The Bruhat graph and the BGG Verma sum in degree k

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$, positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$, positive system ${\mathrm{\Phi }}^{+}$, simple roots ${\alpha }_1,\dots ,{\alpha }_r$, Weyl group $W$ and Weyl vector $\rho$, as in Finite Weyl root system, lattice and chamber conventions and The root set is a reduced crystallographic root system. Let ${\mathrm{\Lambda }}^{+}$ be the dominant integral weights (Integral, dominant, and strictly dominant weights) and let $\lambda \in {\mathrm{\Lambda }}^{+}$. Write $w\circ \lambda =w(\lambda +\rho )-\rho$ for the dot action of The Weyl vector rho for a chosen positive system.

The Bruhat graph has vertex set $W$; an arrow $x\to y$ means that $y⊲x$ is a cover in Bruhat order, i.e. $y<x$ and $\mathrm{\ell }(x)=\mathrm{\ell }(y)+1$ (Bruhat order on a finite Weyl group). Equivalently, $\mathrm{\ell }(x)=\mathrm{\ell }(y)+1$ and $x=y{s}_{\beta }$ for a unique positive root $\beta \in {\mathrm{\Phi }}^{+}$ (Bruhat covers are right multiplication by positive-root reflections). A square is a quadruple $(x,m_1,m_2,y)$ with $x⊳m_1$, $x⊳m_2$, $m_1⊳y$, $m_2⊳y$ and $m_1\mathrm{\ne }{m}_2$.

For $0\le k\le \mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$ put

$C_k(\lambda )={⨁}_{\mathrm{\ell }(w)=k}M(w\circ \lambda ),$

the direct sum of Verma modules (Verma modules) over the elements of $W$ of length $k$, with its fixed direct-sum decomposition indexed by those elements. Each $C_k(\lambda )$ is an object of $\mathcal{O}$ (The classical BGG category O), being a finite direct sum of Verma modules. The endpoints are $C_0(\lambda )=M(\lambda )$ and $C_{\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }}(\lambda )=M(w_0\circ \lambda )$, where $w_0\in W$ is the longest element (Finite Weyl closed chambers and stabilizers); moreover $C_k(\lambda )=0$ for $k>\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$. Because $\lambda +\rho$ is regular (Positive coroot pairings of a dominant integral weight), the weights $w\circ \lambda$ are pairwise distinct: $w\circ \lambda =w^{\mathrm{\prime }}\circ \lambda$ forces $w=w^{\mathrm{\prime }}$. The integral Weyl group of The integral Weyl group of a weight therefore acts by the regular dot orbit $W\circ \lambda$ on the indexing set.