Let $S$ be a surface, and $S^{[n]}$ the Hilbert scheme of $n$ points on $S$. One defines $$ \mathbb H = \bigoplus_n H^*(S^{[n]}). $$ One has Nakajima's operators $\mathfrak q_n(\alpha)$ (with $\alpha \in H^*(S)$, $n \in \mathbb Z$) and their derivatives $\mathfrak q^{(\nu)}_n(\alpha)$ which act on $\mathbb H$.
For a sheaf $F$ one also has operators $\mathfrak ch(F)$ and $\mathfrak c(F)$ that act on $x \in H^*(S^{[n]}) \subset \mathbb H$ by multiplication by $ch(F^{[n]})$ or $c(F^{[n]})$, respectively (where $F^{[n]}$ is a certain sheaf on $X^{[n]}$ constructed from $F$).
In Lehn's paper, he gives formulas for the commutators of the $\mathfrak q$'s with themselves and also the formula (Theorem 4.2) $$ \mathfrak c(F) \cdot \mathfrak q_1(\alpha) \cdot \mathfrak c(F)^{-1} = \sum_{k,\nu \ge 0} \binom{r-k}{\nu} q_1^{(\nu)}(c_k(F)\alpha) $$
It seems reasonable (although now I can't remember if I have seen this written somewhere before or not) to define $\mathfrak{td}$ to be the operator that acts on $x \in H^*(S^{[n]})$ as multiplication by the Todd class of $X^{[n]}$.
QUESTION Is there a formula for $\mathfrak{td}\cdot \mathfrak q_1(\alpha)\cdot \mathfrak{td}^{-1}$, analogous to the formula above?
I hoped to be able to imitate Lehn's proof, plugging in an inductive formula for the tangent bundle of $X^{[n]}$ like Prop. 2.3 of this paper, but so far it isn't working for me.
It does seem like a question that may have been answered already, but I haven't found anything yet.
