Tangent cones to Schubert varieties for Kac--Moody groups

Let $G$ be the affine Kac–Moody group of type ${\tilde{A}}_{n-1}$, $B$ be an Iwahori subgroup in $G$, $\mathcal{F}=G\mathrm{/}B$ be the flag variety, and $W$ be the Weyl group of $G$. Given distinct involutions $w_1$, $w_2\in W$, we prove that the tangent cones $C_{w_1}$, $C_{w_2}$ to the corresponding Schubert subvarieties $X_{w_1}$ and $X_{w_2}$ of $\mathcal{F}$ at the point $p=e\mathrm{m}\mathrm{o}\mathrm{d}B$ do not coincide as subvarieties of the tangent space to $\mathcal{F}$ at the point $p$. This generalizes similar results in the finite-dimensional setting. The main technical tools we used are combinatorics of the embeddings of the Weyl groups of different ranks and coadjoint orbits for the unipotent radical of the group $B$.