The $$\xi $$ ξ -stability on the affine grassmannian

Abstract

We introduce a notion of \(\xi \)-stability on the affine grassmannian \({\fancyscript{X}}\) 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 \({\fancyscript{X}}^{\xi }/T\) of the stable part \({\fancyscript{X}}^{\xi }\) 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 over an algebraic curve. For the group \({\mathrm {SL}}_{d}\), we calculate the Poincaré series of the quotient \({\fancyscript{X}}^{\xi }/T\).