Isotropy representation

Short description: Linear representation of a group on the tangent space to a fixed point of the group.

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:

$\rho :G_o\to \mathrm{GL}(T_{o}M)$

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 $G=G_e$.

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.