Translation to and from a single wall on standard modules

Claim (1). Let $\eta =w^{\mathrm{\prime }}\cdot \mu$ be a label in the dot orbit of $\mu$ with $\dim E_{\eta -w\cdot \lambda }\mathrm{\ne }0$, and put $\gamma :=\eta -w\cdot \lambda$; then $w^{\mathrm{\prime }}\cdot \mu =w\cdot \lambda +\gamma$ and $\gamma$ is a weight of $E$, so [F5] gives $\eta =w^{\mathrm{\prime }}\cdot \mu =w\cdot \mu$ and $\gamma =w(\mu -\lambda )$, which occurs in $E$ with multiplicity one. Hence in the flag of $E\otimes \mathrm{\Delta }(w\cdot \lambda )$ supplied by [F2], the only label of central character ${\chi }_{\mu }$ (equivalently, the only label in the dot orbit of $\mu$, by [F1]) is $\eta =w\cdot \mu$, with multiplicity one.

Claim (2). Let $\eta =w^{\mathrm{\prime }}\cdot \lambda$ be a label in the dot orbit of $\lambda$ with $\dim E_{\eta -w\cdot \mu }^{\ast }\mathrm{\ne }0$, and set $\gamma :=w\cdot \mu -\eta$; by [F2] the weight $\gamma$ is a weight of $E$ and $\mid \gamma \mid \le \mid \nu \mid$. Since $w^{\mathrm{\prime }}\cdot \lambda =w^{\mathrm{\prime }}({\lambda }_{\bullet })-\rho$ and $w\cdot \mu =w({\mu }_{\bullet })-\rho$, setting $x=(w^{\mathrm{\prime }}{)}^{-1}w$ gives $(w^{\mathrm{\prime }}{)}^{-1}(-\gamma )={\lambda }_{\bullet }-x{\mu }_{\bullet }$, so $\mid {\lambda }_{\bullet }-x{\mu }_{\bullet }\mid =\mid \gamma \mid$. By [F2] $\mid \nu \mid =\mid {\mu }_{\bullet }-{\lambda }_{\bullet }\mid$, while [F3] applied to the dominant weights $\xi =-{\lambda }_{\bullet }$ and ${\eta }^{\mathrm{\prime }}=-{\mu }_{\bullet }$ gives $\mid {\mu }_{\bullet }-{\lambda }_{\bullet }\mid \le \mid {\lambda }_{\bullet }-x{\mu }_{\bullet }\mid$. Hence equality holds throughout, and the equality case of [F3] gives $x{\eta }^{\mathrm{\prime }}\in {\mathrm{Stab}}_W(\xi ){\eta }^{\mathrm{\prime }}$, and regularity of ${\lambda }_{\bullet }$ makes ${\mathrm{Stab}}_W(\xi )=\{1\}$. Consequently $x{\mu }_{\bullet }={\mu }_{\bullet }$, and the datum ${\mathrm{Stab}}_W({\mu }_{\bullet })=\{1,s\}$ forces $x\in \{1,s\}$. Therefore $\eta =w^{\mathrm{\prime }}\cdot \lambda =w{x}^{-1}\cdot \lambda$ equals $w\cdot \lambda$ or $ws\cdot \lambda$, and in both cases $(w^{\mathrm{\prime }}{)}^{-1}(-\gamma )={\lambda }_{\bullet }-{\mu }_{\bullet }$ (using $s{\mu }_{\bullet }={\mu }_{\bullet }$ when $x=s$), so $-\gamma =w^{\mathrm{\prime }}({\lambda }_{\bullet }-{\mu }_{\bullet })$ and $\gamma =w^{\mathrm{\prime }}(\mu -\lambda )$ lies in $W(\mu -\lambda )=W\nu$; by [F2] it occurs in $E$ with multiplicity one. Conversely, taking $w^{\mathrm{\prime }}=w$ or $w^{\mathrm{\prime }}=ws$ gives $\gamma =w^{\mathrm{\prime }}(\mu -\lambda )\in W\nu$, so both proposed factors occur once; their labels are distinct because ${\lambda }_{\bullet }$ is regular.

For claim (1), the object $T_{\lambda }^{\mu }\mathrm{\Delta }(w\cdot \lambda )={\mathrm{pr}}_{{\chi }_{\mu }}(E\otimes \mathrm{\Delta }(w\cdot \lambda ))$ is Verma-filtered by [F4], and by step 1.1 its only nonzero multiplicity is $(T_{\lambda }^{\mu }\mathrm{\Delta }(w\cdot \lambda ):\mathrm{\Delta }(w\cdot \mu ))=1$; by [F4] it is therefore isomorphic to $\mathrm{\Delta }(w\cdot \mu )$.

For claim (2), the object $T_{\mu }^{\lambda }\mathrm{\Delta }(w\cdot \mu )={\mathrm{pr}}_{{\chi }_{\lambda }}(E^{\ast }\otimes \mathrm{\Delta }(w\cdot \mu ))$ is Verma-filtered by [F4]; by step 1.2 its nonzero multiplicities among labels of central character ${\chi }_{\lambda }$ are exactly one at $\mathrm{\Delta }(w\cdot \lambda )$ and one at $\mathrm{\Delta }(ws\cdot \lambda )$, and all other multiplicity vanish because their labels have different central character. Hence it has a finite Verma flag with exactly these two factors, each once, and its class in the Grothendieck group is $[\mathrm{\Delta }(w\cdot \lambda )]+[\mathrm{\Delta }(ws\cdot \lambda )]$.

Steps 2.1 and 2.2 prove the two claims of the statement; with [F6] recording that the two translation functors are exact, the theorem follows.