Quasi-parabolic subgroups of the Weyl group of type

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 ( 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.