Upper and Lower Bounds for Kazhdan-Lusztig Polynomials

We give upper and lower bounds for the Kazhdan-Lusztig polynomials of any Coxeter group W. If W is nite we prove that, for any k 0, the k-th coeecient of the Kazhdan-Lusztig polynomial of two elements u, v of W is bounded from above and below by a polynomial (which depends only on k) in l(v)?l(u). In particular, this implies the validity of Lascoux-Schutzenberger's conjecture for all suuciently long intervals, and gives supporting evidence in favor of the Dyer-Lusztig conjecture.