We prove that, when n goes to infinity, the expression, with respect to the dual Kazhdan-Lusztig basis, of the product \(\hat{\underline{H}}_x\underline{H}_y\) 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 \(\mathfrak {sl}_\infty \) 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 Math. J. 19(4), 655–693 (2019).
