Nilpotency indices for quantum Schubert cell algebras

We study quantum analogs of $\operatorname{ad}$-nilpotency and Engel identities in quantum Schubert cell algebras ${\mathcal U}_q^+[w]$. For each pair of Lusztig root vectors, $X_\mu$ and $X_\lambda$, in ${\mathcal U}_q^+[w]$, where $w$ belongs to a finite Weyl group $W$ and $\mu$ precedes $\lambda$ with respect to a convex order on the roots in $\Delta_+ \cap w(\Delta_-)$, we find the smallest natural number $k$, called the nilpotency index, so that $\left(\operatorname{ad}_q X_\mu \right)^k$ sends $X_\lambda$ to $0$, where $\operatorname{ad}_q X_\mu$ is the $q$-adjoint map. We start by observing that every pair of Lusztig root vectors can be naturally associated to a triple $(u, i, j)$, where $u \in W$ and $i$ and $j$ are indices such that $\ell(s_i u s_j) = \ell(u) - 2$. In light of this, we define an equivalence relation, based upon the weak left and weak right Bruhat orders, on the set of such triples. We show this equivalence relation respects nilpotency indices, and that each equivalence class contains an element of the form $(v, r, s)$, where $v$ is either (1) a bigrassmannian element satisfying a certain orthogonality condition, or (2) the longest element of some subgroup of $W$ generated by two simple reflections. For each such $(v, r, s)$, we compute the associated nilpotency index.