the structure of the Poincaré series.
• Applies the lemma to invert the recursion relation previously derived by Atiyah and Bott and by Harder and Narasimhan.
• Employs techniques from algebraic geometry and representation theory to relate the cohomology of $G$-bundle stacks to automorphic data.
• Uses the structure of the stack of semi-stable $G$-bundles to derive the Poincaré series via recursive relations.
• Applies the inversion technique to obtain the full Poincaré series of the stack of all $G$-bundles.
• Relies on the theory of reductive groups and geometric invariant theory to define and analyze the stack of $G$-bundles.

Experimental results

Research questions

• RQ1 How can the Langlands lemma be applied to invert the recursion for the Poincaré series of semi-stable $G$-bundles on a curve?
• RQ2 What is the explicit form of the Betti numbers of the stack of $G$-bundles on a curve?
• RQ3 Can the recursion relation from Atiyah-Bott and Harder-Narasimhan be inverted using automorphic methods?
• RQ4 What topological invariants of the stack of $G$-bundles are determined by this inversion process?
• RQ5 How does the Langlands lemma facilitate the computation of Betti numbers in the context of algebraic stacks?

Key findings

• The Langlands lemma successfully inverts the recursion relation for the Poincaré series of semi-stable $G$-bundles on a curve.
• The method yields an explicit formula for the Poincaré series of the stack of all $G$-bundles on a curve.
• The Betti numbers of the stack of $G$-bundles are determined through this inversion process.
• The approach establishes a direct link between automorphic forms (via Langlands lemma) and the cohomology of moduli stacks.
• The result provides a new topological invariant computation for $G$-bundle stacks using tools from Eisenstein series theory.
• The method generalizes and refines earlier results by Atiyah-Bott and Harder-Narasimhan by providing a closed-form solution.