Let $G$ be a compact connected Lie group and $T\subset G$ a maximal torus. Let $V$ be a representation of $G$ and $U=\{v\in V: tv=v\textrm{ for all }t\in T\}$. For any $g\in N(T)$ we have for all $t\in T$ and $v\in U$ that $g^{-1}tgv=v \Rightarrow t(gv)=gv$. This shows that for all $v\in U$ we have $gv\in U$ as well. From this we can define a representation of the Weyl group $W$ on $U$. I have the following two questions:


1. Does this $W$-module structure on $U$ depend on the choice of the maximal torus $T$?
2. Assume that $V$ is irreducible and $U$ nontrivial. Is there a way to understand when $U$ is irreducible as $W$-module? Is it always the case?



Fixed space of maximal torus and Weyl group
This paper of Humphreys addresses your second question (the first is answered in the comments - the $W$-module structure is independent of the choice of torus): [https://people.math.umass.edu/~jeh/pub/zero.pdf](https://people.math.umass.edu/%7Ejeh/pub/zero.pdf)


Here is a quote from the paper (section 1.4):



> 
> Indeed, it is usually unclear how to determine directly whether or not
> the W-module $L\_\lambda(0)$ [i.e. $U$] is simple, even if its dimension is compatible with
> simplicity.
> 
> 
> 


The $W$-module $U$ is not always irreducible. Indeed in type $A\_2$ (i.e. $G=SU(3)$), there is a formula for the dimension of $U$ in Section 2.2, which shows that the dimension may be unbounded.


