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



stabilizer  of convex cones in a linear space
As Willie already said, you should look at the action of your group on the the extreme rays of C. Look at the kernel of this action, K. 
The group GL(C)/K is then finite (and bounds on its order 
can be derived from the fact that it will be a permutation action, that is realised in 
a subspace of certain dimension...)


Regarding $K$, there will be a partition of the set of extreme rays into components $I\_1,\dots,I\_k$, so that each basis of the linear span of $C$ consisting of extreme rays will have $\dim (V\_{I\_j})$ elements from $V\_{I\_j}$, the linear span of $I\_j$. Then $K$ will induce the multiplication by positive scalars action on each $V\_{I\_j}$, and will be the direct product of these actions. 


(Thanks to David Speyer for pointing the error in the original description of $K$).


