Upper and Lower Bounds for Kazhdan–Lusztig Polynomials

We give upper and lower bounds for the Kazhdan–Lusztig polynomials of any Coxeter groupW. IfWis finite we prove that, for any k ≥ 0, thekth coefficient of the Kazhdan–Lusztig polynomial of two elementsu,vofWis bounded from above by a polynomial (which depends only onk) inl(v) − l(u). In particular, this implies the validity of Lascoux–Schutzenbergerʹs conjecture for all sufficiently long intervals, and gives supporting evidence in favour of the Dyer–Lusztig conjecture.