Nonintegral sl2 central characters split into two simple blocks

For ${\mathfrak{s}\mathfrak{l}}_2$, the dot orbit of $\lambda$ is exactly $\{\lambda ,-\lambda -2\}$. Its two labels are nonintegral and distinct: equality would imply $\lambda =-1$. By F8 they have the same central character. Conversely, if a simple object belongs to ${\mathcal{O}}_{{\chi }_{\lambda }}$, F6 writes it as $L(\mu )$. Its highest vector is killed by a power of every $z-{\chi }_{\lambda }(z)$ by F5, while F7 says that $z$ acts on it as ${\chi }_{\mu }(z)$. Hence ${\chi }_{\mu }={\chi }_{\lambda }$, and F8 forces $\mu \in \{\lambda ,-\lambda -2\}$. To see that both Vermas are simple, fix either nonintegral label $\eta$. The relations $[h,f]=-2f$ and $[e,f]=h$ give $h{f}^r{v}_{\eta }=(\eta -2r)f^r{v}_{\eta }$ and, by induction using $e{f}^{r+1}{v}_{\eta }=fe{f}^r{v}_{\eta }+h{f}^r{v}_{\eta }$, the coefficient formula $e{f}^r{v}_{\eta }=r(\eta -r+1)f^{r-1}{v}_{\eta }$ for $r\ge 1$. Every nonzero submodule of $M(\eta )$ contains a nonzero weight vector: project a finite sum of distinct weight components using a polynomial in $h$. Choose its least occurring PBW index $r$. If $r>0$, the derived formula makes $e{f}^r{v}_{\eta }=r(\eta -r+1)f^{r-1}{v}_{\eta }\mathrm{\ne }0$, contradicting minimality; hence $r=0$ and the submodule contains $v_{\eta }$, so it equals $M(\eta )$. The shifted simple-root pairings of the two labels are $\lambda +1$ and $-(\lambda +1)$, both nonintegral. Hence their integral-reflection linkage classes are singletons by The integral Weyl group of a weight.

The two labels lie in distinct root-lattice cosets: their difference is $2(\lambda +1)$ in $h$-weight coordinates, whereas the ${\mathfrak{s}\mathfrak{l}}_2$ root lattice has even integral coordinates; $\lambda +1\mathrm{\notin }\mathbb{Z}$. For each $X\in {\mathcal{O}}_{{\chi }_{\lambda }}$, group its weight spaces by every coset of $Q$. Root operators preserve each group, so F1 makes these groups submodules whose direct sum is $X$. Any nonzero group has a simple composition factor by F9. Step 1.1 allows only the two labels, whose simple modules have weights in their respective cosets, so all other groups vanish. Hence $X=X_{\lambda +Q}\oplus X_{-\lambda -2+Q}$ functorially, and every nonzero summand has factors only of its corresponding label. Both summands are nonzero because they contain their simple Verma modules.

Fix one label $\eta$ and put $S=M(\eta )=L(\eta )$. Every object in its coset summand has a finite composition series by F9, and all its factors are $S$ by step 2.1. Every extension of $S$ by itself splits by F2. More generally an extension $0\to S^r\to E\to S\to 0$ splits: push out along each coordinate projection $S^r\to S$, obtaining $E_i=(E\oplus S)\mathrm{/}\{(a,-a_i):a\in S^r\}$. Each has a retraction to its kernel $S$. Composing these retractions with $E\to E_i$ and collecting coordinates gives a retraction $E\to S^r$, hence a splitting. The case $r=0$ is immediate.

Induction on a composition series now expresses every object in either coset summand as a finite direct sum of its sole simple $S$. Since $\mathrm{End}(S)=\mathbb{C}$, maps between $S^r$ and $S^s$ are exactly complex matrices. Thus $V\mapsto S\otimes V$ (with trivial action on the finite-dimensional multiplicity space) and $X\mapsto \mathrm{Hom}(S,X)$ are inverse equivalences for each summand. Zero corresponds to the zero-dimensional vector space. Each summand is indecomposable as a category because every nonzero object contains its sole simple $S$ and all objects are sums of $S$. By F10, ${\mathcal{O}}_{{\chi }_{\lambda }}$ itself is a direct summand of $\mathcal{O}$, so these are precisely its two blocks in $\mathcal{O}$.