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 ${\displaystyle (G,\sigma )}$ 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,

${\displaystyle (d\sigma _{gh})_{o}=d(\sigma _{g}\circ \sigma _{h})_{o}=(d\sigma _{g})_{o}\circ (d\sigma _{h})_{o}}$

and thus there is a representation:

${\displaystyle \rho :G_{o}\to \operatorname {GL} (T_{o}M)}$

given by

${\displaystyle \rho (g)=(d\sigma _{g})_{o}}$.

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 ${\displaystyle G=G_{e}}$.

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.