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


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