Bruhat order on the involutions of classical Weyl groups

It is well known that a Coxeter group W, partially ordered by the Bruhat order, is a graded poset, with rank function given by the length, and that it is EL-shellable, hence Cohen–Macaulay, and Eulerian. We ask whether Invol(W), the subposet of W induced by the set of involutions, is endowed with similar properties. If W is of type A or B, we proved, respectively in [F. Incitti, The Bruhat order on the involutions of the symmetric group, J. Algebraic Combin. 20 (2004), 243–261] and [F. Incitti, The Bruhat order on the involutions of the hyperoctahedral group, European J. Combin. 24 (2003), 825–848], that Invol(W) is graded, EL-shellable and Eulerian. In this work we complete the investigation on the classical Weyl groups, extending these results to type D and providing a unified description for the rank function.