Affine realization of tilted Richardson varieties

Abstract

Richardson varieties provide geometric representatives for products of Schubert classes, with stratifications indexed by the subintervals of Bruhat intervals. We construct analogous geometric models for tilted Bruhat intervals for all split connected reductive groups. Our construction uses Richardson varieties in affine flag varieties and yields abstract tilted Richardson varieties whose isomorphism classes, together with their labeled stratifications, are independent of the choices involved. We obtain an explicit formula for the point-counting polynomials of their open parts in terms of increasing minimal weight paths, extending Dyer's formula for Kazhdan--Lusztig $R$-polynomials. Over the real numbers, their totally nonnegative parts are regular CW complexes homeomorphic to closed balls, which gives a natural geometric realization of the posets of tilted Bruhat subintervals in all Lie types. Over the complex numbers, we prove that their fundamental classes push forward under a continuous loop evaluation map to the Poincaré duals of the minimal-degree coefficients of quantum Schubert products.