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.

Given a Lie group action ${\displaystyle (G,\sigma )}$ on a manifold M, if G_o is the stabilizer of a point o (isotropy subgroup at o), then, for each g in G_o, ${\displaystyle \sigma _{g}:M\to M}$ fixes o and thus taking the derivative at o gives the map ${\displaystyle (d\sigma _{g})_{o}:T_{o}M\to T_{o}M.}$ By the chain rule,

and thus there is a representation:

given by

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