Quasi-parabolic subgroups of the Weyl group of type D

We consider quasi-parabolic subgroups of the Weyl group W (D_n) of type D_n, which are intersections of W (D_n) with quasi-parabolic subgroups of the Weyl group W (B_n) of type B_n (see [J. Du, L. Scott, The q-Schur² algebra, Trans. Amer. Math. Soc. 352 (2000) 4325-4353] and [C.K. Mak, Quasi-parabolic subgroups of G (m, 1, r), J. Algebra 246 (2001) 471-490]). We study the properties of cosets of these subgroups in W (D_n). A length function formula of type D_n 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 B_n and type D_n 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 (D_n) is also obtained, and we show that each of these representatives is of minimum length with respect to type B_n length functions in the double coset it belongs to.