Isotropy representation
# 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 $G_o$ is the stabilizer of a point o (isotropy subgroup at o), then, for each g in $G_o,$ $\sigma _{g}:M\to M$ 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 $G=G_{e}$.

