Tensor ideals in the category of tilting modules

Abstract

We study the tensor category $\cQ$ of tilting modules over a quantum group $U_q$ with divided powers. The set $X_+$ of dominant weights is a union of closed alcoves $\oC_w$ numbered by the elements $w\in W^f$ of a certain subset of affine Weyl group $W$. G.Lusztig and N.Xi defined a partition of $W^f$ into canonical right cells and the right order $\le_R$ on the set of cells. For a cell $A\subset W^f$ we consider a full subcategory $\cQ_{<A}$ formed by direct sums of tilting modules $Q(\lambda)$ with highest weights $\lambda \in \bigcup_{w\in B<_RA} \oC_w$. We prove that $\cQ_{<A}$ is a tensor ideal in $\cQ$, generalizing H.Andersen's Theorem about the ideal of negligible modules which in our notations is nothing else then $\cQ_{<\{ e\}}$. The proof is an application of a recent result by W.Soergel who has computed the characters of tilting modules.