Standard conjectures for the arithmetic Grassmannian G(2,N) and Racah polynomials

Abstract

We prove the arithmetic Hodge index and hard Lefschetz conjectures for the Grassmannian $G=G(2,N)$ parametrizing lines in projective space, for the natural arithmetic Lefschetz operator defined via the Pl\"ucker embedding of $G$ in projective space. The analysis of the Hodge index inequalities involves estimates on certain Racah polynomials.