Convexity, stratification and faces of the relative cone

The intersection of a convex cone with the linear subspace ${\mathrm{\Theta }}_J$ is a convex cone. Every point of $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}$ lies in some translate of the closed fundamental chamber, which is the disjoint union of its coordinate faces $C_L$. If $x{C}_L$ and $y{C}_K$ meet, [F3] applied to the corresponding points in the closed fundamental chamber gives $L=K$ and $y^{-1}x\in W_L$; conversely $W_L$ fixes $C_L$ pointwise by [F1]. Thus the full-cone face translates are disjoint when indexed by right cosets $W\mathrm{/}{W}_L$.

For $f\in C_L$, a root $\beta$ has $f(\beta )=0$ exactly when all its nonzero simple-root coordinates are indexed by $L$: its coordinates have one sign by [F2], while $f$ vanishes on $L$ and is strictly positive elsewhere. Therefore whether $xf$ lies in ${\mathrm{\Theta }}_J$ is independent of the choice of $f\in C_L$, since it is determined by the supports of the roots $x^{-1}{\alpha }_j$, $j\in J$. Each full-cone stratum is consequently either contained in or disjoint from ${\mathrm{\Theta }}_J$. Intersecting the union in 1.1 with ${\mathrm{\Theta }}_J$ gives the asserted disjoint stratification.

Suppose $x{C}_L\subseteq {\mathrm{\Theta }}_J$. Its linear span is $x{\mathrm{\Theta }}_L$, so $x{\mathrm{\Theta }}_L\subseteq {\mathrm{\Theta }}_J$ and $\mathrm{\mid }L\mathrm{\mid }\ge \mathrm{\mid }J\mathrm{\mid }$. If equality holds, the two subspaces have the same dimension and hence coincide. A point $xf$ with $f\in C_L$ lies on $H_{\alpha }$ only if $x^{-1}\alpha$ is supported on $L$, by the sign argument in 2.1. Then $H_{\alpha }$ contains $x{\mathrm{\Theta }}_L={\mathrm{\Theta }}_J$, so $xf$ avoids every relative hyperplane. By [F4] and [F5], the chamber containing $xf$ is some labelled face translate; uniqueness in 1.1 makes that translate $x{C}_L$.

Conversely, a chamber is $x{C}_L$ for a label in [F4]. The equality $W_J=x{W}_L{x}^{-1}$ implies equality of their fixed subspaces. The fixed subspace of $W_K$ is ${\mathrm{\Theta }}_K$: each generating reflection fixes its simple-root hyperplane pointwise, and a point fixed by that reflection lies on that hyperplane. Thus ${\mathrm{\Theta }}_J=x{\mathrm{\Theta }}_L$, and comparing dimensions gives $\mathrm{\mid }J\mathrm{\mid }=\mathrm{\mid }L\mathrm{\mid }$. Together with 3.1 this proves the chamber characterization.

Let $x{C}_L$ be a chamber. By 4.1, $x{\mathrm{\Theta }}_L={\mathrm{\Theta }}_J$. In the coordinates $f({\alpha }_i)$ for $i\mathrm{\notin }L$, the set $C_L$ is the positive orthant in ${\mathrm{\Theta }}_L$, and its closure is the nonnegative orthant. Its codimension-one faces are obtained by setting exactly one of these coordinates to zero while keeping the others positive. Applying the linear isomorphism $x$ gives the faces $x{C}_{L+i}$. Their closures are $x(\overline{C_L}\cap H_{{\alpha }_i})=\overline{x{C}_L}\cap H_{x{\alpha }_i}$. Since $x{\alpha }_i$ is not in the span of $\{x{\alpha }_l:l\in L\}$, this hyperplane does not contain ${\mathrm{\Theta }}_J$; its trace is therefore a relative hyperplane supporting the wall. Every codimension-one wall of the orthant is one of these facets.

If $J=\mathrm{\varnothing }$, then ${\mathrm{\Theta }}_J=\mathrm{\Theta }$ and every face translate satisfies the containment condition. The stratification from 2.1 is therefore the full Tits-cone stratification. Steps 1.1–5.1 establish all the assertions.