# Fixed space of maximal torus and Weyl group

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?

• For (1). We consider $U=U(G,T)$ and $W=W(G,T)$. If $T'\subset G$ is another maximal torus, then there exists $g\in G$ such that $T'=gTg^{-1}$. The isomorphism $g\colon (G,T)\to (G,T')$ induces compatible isomorphisms $U(G,T)\overset{\sim}{\to} U(G,T')$ and $W(G,T)\overset{\sim}{\to} W(G,T')$. Therefore, the answer to Question (1) is No. – Mikhail Borovoi Aug 7 '20 at 9:54

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
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.