I get a feeling what I am going to ask is very standard and classic, but I am not able to find any reference. Any answer or reference would be appreciated.
Let us assume that $G$ is a simply connected compact Lie subgroup of $GL(n,\mathbb{C})$. Let $T$ be a maximal torus of $G$. Let $\mathfrak{g}$ and $\mathfrak{t}$ denote the Lie algebras of $G$ and $T$ respectively.
We know that the Weyl group $W$ of $G$ acts freely transitively on the following three sets,
- $N_G(T)/T$ where $N_G$ is the normalizer in $G$
- Borel subgroups containing $T$
- The unit eigenvectors for $\mathfrak{t}$ in $\mathfrak{g}$ (Trace norm)
We can use any of the three to define the Weyl group. The correspondence between the first two is pretty explicit and can be found in standard texts, say Humphreys.
I wanted to ask if one can give an explicit correspondence between 1 and 3 or 2 and 3?
I tried exponentiating the roots but to no avail. Even for the simple case of $G=SL(n,\mathbb{C})$ and $T$ being the diagonal matrices, the exponentials of the roots are $1 + E_{ij}$ but these do not lie in distinct Borel subgroups nor are they elements of the normalizer.