Dimensions of quantized tilting modules

Let $U$ be the quantum group with divided powers at $p$-th root of unity for prime $p$. To any two-sided cell $A$ in the corresponding affine Weyl group, one associates the tensor ideal in the category of tilting modules over $U$. In this note we show that for any cell $A$ there exists a tilting module $T$ from the corresponding tensor ideal such that the greatest power of $p$ which divides $\dim T$ is $p^{a(A)}$, where $a(A)$ is Lusztig's $a$-function. This result is motivated by a conjecture of J. Humphreys.