The $\xi$-stability on the affine grassmannian

TL;DR

This paper introduces a new notion of $\xi$-stability on the affine Grassmannian for classical groups, constructs the stable quotient as an ind-scheme, and computes its Poincaré series for $ ext{SL}_d$, linking local stability to Higgs bundle moduli.

Contribution

It defines $\xi$-stability on the affine Grassmannian, proves the existence of the stable quotient as an ind-scheme, and provides explicit calculations for $ ext{SL}_d$.

Findings

The stable quotient $ ext{X}^\xi / T$ exists as an ind-scheme.

A reduction process analogous to Harder-Narasimhan is established.

The Poincaré series of the quotient for $ ext{SL}_d$ is explicitly computed.

Abstract

We introduce a notion of $\xi$-stability on the affine grassmannian $\xx$ for the classical groups, this is the local version of the $\xi$-stability on the moduli space of Higgs bundles on a curve introduced by Chaudouard and Laumon. We prove that the quotient $\xx^{\xi}/T$ of the stable part $\xx^{\xi}$ by the maximal torus $T$ exists as an ind-$k$-scheme, and we introduce a reduction process analogous to the Harder-Narasimhan reduction for vector bundles. For the group $\mathrm{SL}_{d}$, we calculate the Poincar\'e series of the quotient $\xx^{\xi}/T$.