A GEOMETRIC REALIZATION OF THE CENTRE OF THE SMALL QUANTUM GROUP

Abstract We propose a new geometric model for the centre of the small quantum group using the cohomology of certain affine Springer fibres. More precisely, we establish an isomorphism between the equivariant cohomology of affine Spaltenstein fibres for a split element and the centre of the deformed graded modules for the small quantum group. We also obtain an embedding from the invariant part of the non-equivariant cohomology under the action of the extended affine Weyl group to the G ˇ ${\check G}$ ModifyingAbove upper G With ˇ -invariant part of the centre of the small quantum group, which we conjecture to be an isomorphism. Finally, we give a dimension formula for the invariants on the cohomology side, thus providing a lower bound for the dimension of the centre.