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In this paper, we study the decomposition of Bruhat intervals in a Coxetergroup with respect to cosets of a parabolic subgroup. Our main result is thatthe intersection of a lower Bruhat interval with a parabolic coset contains aunique maximal element. As an application, we give a decomposition formula forthe Poincar\'{e} polynomial of a Coxeter group element. We also show that thefibers of standard parabolic projection maps on Schubert varieties arethemselves Schubert varieties.

The parabolic coset structure of Bruhat intervals in Coxeter groups