isotropy representation

Let $𝔤$ be a Lie algebra, and $𝔥\subset 𝔤$ a subalgebra. Theisotropy representation of $𝔥$ relative to $𝔤$ is the naturallydefined action of $𝔥$ on the quotient vector space $𝔤/𝔥$.

Here is a synopsis of the technical details. As is customary, we willuse

$$b+𝔥,b\in 𝔤$$

to denote the coset elements of $𝔤/𝔥$.Let $a\in 𝔥$ be given. Since $𝔥$ is invariant with respect to${ad}_{𝔤}(a)$, the adjoint action factors through the quotient togive a well defined endomorphism of $𝔤/𝔥$. The action is givenby

$$b+𝔥\mapsto [a,b]+𝔥,b\in 𝔤.$$

This is the actionalluded to in the first paragraph.