Dot-Weyl facets and single-wall translation data

Assume the Axiom of Choice (The Axiom of Choice). Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$, with the chosen positive system ${\mathrm{\Phi }}^{+}$ and Weyl vector $\rho$ of The Weyl vector rho for a chosen positive system and Weyl group $W$ acting by the root reflections $s_{\alpha }(\lambda )=\lambda -\lambda ({\alpha }^{\vee })\alpha$ of Root reflections and the Weyl group action. Write the dot action as $w\cdot \lambda =w(\lambda +\rho )-\rho$ and write $⟨\zeta ,{\alpha }^{\vee }⟩$ for the pairing with the coroot (The root set is a reduced crystallographic root system).

Put $R={\mathrm{span}}_{\mathbb{R}}\mathrm{\Phi }\subseteq {\mathfrak{h}}^{\ast }$. For a weight $\lambda \in R$, the dot-Weyl facet of $\lambda$ is the set of weights $F_{\lambda }=\{\zeta \in R:\mathrm{sgn}⟨\zeta +\rho ,{\alpha }^{\vee }⟩=\mathrm{sgn}⟨\lambda +\rho ,{\alpha }^{\vee }⟩\text{ for every root }\alpha \in \mathrm{\Phi }\},$ where $\mathrm{sgn}$ takes the values positive, zero and negative. Its upper closure is $F_{\lambda }^{+}=\{\zeta \in R:⟨\zeta +\rho ,{\alpha }^{\vee }⟩\text{ is positive, zero or nonpositive according as }⟨\lambda +\rho ,{\alpha }^{\vee }⟩\text{ is positive, zero or negative, for every }\alpha \in {\mathrm{\Phi }}^{+}\}\mathrm{.}$ The upper-closure test uses only positive roots: imposing it also on their negatives would incorrectly exclude wall points from the upper closure of the antidominant chamber. Thus $F_{\lambda }\subseteq F_{\lambda }^{+}$, the facets refine the closures of the open Weyl chambers, and $F_{\lambda }$ depends only on the wall-sign pattern of $\lambda +\rho$.

Two special positions are used throughout. A weight $\lambda$ is dot-regular when $⟨\lambda +\rho ,{\alpha }^{\vee }⟩\mathrm{\ne }0$ for every root $\alpha$, so that $F_{\lambda }$ is an open chamber; it is dot-antidominant when $⟨\lambda +\rho ,{\alpha }^{\vee }⟩\le 0$ for every positive root $\alpha$. Integrality of weights is the notion of Integral, dominant, and strictly dominant weights.

A single-wall translation datum is a triple $(\lambda ,\mu ,\alpha )$ consisting of integral dot-antidominant weights $\lambda ,\mu$ and a positive root $\alpha$ such that:

1. $\lambda$ is dot-regular, so $F_{\lambda }$ is an open chamber;
2. $\mu +\rho$ lies in the closure of the chamber of $\lambda +\rho$: one has $⟨\mu +\rho ,{\alpha }^{\vee }⟩=0$, and $⟨\mu +\rho ,{\beta }^{\vee }⟩$ has the same sign as $⟨\lambda +\rho ,{\beta }^{\vee }⟩$ for every root $\beta$ different from $\alpha$ and from its multiples;
3. the dot stabilizer $S_{\mu }:=\{w\in W:w\cdot \mu =\mu \}$ is exactly $\{1,s_{\alpha }\}$.

Equivalently, $F_{\mu }$ is a codimension-one facet in the closure of the open antidominant chamber $F_{\lambda }$, with $S_{\mu }=\{1,s_{\alpha }\}$. This dot stabilizer and the integral-reflection group $W_{\mu }$ of The integral Weyl group of a weight are defined by different conditions: since $\mu$ is integral, all simple reflections belong to $W_{\mu }$, so $W_{\mu }=W$, whereas $S_{\mu }=\{1,s_{\alpha }\}$. The groups coincide in rank one and differ when the rank is greater than one. In this datum the translating weight $\nu$ is the unique dominant weight of the linear Weyl orbit $W(\mu -\lambda )$; it exists and is unique by Finite Weyl closed chambers and stabilizers, and it is integral because $\mu -\lambda$ is integral and $W$ preserves the weight lattice. The wall reflection is $s:=s_{\alpha }$.

The basic example is ${\mathfrak{s}\mathfrak{l}}_2$ with the datum $(\lambda ,\mu )=(-2,-1)$: here $\rho =1$, $\mu =-\rho$, $\lambda +\rho =-1$ spans the open negative chamber, $\mu +\rho =0$ is the single wall, and the dot-stabilizer of $\mu$ is $\{1,s\}$. The pair $(0,-1)$ uses the dominant regular representative rather than the antidominant representative required here. It defines the same translation functors: $0$ and $-2$ have the same central character, and both weight differences have dominant representative $1$ with translating module $L(1)$, which is self-dual. These functors are defined in the next definition on this page.