The wall-crossing functor and its unit and counit

Assume the Axiom of Choice, inherited from the published block and translation theory (The Axiom of Choice, Translation functors by tensoring and projection).

Fix the dominant regular integral representative $\lambda$ of a regular integral block ${\mathcal{O}}_{\lambda }$ (The regular integral block and its antidominant weight) and an integral weight $\mu$ such that $\lambda +\rho$ is regular dominant and $\mu +\rho$ is dominant with dot stabilizer exactly $\{1,s\}$ for a reflection $s=s_{\alpha }$; this is the dominant-chamber version of single-wall translation data, the published Dot-Weyl facets and single-wall translation data being stated with antidominant representatives, and the dominant standard counts below are proved through the norm-distance and extremal-weight bounds (A dominant vector minimises its distance to a dominant weight, Weights of a finite-dimensional simple module lie in the norm ball) rather than by applying that datum literally. Let $\nu$ be the unique dominant element of the linear Weyl orbit of $\mu -\lambda$ and let $E=L(\nu )$ be the finite-dimensional simple module of highest weight $\nu$, so that $\mu -\lambda$ is an extremal weight of $E$; write $E^{\ast }=L(-w_0\nu )$ for its dual (Highest weight of the dual representation).

With ${\chi }_{\lambda },{\chi }_{\mu }$ the generalized central characters and $\mathrm{pr}$ the exact central projections of Translation functors by tensoring and projection, define the onto-wall and out-of-wall translations $T_{\mathrm{o}\mathrm{n}}:={\mathrm{pr}}_{{\chi }_{\mu }}\circ (E\otimes -){\mid }_{{\mathcal{O}}_{{\chi }_{\lambda }}},T_{\mathrm{o}\mathrm{u}\mathrm{t}}:={\mathrm{pr}}_{{\chi }_{\lambda }}\circ (E^{\ast }\otimes -){\mid }_{{\mathcal{O}}_{{\chi }_{\mu }}},$ and the wall-crossing functor ${\theta }_s:=T_{\mathrm{o}\mathrm{u}\mathrm{t}}\circ T_{\mathrm{o}\mathrm{n}}\text{\hspace{0.17em}⁣}:{\mathcal{O}}_{\lambda }⟶{\mathcal{O}}_{\lambda }\mathrm{.}$

Basic properties. Both translations are exact, and $T_{\mathrm{o}\mathrm{u}\mathrm{t}}$ is both a left and a right adjoint of $T_{\mathrm{o}\mathrm{n}}$ (Translation functors are exact and biadjoint); hence ${\theta }_s$ is exact and, as a composite of restriction-of-scalars and central block projection applied to tensoring by $E^{\ast }\otimes E$, it is a natural direct summand of the endofunctor ${\mathrm{pr}}_{{\chi }_{\lambda }}\circ ((E^{\ast }\otimes E)\otimes -)\circ {\mathrm{incl}}_{{\chi }_{\lambda }}$, so it is a projective functor on ${\mathcal{O}}_{\lambda }$ in the sense of Projective functors on a regular integral block. On standard objects the dominant-chamber counts are $T_{\mathrm{o}\mathrm{n}}\mathrm{\Delta }(w\cdot \lambda )\cong \mathrm{\Delta }(w\cdot \mu )$ and the two-factor flag $[\mathrm{\Delta }(w\cdot \lambda )]+[\mathrm{\Delta }(w\cdot s\cdot \lambda )]$ for $T_{\mathrm{o}\mathrm{u}\mathrm{t}}\mathrm{\Delta }(w\cdot \mu )$, as follows directly from the tensor flag and the two norm bounds above. Put $v=\lambda +\rho$, $u=\mu +\rho$. A retained onto-wall label satisfies $wv+\gamma =xu$ with $\mathrm{\mid }\gamma \mathrm{\mid }\le \mathrm{\mid }u-v\mathrm{\mid }$. Thus $\mathrm{\mid }\gamma \mathrm{\mid }=\mathrm{\mid }v-w^{-1}xu\mathrm{\mid }\ge \mathrm{\mid }v-u\mathrm{\mid }$; equality and regularity of $v$ give $w^{-1}xu=u$, leaving the single label $w\cdot \mu$, with its extremal weight of multiplicity one. For the return translation, $wu+\delta =xv$ implies equality in $\mathrm{\mid }u-w^{-1}xv\mathrm{\mid }\ge \mathrm{\mid }u-v\mathrm{\mid }$, so $w^{-1}xv\in \{v,sv\}$ and $x\in \{w,ws\}$, again with extremal weights of multiplicity one. These calculations use the Verma tensor flag of Finite-dimensional tensoring preserves Verma flags.

Unit and counit. The two tensor-block adjunctions supply natural transformations $\eta \text{\hspace{0.17em}⁣}:{\mathrm{id}}_{{\mathcal{O}}_{\lambda }}⟶{\theta }_s,\epsilon \text{\hspace{0.17em}⁣}:{\theta }_s⟶{\mathrm{id}}_{{\mathcal{O}}_{\lambda }},$ where $\eta$ is the unit of the adjunction $T_{\mathrm{o}\mathrm{n}}⊣T_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and $\epsilon$ the counit of the adjunction $T_{\mathrm{o}\mathrm{u}\mathrm{t}}⊣T_{\mathrm{o}\mathrm{n}}$; each satisfies the triangle identities with its companion counit or unit on the wall category, and these identities are part of the data. The pair $({\theta }_s,\eta ,\epsilon )$ is determined by the choice of the biadjunction maps, which are fixed once and for all.

Independence of the auxiliary weight. Up to natural equivalence the functor ${\theta }_s$ depends only on the wall $s$ and the block ${\mathcal{O}}_{\lambda }$, not on the chosen ambient weight $\mu$: this well-definedness statement is discharged forward by Indecomposable projective functors are indexed by the Weyl group ↗, which lifts the equality of the evaluations on standard objects to a natural equivalence; it is recorded in justified_by and is not used to construct ${\theta }_s$ or to generate objects.