Let $X:=X_{18}$ be an index one smooth prime Fano threefold of degree 18.
Consider its semi-orthogonal decomposition: $D^b(X)=\langle\mathcal{O}_X,\mathcal{E}^{\vee},\mathcal{A}_X\rangle=\langle\mathcal{Q}^{\vee},\mathcal{O}_X,\mathcal{A}_X\rangle$, where $\mathcal{E},\mathcal{Q}$ are tautological sub and quotient bundle on $X$ coming from Grassmannian $\mathrm{Gr}(2, 7)$. Note that $\mathcal{A}_X\cong ^{\perp}\langle Q^{\vee},\mathcal{O}_X\rangle$ or $\mathcal{A}_X\cong ^{\perp}\langle \mathcal{O}_X, \mathcal{E}^{\vee}\rangle$. It is known that $\mathcal{A}_X\cong D^b(C_2)$ where $C_2$ is a smooth genus 2 curve(hyperelliptic curve). It is also known that the group of auto-equivalences of $D^b(C_2)$ is generated by $Aut(C_2), [1], -\otimes\mathcal{L}$(automorphism of the curve, shift functor and tensoring with line bundles).
There is a natural involution $\tau\in Aut(C_2)$(hyperelliptic involution) inducing an auto-equivalence on $D^b(C_2)$(still denoted by $\tau$). My question: Is there any way to write the auto-equivalence $\tau:\mathcal{A}_X\rightarrow\mathcal{A}_X$ purely in terms of composition of functors associated to the objects in $D^b(X)$?
For example, if $X:=X_{10}$ is a special Gushel-Mukai threefold, its semi-orthogonal decomposition is given as $D^b(X)=\langle \mathcal{B}_X, \mathcal{E},\mathcal{O}_X\rangle$, where $\mathcal{B}_X\cong\langle\mathcal{E},\mathcal{O}_X\rangle^{\perp}$, the geometric involution $\tau$ on $X$ gives an auto-equivalence of $\mathcal{B}_X$ and one can write $\tau^{-1}$ as $\mathrm{L}_{\mathcal{E}}\circ\mathrm{L}_{\mathcal{O}_X}\circ(-\otimes\mathcal{O}_X(H))[-1]$.
