Choosing canonical representatives of Weyl group elements, some questions

Let $G$ be a connected, reductive group which is quasisplit over a field $k$ of characteristic zero. Let $B$ be a Borel subgroup defined over $k$, containing a maximal torus $T$ defined over $k$. Let $\Delta$ be the base of $\Phi(G,T)$ corresponding to $B$, and let $\Gamma = \textrm{Gal}(\overline{k}/k)$. A $k$-splitting is a choice of isomorphisms of algebraic groups $X_{\alpha}: \mathbf G_{a,\overline{k}} \rightarrow U_{\alpha} : \alpha \in \Delta$, where $U_{\alpha}$ is the root subgroup of $U_{\overline{k}}$ corresponding to $\alpha$. We say that $\Gamma$ preserves the splitting if $\gamma.X_{\alpha} = X_{\gamma.\alpha}$. There exists a $k$-splitting preserved by $\Gamma$.

In the paper On the Definition of Transfer Factors by Langlands and Shelstad, a method to choose canonical Weyl group representatives is given:

One first chooses representatives for the simple reflections $\varpi(\alpha) : \alpha \in \Delta$, and then representatives for the other reflections by means of a reduced decomposition.

I am unfamiliar with the way the authors are thinking of these things, and had a few questions. I would appreciate any explanation or references.

1 . If $H_{\alpha}: \mathbf G_{m,\overline{k}} \rightarrow T$ is the coroot attached to $\alpha$, how can $H_{\alpha}$ be regarded as an element of the Lie algebra $\mathfrak t$ of $T$?

2 . What is meant by $\textrm{exp } X_{\alpha}$? Does this have something to do with the tangent space map associated to $X_{\alpha}$?

3 . What is meant precisely by "the homomorphism $\textrm{SL}(2) \rightarrow G$ attached to the Lie triple $\{X_{\alpha},H_{\alpha}, X_{-\alpha}\}$? I assume this has something to do with a choice of an isogeny $\textrm{SL}_2 \rightarrow [Z_G((\textrm{Ker } \alpha)^0),Z_G((\textrm{Ker } \alpha)^0)]$.

• (1) $H_\alpha$ as an element of $\mathfrak t$ is $\mathrm dH_\alpha(1)$ in your sense. (2) Yes, precisely; it uses an identification of the root subspace with the root subgroup. This is canonical only up to conjugation by $T$. (3) This is the homomorphism coming from the theory of Jacobson–Morosov triples. It too is canonical only up to conjugation by something or other. Jun 17, 2017 at 3:12
• (2) $X_\alpha$ is a nilpotent matrix, so in char. 0 $\exp X_\alpha$ is defined. (3) You have $H_\alpha$, then you choose $X_\alpha$ and get $X_{-\alpha}$ such that $[X_\alpha,X_{-\alpha}]=H_\alpha$. This triple $(X_\alpha, H_\alpha,X_{-\alpha})$ defines an embedding ${\rm Lie\,SL}(2)\to {\rm Lie}\,G$, which is the differential of some homomorphism ${\rm SL}(2) \to G$. This homomorphism is what you need. Jun 18, 2017 at 13:01