A stability phenomenon in Kazhdan-Lusztig combinatorics

We prove that, when $n$ goes to infinity, the expression, with respect to the dual Kazhdan-Lusztig basis, of the product ${\widehat{\underset{‾}{H}}}_x{\underset{‾}{H}}_y$ of elements of the dual and the usual Kazhdan-Lusztig bases in the Hecke algebra of the symmetric group $S_n$ stabilizes. As an application, we define the action of projective functors on the principal block of category $\mathcal{O}$ for ${\mathfrak{s}\mathfrak{l}}_{\mathrm{\infty }}$ and show that the subcategory of finite length objects is stable under this action. As a bonus, we also prove that this latter block is Koszul, answering, for this block, a question from \cite{CP}.