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 formula_1 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", formula_2 fixes "o" and thus taking the derivative at "o" gives the map formula_3 By the chain rule,
formula_4
and thus there is a representation:
formula_5
given by
formula_6.
It is called the isotropy representation at "o". For example, if formula_7 is a conjugation action of "G" on itself, then the isotropy representation formula_8 at the identity element "e" is the adjoint representation of formula_9.