Boundedness of Lusztig's $\mathbf{a}$-function for Coxeter groups of finite rank

Abstract

We prove that there exists a bound $N_L^{\prime }(W)$ for a positively weighted Coxeter group $(W,S,L)$ of finite rank. In particular, Lusztig's $\mathbf{a}$-function of $(W,S,L)$ is bounded.

Final version