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)

References

[a1]	S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4