Residue translation on small parabolic category-O standards

The moves. Apply the Levi tensor decomposition to each one-box column and add the two valid positions: tensoring with the natural module moves an entry $0$ to $1$ in either position, giving $f_0{M}_{00}=M_{01}+M_{10}$; there is no entry $0$ in $M_{11}$, giving $f_0{M}_{11}=0$; and the unique entry $0$ in either mixed monomial changes to $1$, giving $f_0{M}_{01}=f_0{M}_{10}=M_{11}$. The dual functor $e_0$ is computed verbatim with the dual natural module and gives the displayed reverse moves.

The commutator. In the ordered basis $(M_{00},M_{01},M_{10},M_{11})$ the matrices of $f_0$ and $e_0$ are $f_0=\left(\begin{array}{cccc}0& 0& 0& 0\\ 1& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 1& 1& 0\end{array}\right),e_0=\left(\begin{array}{cccc}0& 1& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right),$ and direct multiplication gives $[e_0,f_0]=\mathrm{diag}(2,0,0,-2)$, the ${\mathfrak{s}\mathfrak{l}}_2$-Cartan action on the Fock basis.

The actual simple classes. Here $\mathfrak{g}={\mathfrak{g}\mathfrak{l}}_2$ and the parabolic is the Borel. The tuple formula gives highest weights $(0,1),(0,2),(1,1),(1,2)$ for $00,01,10,11$, with root highest weights $-1,-2,0,-1$. On a Verma vector of root highest weight $m$, induction using [F3] gives $e{f}^{k}v=k(m-k+1)f^{k-1}v$. For $m=-1,-2$ every coefficient for $k>0$ is nonzero, so a nonzero weight submodule contains $v$ and the Verma is simple. For $m=0$, the span of $f^{k}v$, $k\ge 1$, is the simple Verma of highest weight $(0,2)$, and the quotient is the one-dimensional simple of weight $(1,1)$. Thus $L_{00}=M_{00}$, $L_{01}=M_{01}$, $[L_{10}]=[M_{10}]-[M_{01}]$ and $L_{11}=M_{11}$. Both singular endpoint blocks have one label, so BGG reciprocity makes $L_{00}=P_{00}$ and $L_{11}=P_{11}$ projective-injective. In the mixed block reciprocity gives $[P_{01}]=[M_{01}]+[M_{10}]$ and $[P_{10}]=[M_{10}]$.

Crystal and head/socle comparison. The moves in step 1.1 give $[f_0{P}_{00}]=[e_0{P}_{11}]=[P_{01}]$; these objects are projective, so finite decomposition into projective covers and independence of their classes identify both with $P_{01}$. They are self-dual because the endpoint simples are self-dual and the residue functors commute with duality. Hence they have simple head and socle $L_{01}$ and class $2[L_{01}]+[L_{10}]$. Also $f_0{L}_{01}=L_{11}$ and $e_0{L}_{01}=L_{00}$ by the standard moves, while exactness gives $[f_0{L}_{10}]=[e_0{L}_{10}]=0$, so these two translates vanish. The endpoint outward translates vanish as well. Independently, the source signature cancels $-+$ pairs, leaving the component $00\to 01\to 11$ and isolated $10$. These direct rank-one categorical computations establish precisely its head, socle and vanishing labels without using a general tensor-to-simple crystal identification.

Conclusion. On the four standard monomials the residue functors act by the displayed moves, their commutator is $\mathrm{diag}(2,0,0,-2)$, the source crystal is $00\to 01\to 11$ with isolated $10$, and the categorical translates have the corresponding unique heads and socles; all computations use the actual generalized standards and exact functors.