Towards a stable splitting of GL_n in A1-homotopy theory

We give a brief overview of work of H. Miller and T. Frankel which shows that there is a filtration on the Lie group G(n) (where G(n) is one of O(n), U(n), or Sp(n)), whose strata are certain vector bundles over Grassmannians. It can be shown that this filtration splits stably, so that the group G(n) stably decomposes as a wedge sum of Thom spaces associated to these vector bundles over Grassmannians.  We then give an account of ongoing work to extend these results to the setting of motivic homotopy theory. To this end, we describe the automorphism bundle of the tautological vector bundle on the Grassmannians of k-planes in n-space, A(k,n), and construct a map A(k,n) -> GL_n with image the k-th stage of the filtration.