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 , σ ) {\displaystyle (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, σ g : M → M {\displaystyle \sigma _{g}:M\to M} fixes o and thus taking the derivative at o gives the map ( d σ g ) o : T o M → T o M . {\displaystyle (d\sigma _{g})_{o}:T_{o}M\to T_{o}M.} By the chain rule,
( d σ g h ) o = d ( σ g ∘ σ h ) o = ( d σ g ) o ∘ ( d σ h ) o {\displaystyle (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:
ρ : G o → GL ( T o M ) {\displaystyle \rho :G_{o}\to \operatorname {GL} (T_{o}M)}
given by
ρ ( g ) = ( d σ g ) o {\displaystyle \rho (g)=(d\sigma _{g})_{o}} . It is called the isotropy representation at o. For example, if σ {\displaystyle \sigma } is a conjugation action of G on itself, then the isotropy representation ρ {\displaystyle \rho } at the identity element e is the adjoint representation of G = G e {\displaystyle G=G_{e}} .
References http://www.math.toronto.edu/karshon/grad/2009-10/2010-01-11.pdf https://www.encyclopediaofmath.org/index.php/Isotropy_representation Kobayashi, Shoshichi; Nomizu, Katsumi (1996). Foundations of Differential Geometry, Vol. 1 (New ed.). Wiley-Interscience. ISBN 0-471-15733-3.
