# Functoriality of the adjoint representation

Just a simply question. I came across the following statement which is used for deriving Weyl's integral formula:

''$\text{Ad}_G(h)|_{\mathfrak{h}} = \text{Ad}_H(h)$ due to functoriality in the Lie group for the adjoint representation''.

Therefore we have $\text{Ad}_G(t_0^{-1})$ mod $\mathfrak{t}$ = Ad$_{G/T}(t_0^{-1})$.

Could somebody define for me the ''functoriality'' in the context of adjoint representation?

Thank you very much for your attention!

What is meant here is that the adjoint representation is nicely compatible with homomorphisms of Lie groups. For a homomorphism $\phi:H\to G$, let $\phi':\mathfrak h\to\mathfrak g$ be the derivative. Then it is a basic fact of Lie theory that $\phi(exp(tX))=\exp(t\phi'(X))$ for all $X\in\mathfrak h$. Now for $h\in H$ and $X\in\mathfrak h$, you get Ad$(h)(X)$ as the derivative at $t=0$ of the curve $h\exp(tX)h^{-1}$. Using the above fact, you get $\phi(h\exp(tX)h^{-1})=\phi(h)\exp(t\phi'(X))\phi(h)^{-1}$. Differentiating at $t=0$, we obtain $\phi'(Ad(h)(X))=Ad(\phi(h))(\phi'(X))$, which is the functorial property the argument refers to. You just have to apply this to the inclusion $i:H\to G$ whose derivative $i'$ is the inclusion $\mathfrak h\hookrightarrow\mathfrak g$.