Translation through the sl2 wall

Given: The Axiom of Choice, $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2$ with the coordinate $l=⟨\lambda ,{\alpha }^{\vee }⟩$, the wall $l=-1$, the single-wall datum $(\lambda ,\mu )=(-2,-1)$ with translating weight $\nu =1$ and wall reflection $s$, the regular integral block $C$ with simple labels $0,-2$ and its projective objects $P(0),P(-2)$, the wall block ${\mathcal{O}}_{{\chi }_{-1}}$, and $Q=T_{-1}^0\mathrm{\Delta }(-1)$.

For ${\mathfrak{s}\mathfrak{l}}_2$ the pair $(-2,-1)$ is a single-wall translation datum with $\lambda +\rho =-1$ spanning the negative chamber, $\mu +\rho =0$ on the wall, dot-stabilizer $\{1,s\}$ of $\mu$, wall reflection $s$ acting by $s\cdot l=-l-2$, and translating weight $\nu =1$; the reverse pair $(0,-1)$ realizes the same central characters and the same functors, and $T_0^{-1}=T_{-2}^{-1}$, $T_{-1}^0=T_{-1}^{-2}$ (Dot-Weyl facets and single-wall translation data, Translation functors by tensoring and projection).

Two weights have the same central character exactly when they lie in one dot orbit, and $s\cdot (-2)=0$, $s\cdot (-1)=-1$; in particular ${\chi }_0={\chi }_{-2}$ and the dot orbit of $-1$ is $\{-1\}$ (Central characters are dot-Weyl orbits, Dot-Weyl facets and single-wall translation data).

The rank-one model of the parent example has basis $v_k=f^k{v}_0$ with $e\cdot v_k=k(\lambda (h)-k+1)v_{k-1}$; with $\lambda (h)=-1$ this is $e\cdot v_k=-k^2{v}_{k-1}\mathrm{\ne }0$ for all $k\ge 1$, so the only singular vector of $M(-1)$ is its top; every nonzero submodule of a Verma module contains a singular vector, so $M(-1)$ is simple and $\mathrm{\Delta }(-1)=M(-1)=L(-1)$ (The two projectives in the principal sl2 block, Every nonzero Verma submodule contains a singular vector, Finite Verma flags and their multiplicities).

The blocks are the full subcategories of objects whose simple composition factors have labels in one linkage class, and the block of ${\chi }_{-1}$ is the full subcategory of objects whose simple composition factors are $L(\eta )$ with ${\chi }_{\eta }={\chi }_{-1}$; by [F2] these are exactly the objects with all composition factors $L(-1)$, that is, the truncation ${\mathcal{O}}_{\mathrm{\Gamma }}$ at the one-element finite downward-closed ideal $\mathrm{\Gamma }=\{-1\}$, in which $-1$ is maximal (Central-character summands refine into linkage blocks, Truncation at a finite downward-closed ideal of a linkage class, Central characters are dot-Weyl orbits).

Let $\mathrm{\Gamma }$ be a finite downward-closed ideal of a linkage class and $\lambda \in \mathrm{\Gamma }$ maximal. Then $\mathrm{\Delta }(\lambda )=M(\lambda )$ is projective in ${\mathcal{O}}_{\mathrm{\Gamma }}$ (A maximal-label Verma is projective in its truncation).

The regular integral block $C$ has simple labels $0$ and $-2$; $\mathrm{\Delta }(0)=M(0)$ is projective and is the projective cover $P(0)$ of $L(0)$; $L(-2)=\mathrm{\Delta }(-2)=M(-2)$ has the projective cover $P(-2)$, which fits into the nonsplit sequence $0\to \mathrm{\Delta }(0)\to P(-2)\to \mathrm{\Delta }(-2)\to 0$; and the flag multiplicities are $(P(0):\mathrm{\Delta }(0))=1$, $(P(-2):\mathrm{\Delta }(0))=1$ and $(P(-2):\mathrm{\Delta }(-2))=1$, while $(P(0):\mathrm{\Delta }(-2))=0$ because $P(0)=\mathrm{\Delta }(0)$ has the one-step flag $0\subseteq \mathrm{\Delta }(0)$ (The two projectives in the principal sl2 block, Finite Verma flags and their multiplicities).

$T_{\lambda }^{\mu }\mathrm{\Delta }(w\cdot \lambda )\cong \mathrm{\Delta }(w\cdot \mu )$ and $T_{\mu }^{\lambda }\mathrm{\Delta }(w\cdot \mu )$ has a finite Verma flag with exactly the two factors $\mathrm{\Delta }(w\cdot \lambda )$ and $\mathrm{\Delta }(ws\cdot \lambda )$, each with multiplicity one, for every $w\in W$ (Translation to and from a single wall on standard modules).

The functors $T_{\lambda }^{\mu }$, $T_{\mu }^{\lambda }$ are exact, both send projectives to projectives, and $T_{\mu }^{\lambda }$ is left adjoint to $T_{\lambda }^{\mu }$ (Translation functors are exact and biadjoint).

Every object of $\mathcal{O}$ has finite length, hence is a finite direct sum of indecomposables; a direct summand of a projective object is projective; every indecomposable projective object has a unique maximal proper subobject, so its head is simple and the object is a projective cover of that head; and any two indecomposable projectives with isomorphic heads are isomorphic (Every object of O has finite length, Fitting decomposition in a finite-length abelian category, Projective covers in O are indecomposable and unique, Projective object characterisations).

The multiplicity $(X:\mathrm{\Delta }(\mu ))$ is well defined for every Verma-filtered $X$ and is additive over direct sums: a Verma flag of $X$ and one of $Y$ concatenate to a Verma flag of $X\oplus Y$ with the combined factors (Finite Verma flags and their multiplicities, Verma-flag multiplicities are independent of the flag).