The Soergel functor on the small regular sl_2 block

Assume the Axiom of Choice inherited from the category-$\mathcal{O}$ suppliers. Work in the dominant regular integral ${\mathfrak{s}\mathfrak{l}}_2$ block at $\lambda =0$, with $\rho =1$, $W=\{e,s\}$ and $s\cdot 0=-2$ (The regular integral block and its antidominant weight). Set $\mathbb{k}=\mathbb{C}$. Put ${\mathrm{\Delta }}_e=M(0)$, ${\mathrm{\Delta }}_s=M(-2)=L(-2)$, $P_e=P(0)={\mathrm{\Delta }}_e$ and $P_s=P(-2)$; the block is described by the nonsplit sequences $0\to L(-2)\to {\mathrm{\Delta }}_e\to L(0)\to 0,0\to {\mathrm{\Delta }}_e\to P_s\to {\mathrm{\Delta }}_s\to 0,$ the second being the one-letter case of Wall-crossing acts on standard modules by a two-step extension and Wall-crossing compositions produce the projective with the top standard label, and $P_e={\mathrm{\Delta }}_e$ being the dominant Verma (BGG reciprocity, Verma modules).

The values of $V$. Let $C=\mathbb{k}[x]\mathrm{/}(x^2)$ with $\mathrm{deg}x=2$ be the coinvariant algebra and $V=\mathrm{Hom}(P_s,-)$ (The antidominant projective and the Soergel functor), so that $\mathrm{End}(P_s)\cong C$ by The endomorphisms of the antidominant projective form the coinvariant algebra. Since $P_s$ has indecomposable head $L(-2)$ and $L(-2)=M(-2)$ is simple, the left $C$-module values are $V(P_e)=\mathbb{k}=C\mathrm{/}(x),V(P_s)=C,V({\mathrm{\Delta }}_e)=V({\mathrm{\Delta }}_s)=\mathbb{k},V(L(0))=0,V(L(-2))=\mathbb{k}\mathrm{.}$ Indeed $V({\mathrm{\Delta }}_e)$ and $V({\mathrm{\Delta }}_s)$ are one-dimensional by exactness applied to the first sequence and $V(L(-2))=\mathbb{k}$ from the head quotient, while $V(L(0))=0$ because $L(0)$ is not a quotient of $P_s$; the unique one-dimensional $C$-module is $\mathbb{k}$ with $x$ acting by zero.

The projective Hom table. With rows indexed by the source and columns by the target, the four Hom spaces in the ordered bases $(P_e,P_s)$ are $\left(\begin{array}{cc}\mathrm{Hom}(P_e,P_e)& \mathrm{Hom}(P_e,P_s)\\ \mathrm{Hom}(P_s,P_e)& \mathrm{Hom}(P_s,P_s)\end{array}\right)=\left(\begin{array}{cc}\mathbb{k}& \mathbb{k}\\ \mathbb{k}& C\end{array}\right),\text{dimensions }\left(\begin{array}{cc}1& 1\\ 1& 2\end{array}\right)\mathrm{.}$ Choose the nonzero maps $i\text{\hspace{0.17em}⁣}:P_e\to P_s$ and $p\text{\hspace{0.17em}⁣}:P_s\to P_e$; under the Struktursatz Soergel's Struktursatz: the functor V is fully faithful on projectives their images in ${\mathrm{Hom}}_C(\mathbb{k},C)=\mathbb{k}{i}_0$ and ${\mathrm{Hom}}_C(C,\mathbb{k})=\mathbb{k}{p}_0$ are the generators $i_0(1)=x$ and $p_0(a+bx)=a$, and the compositions compute $p_0{i}_0=0$ on $\mathbb{k}$ while $i_0{p}_0=m_x$, multiplication by $x$ on $C$ (equivalently $pi=0$ and $ip$ is a nonzero square-zero endomorphism; rank one refers to its image under $V$ on $C$). In particular $\mathrm{End}(P_s)\cong C$ is recovered as the full two-dimensional endomorphism algebra, and the full-faithfulness comparison is computed on the four projective Hom spaces rather than claimed for the whole block.

Polynomial lifts and standard ranks. For $S=\mathbb{k}[a]$ with $s(a)=-a$, the empty-word lift is $B_e=S$ and the rank-one lift is $B_s=S{\otimes }_{\mathbb{k}[a^2]}S(1)$ (The Soergel bimodule $B_i$ of a simple reflection); their right origin reductions are $\mathbb{k}$ and $C(1)$, whose centered auxiliary degrees are $-1,1$, and the labels $e^{-1}=e$, $s^{-1}=s$ agree with the dictionary of Indecomposable projectives correspond to indecomposable Soergel modules. Before reduction the standard graph ranks, in the order $(e,s)$, are $(1,0)$ for $B_e$ and $(1,1)$ for $B_s$: in the two-variable model $B_s=\mathbb{k}[l,r]\mathrm{/}(l^2-r^2)(1)$ the submodule generated by $l-r$ is free of rank one on the graph $l=-r$ with annihilator $(l+r)$, and the quotient is free of rank one on the graph $l=r$, with annihilator $(l-r)$ (Standard graph bimodules, support filtrations and characters). These ranks agree with the Verma-flag multiplicities $(1,0)$ of $P_e$ and $(1,1)$ of $P_s$ at $v=1$, as predicted by Verma-flag multiplicities are standard-filtration ranks under V; they are taken before the graph supports coalesce at the origin, and no grading on arbitrary category-$\mathcal{O}$ objects or intrinsic separated-support filtration on $V(P)$ is asserted.