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

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

2002-10-20

arxiv.org/abs/math/0210311v1

Mathematics

Representation Theory

14 pages

Scientific paper

It is shown that there is an order isomorphism $\phi'$ 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 $\hat{W}$ (in general neither finite nor affine). Moreover, $\phi'$ 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 $\hat{W}$.

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