Quasi-parabolic subgroups of the Weyl group of type DD

We consider quasi-parabolic subgroups of the Weyl group W(Dn)W(Dn) of type DnDn, which are intersections of W(Dn)W(Dn) with quasi-parabolic subgroups of the Weyl group W(Bn)W(Bn) of type BnBn (see [J. Du, L. Scott, The qq-Schur22 algebra, Trans. Amer. Math. Soc. 352 (2000) 4325–4353] and [C.K. Mak, Quasi-parabolic subgroups of G(m,1,r)G(m,1,r), J. Algebra 246 (2001) 471–490]). We study the properties of cosets of these subgroups in W(Dn)W(Dn). A length function formula of type DnDn is derived. A complete set of right coset representatives of these subgroups is constructed. We show that each of these representatives is of minimum length (with respect to both type BnBn and type DnDn length functions) in the coset it belongs to. Characterizations of these representatives via certain tableaux are given. Finally, a complete set of double coset representatives of quasi-parabolic subgroups in W(Dn)W(Dn) is also obtained, and we show that each of these representatives is of minimum length with respect to type BnBn length functions in the double coset it belongs to.