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Kazhdan-Lusztig basis elements appearing in product with distinguished involution

My apologies if the below is too malformed to make sense. Let $(W,S)$ be the affine Weyl group of a reductive group $G$, and let $\{C_w\}$ be the Kazhdan-Lusztig $C$-basis (an answer in terms of the $...
Stefan  Dawydiak's user avatar