[Paper Review] Higgs bundles and indecomposable parabolic bundles over the projective line

Emmanuel Letellier|arXiv (Cornell University)|Sep 15, 2016

Algebraic Geometry and Number Theory18 references3 citations

TL;DR

This paper establishes a cohomological conjecture linking the count of geometrically indecomposable parabolic bundles over finite fields to the Hodge theory of moduli spaces of meromorphic Higgs bundles on the projective line. Using Hall-Littlewood symmetric functions and Fourier transforms of Deligne-Lusztig characters, it proves the conjecture in the rank 2 case and for uniform vector bundles, showing that the number of such bundles matches the weighted trace of Frobenius on cohomology of the Higgs moduli space.

ABSTRACT

In this paper we count the number of isomorphism classes of geometrically indecomposable quasi-parabolic structures of a given type on a given vector bundle on the projective line over a finite field. We give a conjectural cohomological interpretation for this counting using the moduli space of Higgs fields on the given vector bundle over the complex projective line with prescribed residues. We prove a certain number of results which bring evidences to the main conjecture. We detail the case of rank 2 vector bundles.

Motivation & Objective

• To count isomorphism classes of geometrically indecomposable parabolic structures on vector bundles over the projective line over finite fields.
• To provide a cohomological interpretation of this count using the moduli space of Higgs bundles with prescribed residues over the complex projective line.
• To prove the conjectured correspondence between character sums from finite group representation theory and the number of indecomposable parabolic bundles.
• To verify the conjecture in the rank 2 case and for uniform bundles via explicit computation and character formulae.
• To establish a link between the geometry of Higgs moduli spaces and arithmetic counts over finite fields through Fourier transforms of Deligne-Lusztig characters.

Proposed method

• Uses Hall-Littlewood symmetric functions to express the number of indecomposable parabolic bundles as a generating function.
• Applies Harish-Chandra induction and character formulas to relate counts over finite fields to representations of GL_n(F_q).
• Employs Fourier transforms of Deligne-Lusztig characters to connect finite field counts to cohomology of Higgs moduli spaces.
• Introduces a Lie algebra version of Deligne-Lusztig characters and uses commutation relations with Fourier transforms.
• Applies the formula for the coefficient of Y^m in the logarithm of the generating function to extract the number of bundles via character traces.
• Proves the main conjecture in special cases by showing that the trace of Frobenius on cohomology matches the arithmetic count.

Experimental results

Research questions

• RQ1 How can the number of isomorphism classes of geometrically indecomposable parabolic bundles over a finite field be computed?
• RQ2 What is the cohomological meaning of this count in terms of the moduli space of Higgs bundles with prescribed residues?
• RQ3 Can the conjectural link between character sums and arithmetic counts be proven in specific cases?
• RQ4 How do Hall-Littlewood symmetric functions and Fourier transforms of Deligne-Lusztig characters encode the geometry of the moduli space?
• RQ5 Is the conjecture true for rank 2 bundles and uniform vector bundles?

Key findings

• The number of geometrically indecomposable parabolic bundles of type s on a vector bundle E over P^1 over F_q is given by a formula involving Hall-Littlewood symmetric functions.
• For the rank 2 case, explicit formulas are derived for the cases a = b and a > b, showing dependence on the partition type of the weights.
• The conjectural cohomological interpretation is proven in the case where E = O(a)^n, linking the count to the trace of Frobenius on the cohomology of the Higgs moduli space.
• The formula for the coefficient of Y^m in the logarithm of the generating function Ω_D_w(q) matches the character sum involving ε_λ and the size of the set X_{S_λ, D_w}^E.
• The main conjecture is verified in the case of uniform bundles and for trivial w, using a combination of character theory and cohomological trace formulas.
• The proof relies on a Lie algebra version of Deligne-Lusztig characters and a commutation formula between Fourier transforms and induction, extending results from Kazhdan and Springer.