Schubert polynomials for the affine Grassmannian

Let G be a complex simply connected simple group and K a maximal compact subgroup. Let F = C((t)) denote the field of formal Laurent series and O = C[[t]] the ring of formal power series. The quotient Qr = G(F)/G(0) is known as the affine Grassmannian of G. It is known that Qr is homotopy equivalent to the space of based loops ?K into the compact group K (see [9, 30]). The multiplication on QK endows the homology H* (Qr) and cohomology H* (Qr) with structures of dual Hopf algebras. These Hopf algebras were first identified by Bott [4] (another descrip tion is given by Ginzburg [10], which we shall not use). The Bruhat decomposition of G(F) induces a stratification of Qr by Schubert cells ilw, indexed by Grassman nian elements w of the affine Weyl group WaffWe denote by aw G H*(Qr) and aw G H* (Qr) the Schubert classes in homology and cohomology. When G = SL(n,C), Bott identifies H*(Qr) and H*(Qr) with a subring An and a quotient An of the ring of symmetric functions. Our main theorem (Theorem 7.1) identifies the Schubert classes aw G ?* (Qr) and aw G H* (Qr) as explicit symmet ric functions under this isomorphism. These symmetric functions are combinatori ally defined: in homology the aw G H*(Qr) are represented by Lascoux-Lapointe Morse's fc-Schur functions Sw(x) G An and in cohomology the aw G H*(Qr) are represented by the dual fc-Schur functions (or affine Schur functions) Fw(x) G An; see [21, 24, 17]. Our theorem was originally conjectured by Mark Shimozono (the conjecture was made explicit in the cohomology case by Jennifer Morse). Kostant and Kumar [14] studied the topology of homogeneous spaces for arbi trary Kac-Moody groups and in particular calculated the structure constants of H* (Qr) in the Schubert basis, using the algebraic construction of the nilHecke ring. Our connection between the topology and the combinatorics proceeds via the study of three subalgebras of the nilHecke ring. The first subalgebra is the nilCoxeter al gebra Ao, which is the algebra generated by the "divided difference" operators. The second algebra is a certain centralizer algebra ^Aaff (S) which we call the Peterson subalgebra. Peterson [28] constructs an isomorphism j from the T-equivariant ho mology Hr(Qr) of the affine Grassmannian to Z&&ff(S). The third algebra B is a combinatorially defined subalgebra of A0, which we call the (affine) Fomin-Stanley