Suppose $X$ is a prime Fano threefold of index 1 such that $H = -K\_X$ is ample. There is a full classification of the derived category of such threefolds depending on the genus of $X$; in the case that $g \geq 6$ we have
$$
D^b(X) = \langle \operatorname{Ku}(X), \mathcal{E}, \mathscr{O}\rangle
$$
where $\operatorname{Ku}(X) = \langle \mathcal{E}, \mathscr{O}\rangle^\perp$ and $\mathcal{E}$ is the pullback of the rank 2 tautological Grassmannian bundle..


Now suppose I take $\mathcal{D} = \langle \mathscr{O}\rangle^\perp = \langle \operatorname{Ku}(X), \mathcal{E}\rangle$ along with its respective inclusion functor
$$
i : \mathcal{D} \hookrightarrow D^b(X)
$$
Then $\mathcal{D}$ is an admissible subcategory, and in particular [1] shows that
$$
S\_\mathcal{D} = \mathbb{R}\_{\mathscr{O}(-H)} \circ S\_{D^b(X)}
$$
(where $\mathbb{R}$ denotes right mutation). I'm currently using the fact that an admissible subcategory with a Serre functor relates left and right adjoints via
$$
i^{!} = S\_{\mathcal{D}} \circ i^{\ast} \circ S^{-1}\_{D^b(X)}
$$
to attempt to compute $i^! \mathscr{O}$, which should simplify as


$$
i^! \mathscr{O} = \mathbb{R}\_{\langle \mathscr{O} \rangle} \circ S\_{D^b(X)} \circ \mathbb{L}\_{\langle \mathscr{O}\rangle } ( \mathscr{O}(H)[-3] )
$$


(since $\mathbb{L}\_{\langle \mathscr{O} \rangle } = i^\ast$ and $S^{-1}\_{D^b(X)} \mathscr{O} = \mathscr{O}(H)[-3]$).


However, its not exactly clear from the definition what $\mathbb{L}\_{\langle \mathscr{O}\rangle} \mathscr{O}(H)[-3]$ should be — first off it doesnt necessarily commute with shift since mutations aren't always exact equivalences. And taking it as the cone in the triangle
$$
\operatorname{RHom}(\mathscr{O}, \mathscr{O}(H)) \otimes \mathscr{O} \to \mathscr{O}(H) \to \mathbb{L}\_{\langle \mathscr{O} \rangle} \mathscr{O}(H)
$$
isn't any more enlightening. How should one reasonably compute $i^! \mathscr{O}$ in this scenario?


[1] Jacovskis, Liu, Zhang. Brill-Noether Theory for Kuznetsov Components and Refined Categorical Torelli Theorems for Index One Fano Threefolds, 2022



Right adjoint of subcollection of semi-orthogonal decomposition
First, mutations are exact, hence commute with shifts. Second,
$$
\mathrm{RHom}(\mathcal{O},\mathcal{O}(H)) = 
H^\bullet(X, \mathcal{O}(H)) = 
H^\bullet(\mathbb{P}^{g+1}, \mathcal{O}(H)),
$$
hence the evaluation morphism on $X$ is the restriction of the analogous evaluation morphism on $\mathbb{P}^{g+1}$, where its cone is isomorphic to $\Omega(H)[1]$. Thus,
$$
\mathbb{L}\_{\mathcal{O}}(\mathcal{O}(H)) \cong \Omega(H)[1]\vert\_X.
$$


