# Isotropy representation

The natural linear representation of the isotropy group of a differentiable transformation group in the tangent space to the underlying manifold. If $G$ is a group of differentiable transformations on a manifold $M$ and $G _ {x}$ is the corresponding isotropy subgroup at the point $x \in M$, then the isotropy representation $\mathop{\rm Is} _ {x} : G _ {x} \rightarrow \mathop{\rm GL} ( T _ {x} M )$ associates with each $h \in G _ {x}$ the differential $\mathop{\rm Is} _ {x} ( h) = d h _ {x}$ of the transformation $h$ at $x$. The image of the isotropy representation, $\mathop{\rm Is} _ {x} ( G _ {x} )$, is called the linear isotropy group at $x$. If $G$ is a Lie group with a countable base acting smoothly and transitively on $M$, then the tangent space $T _ {x} M$ can naturally be identified with the space $\mathfrak g / \mathfrak g _ {x}$, where $\mathfrak g \supset \mathfrak g _ {x}$ are the Lie algebras of the groups $G \supset G _ {x}$. Furthermore, the isotropy representation $\mathop{\rm Is} _ {x}$ is now identified with the representation $G _ {x} \rightarrow \mathop{\rm GL} ( \mathfrak g / \mathfrak g _ {x} )$ induced by the restriction of the adjoint representation (cf. Adjoint representation of a Lie group) $\mathop{\rm Ad} _ {G}$ of $G$ to $G _ {x}$.

If a homogeneous space $M$ is reductive, that is, if $\mathfrak g = \mathfrak g _ {x} \dot{+} m$, where $m$ is an invariant subspace with respect to $\mathop{\rm Ad} _ {G} ( G _ {x} )$, then $T _ {x} M$ can be identified with $m$, while $\mathop{\rm Is} _ {x}$ can be identified with the representation $h \mapsto ( \mathop{\rm Ad} _ {G} h ) \mid _ {m}$ (see [3]). In this case, the isotropy representation is faithful (cf. Faithful representation) if $G$ acts effectively.

The isotropy representation and linear isotropy group play an important role in the study of invariant objects on homogeneous spaces (cf. Invariant object). The invariant tensor fields on a homogeneous space $M$ are in one-to-one correspondence with the tensors on $T _ {x} M$ that are invariant with respect to the isotropy representation. In particular, $M$ has an invariant Riemannian metric if and only if $T _ {x} M$ has a Euclidean metric that is invariant under the linear isotropy group. There exists on the homogeneous space $M$ a positive invariant measure if and only if $| \mathop{\rm det} A | = 1$ for all $A \in \mathop{\rm Is} _ {x} ( G _ {x} )$. A homogeneous space has an invariant orientation if and only if $\mathop{\rm det} A > 0$ for all $A \in \mathop{\rm Is} _ {x} ( G _ {x} )$. The invariant linear connections on $M$ are in one-to-one correspondence with the linear mappings $\Lambda : \mathfrak g \rightarrow \mathfrak g \mathfrak l ( T _ {x} M )$ with the following properties:

$$\left . \Lambda \right | _ {\mathfrak g _ {x} } = \ ( d \mathop{\rm Is} _ {x} ) _ {e} ,$$

$$\Lambda ( ( \mathop{\rm Ad} h ) X) = \mathop{\rm Is} _ {x} ( h) \Lambda ( X) \mathop{\rm Is} _ {x} ( h) ^ {- 1} \ ( h \in G _ {x} ) .$$

A generalization of the concept of the isotropy representation is the concept of the isotropy representation of order $r$. This is a homomorphism $h \rightarrow j _ {x} ^ {r} h$ of the group $G _ {x}$ into the group $L ^ {r} ( T _ {x} M )$ of invertible $r$-jets of diffeomorphisms of the space $T _ {x} M$ taking the zero to itself. This concept is used in the study of invariant objects of higher orders.

#### References

 [1] R. Sulanke, P. Wintgen, "Differentialgeometrie und Faserbündel" , Deutsch. Verlag Wissenschaft. (1972) [2] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) [3] P.K. Rashevskii, "On the geometry of homogeneous spaces" , Proc. Sem. Vektor. Tenzor. Anal. i Prilozh. k Geom., Mekh. i Fiz. , 9 , Moscow-Leningrad (1952) pp. 49–74 (In Russian) [4] E. Cartan, "La théorie des groupes finis et continus et l'analyse situs" , Gauthier-Villars (1930) [5] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969)