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

It is shown that there is an order isomorphism ′ 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 (in general neither finite nor affine). Moreover, ′ preserves the corresponding Kazhdan–Lusztig polynomials. Springerʹs (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 .