On the Combinatorics of B ×b-orbits on Group Compactifications

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 Ŵ (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 Ŵ .