Let W be the Weyl group of G. In this paper we give an explicit realization of the group ring Z[W ] inside the ring of integer valued constructible functions on Z (with multiplication given by convolution). In this realization, Z[W] comes equipped, in addition to its standard basis, with a remarkable Z-basis ( f w ) w # W which consists of certain explicitly defined (and probably uncomputable) constructible functions on Z. It is likely that this basis coincides with the basis of Z[W ] defined in [KL2] in terms of Springer's representations, hence also with the basis defined in [KT] in terms of multiplicities of characteristic varieties of G-equivariant perverse sheaves on B_B. Our definition also makes sense in the case where W is replaced by an affine Weyl group.
The definition of the functions f w is inspired by the definition [L1, 4.16] of a basis of a certain universal enveloping algebra, given by constructible functions on Lagrangian varieties attached to a quiver. 0.2. Notation. We identify W in the standard way with the set of G-orbits on B_B (diagonal action); let O w be the orbit corresponding to w # W. We have a partition Z= w # W Z w where Z
be the standard partial order on W. Let l: W ร N be the length function; thus, l(w)=dim O w &dim B.