Isotropy representation

In differential geometry, the isotropy representation is a natural linear representation of a Lie group, that is acting on a manifold, on the tangent space to a fixed point.

Construction
Given a Lie group action $$(G, \sigma)$$ on a manifold M, if Go is the stabilizer of a point o (isotropy subgroup at o), then, for each g in Go, $$\sigma_g: M \to M$$ fixes o and thus taking the derivative at o gives the map $$(d\sigma_g)_o: T_o M \to T_o M.$$ By the chain rule,
 * $$(d \sigma_{gh})_o = d (\sigma_g \circ \sigma_h)_o = (d \sigma_g)_o \circ (d \sigma_h)_o$$

and thus there is a representation:
 * $$\rho: G_o \to \operatorname{GL}(T_o M)$$

given by
 * $$\rho(g) = (d \sigma_g)_o$$.

It is called the isotropy representation at o. For example, if $$\sigma$$ is a conjugation action of G on itself, then the isotropy representation $$\rho$$ at the identity element e is the adjoint representation of $$G = G_e$$.