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 $L^2$ 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.