Let $V$ be a finite dimensional vector space over $\mathbb{R}$, and $C\subset V$ a convex cone of the form $C=\mathbb{R}_{\geq0}v_i$ for finitely many $v_i$'s in $V$. How can one describe the stabilizer of $C$ in $GL(V)$?

Here one naturally defines the stabilizer of $C$ to be $GL(C)$ consisting of elements $g\in GL(C)$ such that $gC=C$. Say with respect to a base $(e_i)$ of $V$ and some integer $1\leq r\leq d$ one writes $$C=\sum_{i=1}^{r}\mathbb{R}_{\geq0}e_i +\sum_{j>r}\mathbb{R}e_j$$ then $GL(C)$ is the set of $g=(g_{ij})\in GL_d(\mathbb{R})$ such that $g_{ij}\geq0$ if $i\leq r$ and that the same holds for $ g^{-1} $.

My questions are:

(1) how large could $GL(C)$ be? It is clear that if in the above case with $r=d$ in the expression of $C$ along a basis $e_i$, then using the Bruhat-Tits decomposition in $GL(V)$ one finds large open subset of $GL(C)$ preserving $C$. Can $GL(C)$ be recovered essentially this way by choosing suitable basis?

(2) It seems that to characterize the difference $d-r$ one only needs to find out the split tori contained in $GL(C)$, inspired by the Bruha-Tits decomposition along a suitable basis $(e_i)$. Is this alwas true that the $r-d$ serves as a rank function for $GL(C)$?

(3) Could there be any improvements if one replaces $GL(C)$ by the set of linear maps $a\in End(V)$ such that $aC\subset C$, which is a monoid instead of a group?

thanks

Analysis on symmetric cones. Oxford University Press, 1994. $\endgroup$