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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 on a manifold M, if Go is the stabilizer of a point o (isotropy subgroup at o), then, for each g in Go, fixes o and thus taking the derivative at o gives the map By the chain rule,
and thus there is a representation:
It is called the isotropy representation at o. For example, if is a conjugation action of G on itself, then the isotropy representation at the identity element e is the adjoint representation of .