On involutions in symmetric groups and a conjecture of Lusztig

Let $(W,S)$ be a Coxeter system equipped with a fixed automorphism $\ast$ of order $\le 2$ which preserves $S$. Lusztig (and with Vogan in some special cases) have shown that the space spanned by set of “twisted” involutions was naturally endowed with a module structure of the Hecke algebra of $(W,S)$. Lusztig has conjectured that this module is isomorphic to the right ideal of the Hecke algebra (with Hecke parameter $u^2$) associated to $(W,S)$ generated by the element $X_{\mathrm{\varnothing }}:={\sum }_{w^{\ast }=w}{u}^{-\mathrm{\ell }(w)}{T}_w$. In this paper we prove this conjecture in the case when $\ast =\text{id}$ and $W$ is the symmetric group on $n$ letters.