On the combinatorics of B× B-orbits on group compactifications

Abstract

It is shown that there is an order isomorphism ϕ′ from the poset V of B× B-orbits on the wonderful compactification of a semi-simple adjoint group G with Weyl group W to an interval in reverse Chevalley–Bruhat order on a non-canonically associated Coxeter group W (in general neither finite nor affine). Moreover, ϕ′ preserves the corresponding Kazhdan–Lusztig polynomials. Springer's (partly conjectural) construction of Kazhdan–Lusztig polynomials for the analogues of V for general Coxeter groups W is completed by reducing it by a similar order isomorphism to known results involving a “twisted” Chevalley–Bruhat order on W .