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-Schur~2 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(Dn). 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 Bn and type Dn 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 Bn length functions in the double coset it belongs to.
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