On involutions in symmetric groups and a conjecture of Lusztig

Let (W, S) be a Coxeter system equipped with a fixed automorphism * of order ≤2 which preserves S. Lusztig (and with Vogan in some special cases) have shown that the space spanned by set of "twisted" involutions (i.e., elements w∈W with w*=w-1) was naturally endowed with a module structure of the Hecke algebra of (W, S) with two distinguished bases, which can be viewed as twisted analogues of the well-known standard basis and Kazhdan-Lusztig basis. The transition matrix between these bases defines a family of polynomials Py,wσ which can be viewed as "twisted" analogues of the well-known Kazhdan-Lusztig polynomials of (W, S). Lusztig has conjectured that this module is isomorphic to the right ideal of the Hecke algebra (with Hecke parameter u²) associated to (W, S) generated by the element Xθ:=∑w*=wu-ℓ(w)Tw. In this paper we prove this conjecture in the case when *=id and W=Sn (the symmetric group on n letters). Our methods are expected to be generalised to all the other finite crystallographic Coxeter groups.