Truncated affine grassmannians and truncated affine Springer fibers for ${\mathrm{GL}}_3$

Abstract

We state a conjecture on how to construct affine pavings for cohomologically pure projective algebraic varieties, which admit an action of torus such that the fixed points and $1$-dimensional orbits are finite. Experiments on the affine grassmannian for ${\mathrm{GL}}_3$ under the guideline of this conjecture, together with the work of Berenstein-Fomin-Zelevinsky and Kamnitzer, have led to the conjecture that the truncated affine grassmannians for ${\mathrm{GL}}_{r+1}$ admit affine pavings. For the group ${\mathrm{GL}}_3$, we construct affine pavings for the truncated affine grassmannians, and we use it to study the affine Springer fibers. In particular, we find a family of truncated affine Springer fibers which are cohomologically pure in the sense of Deligne.

22 pages, 3 tikz pictures

References in corpus (2)

• Perverse sheaves on a Loop group and Langlands' duality
• Pureté des fibres de Springer affines pour $G{L}_4$

Cited by in corpus (1)

• On the fundamental domain of affine Springer fibers