On the combinatorics of $B\times B$-orbits on group compactifications

It is shown that there is an order isomorphism ${\varphi }^{\mathrm{\prime }}$ from the poset $V$ of $B\times B$-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 $\widehat{W}$ (in general neither finite nor affine). Moreover, ${\varphi }^{\mathrm{\prime }}$ 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 $\widehat{W}$.