  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.
