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 [math]\displaystyle{ (G, \sigma) }[/math] 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, [math]\displaystyle{ \sigma_g: M \to M }[/math] fixes o and thus taking the derivative at o gives the map [math]\displaystyle{ (d\sigma_g)_o: T_o M \to T_o M. }[/math] By the chain rule,

[math]\displaystyle{ (d \sigma_{gh})_o = d (\sigma_g \circ \sigma_h)_o = (d \sigma_g)_o \circ (d \sigma_h)_o }[/math]

and thus there is a representation:

[math]\displaystyle{ \rho: G_o \to \operatorname{GL}(T_o M) }[/math]

given by

[math]\displaystyle{ \rho(g) = (d \sigma_g)_o }[/math].

It is called the isotropy representation at o. For example, if [math]\displaystyle{ \sigma }[/math] is a conjugation action of G on itself, then the isotropy representation [math]\displaystyle{ \rho }[/math] at the identity element e is the adjoint representation of [math]\displaystyle{ G = G_e }[/math].

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.