We prove the arithmetic Hodge index and hard Lefschetz conjectures for the Grassmannian G(2,N) parametrizing lines in projective space, for the natural arithmetic Lefschetz operator defined via the Plücker embedding of G in projective space. The analysis of the Hodge index inequality involves estimates on values of certain Racah polynomials.
"Standard conjectures for the arithmetic Grassmannian G(2,N) and Racah polynomials." Duke Math. J. 110 (2) 359 - 376, 1 November 2001. https://doi.org/10.1215/S0012-7094-01-11027-2