  We show that the higher Grothendieck-Witt groups, a.k.a. algebraic hermitian
K-groups, are represented by an infinite orthogonal Grassmannian in the
A1-homotopy category of smooth schemes over a regular base for which 2 is a
unit in the ring of regular functions. We also give geometric models for
various P1- and S1-loop spaces of hermitian K-theory.
