# Lately in Lie Groups

So $\mathfrak{g}$ is a complex semisimple lie algebra so can write

$\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{g}_\alpha$

For $X \mathfrak{g}_\alpha$, and $\alpha \ne 0$, then $ad_X$ nilpotent, so $e(X) := \exp (ad_X)$ is well defined.  $E(\mathfrak{h}) \subset GL(\mathfrak{g})$ is the subgroup gen. by these.

Thm D.22 (Fult. and Harr) If $\mathfrak{h}, \mathfrak{h}'$ are Cartan, then they are conjugate, and $E(\mathfrak{h}) = E(\mathfrak{h}') =: E$.

Notes: The actual statement is that there is $g \in E$ such that $g(\mathfrak{h}) = \mathfrak{h'}$.  But $g = \prod_i e(X_i)$.  And using that $\exp ad_X = Ad_{\exp X}$ we see that the action of $g$ is just to conjugate by the elements \exp(X_i)\$.

Also $e(H) = e(X^{-1}) \circ e(H) \circ e(X)$ so for any $g \in E$, it true $E = gEg^{-1}$.

### Weyl Group stuff

(this is coming from around page 201 of Ful and Harr)

When talking about roots $R$, there is the following chain

$R \subset \Lambda_R \subset \Lambda_W \subset \mathfrak{h}^*$

where $\Lambda_R$ is the lattice generated by the roots, and $\Lambda_W$ is the lattice generated by functionals $\beta$ that are integer valued on all $H_\alpha \in \mathfrak{h} \subset \mathfrak{g}$, where $H_\alpha$ is the unique element of $[g_\alpha, g_\alpha]$ such that $\alpha(H_\alpha) = 2$.

Without much motivation define

$\Omega_\alpha = \{\beta \in \mathfrak{h}^*| \beta(H_\alpha) = 0\}$

Its straightforward to verify that this doesn’t intersect the line $\mathbb{C}\cdot \alpha$.  So now define the involution associated to $\alpha \in R$

$W_\alpha(\gamma) = \gamma - \gamma(H_\alpha)\alpha$

The hyperplane $\Omega_\alpha$ and $\mathbb{C}\alpha$ are both preserved by $W_\alpha$.  The Weyl Group associated to $\mathfrak{g}$ is the group generated by $W_\alpha$.  More generally (maybe), one can define

$W_\alpha(\gamma) = \gamma - 2\frac{(\gamma, alpha)}{(\alpha,\alpha)}\alpha$

where $(,)$ is an inner product that makes $W_\alpha$ preserve $\Omega_\alpha$ and reflect the line $\mathbb{C} \alpha$.

On page 492 of Ful and Harr there is a remark saying that all these results about Complex lie algebras can be extended to any char 0 and alg. closed field by some base change argument (what is it?).  Also, given this other results then hold for any char 0 field….

A root $\alpha$ is simple if it cannot be written as a sum $\gamma + \beta$ of other roots.  Now back to material from appendix D.  A cool fact is

Lemma D.27 The Weyl group is generated by $W_\alpha$ where $\alpha$ is a simple root.

This result uses lemma D.25, which says for such $\alpha$, $W_\alpha$ maps $R^+ - \alpha$ to itself.

the roots we’re interested in look like $\beta = \sum_i n_i\alpha_i$ with $\alpha_i$ simple and $n_i$ nonnegative.  Now $W_\alpha(\beta) \in \mathfrak{h}^*$ but why is it in $R$?  Also, if $\beta \ne \alpha$ is simple and positive, then $W_\alpha(\beta)$ is $\beta$ minus some integer multiple of $\alpha$, so

$l(W_\alpha(\beta) ) = l(\beta) - m l(\alpha)$

why is this still positive