Dimensions of quantized tilting modules

V. V. Ostrik

Massachusetts Institute of Technology

Abstract: 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.

Key words and phrases: Quantum groups at roots of unity, tilting modules, special representations of Weyl groups.

MSC: Primary 20G05; Secondary 17B37

Received: September 12, 2000; in revised form December 4, 2000

DOI: 10.17323/1609-4514-2001-1-1-65-71

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