The space L2 on semi-infinite Grassmannian over finite field

We construct a GL-invariant measure on a semi-infinite Grassmannian over a finite field, describe the natural group of symmetries of this measure, and decompose the space L2 over the Grassmannian on irreducible representations. The spectrum is discrete, spherical functions on the Grassmannian are given in terms of the Al Salam-Carlitz orthogonal polynomials. We also construct an invariant measure on the corresponding space of flags.