The wall-crossing functor on Verma modules in rank one

Assume the Axiom of Choice. Work in ${\mathfrak{s}\mathfrak{l}}_2$ with the conventions of The classical BGG category O: $\lambda =0$, $\rho =1$, so the regular standards are ${\mathrm{\Delta }}_1=M(0)$ and ${\mathrm{\Delta }}_s=M(-2)$ for the two elements of $W=\{1,s\}$, and the singular wall weight is $\mu =-1$, whose shifted value is $\mu +\rho =0$. The dominant regular integral block is that of $\lambda =0$ (The regular integral block and its antidominant weight), and the wall-crossing functor is that of The wall-crossing functor and its unit and counit built from the two-dimensional translating module $E=L(1)$ with highest weight $\mu -\lambda =-1$ up to Weyl conjugacy.

Onto the wall. Tensoring the two regular standards with the two-dimensional module and projecting to the central character of the wall preserves the unique wall standard: $T_{\mathrm{o}\mathrm{n}}{\mathrm{\Delta }}_1\cong M(-1)$ and $T_{\mathrm{o}\mathrm{n}}{\mathrm{\Delta }}_s\cong M(-1)$, since ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_s$ differ by the single simple-root lowering and the tensor filtration retains exactly the labels of weight $-1$ in the shifted parametrization, as in Translation to and from a single wall on standard modules.

Back out of the wall. The wall block at $\mu =-1$ has a single simple object $M(-1)=L(-1)$ and is semisimple; hence $M(-1)$ is projective and injective there, and the exact biadjoint return translation $T_{\mathrm{o}\mathrm{u}\mathrm{t}}$ sends it to a projective object of the regular block whose Verma flag has the two factors $M(0)$ and $M(-2)$, once each, by Translation to and from a single wall on standard modules. By biadjunction, its Hom to $L(-2)$ is one-dimensional, since $T_{\mathrm{o}\mathrm{n}}L(-2)=M(-1)$; its Hom to $L(0)$ is zero, since $L(1)\otimes L(0)=L(1)$ has a different central character from $M(-1)$. Thus its only simple quotient is $L(-2)$, once, and the projective-cover decomposition identifies it with $P(-2)$.

Conclusion. Both ${\theta }_s{\mathrm{\Delta }}_1$ and ${\theta }_s{\mathrm{\Delta }}_s$ are therefore isomorphic to the same nonsplit two-step extension $0⟶M(0)⟶P(-2)⟶M(-2)⟶0,$ with class $[{\mathrm{\Delta }}_1]+[{\mathrm{\Delta }}_s]$ in the Grothendieck group, in agreement with Wall-crossing acts on standard modules by a two-step extension. Maximal-weight peeling puts $M(0)$ first in the flag; the extension is nonsplit because $P(-2)$ is indecomposable as a projective cover. The wall sits at highest weight $-1$, equivalently at shifted parameter $\mu +\rho =0$; the regular highest weight $0$ is the dominant maximum of the block; the antidominant minimum is $-2$.