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The natural linear representation of the [[Isotropy group|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|Adjoint representation of a Lie group]])  $   \mathop{\rm Ad} _ {G} $
of  $  G $
to  $  G _ {x} $.

If a [[Homogeneous space|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 [[#References|[3]]]). In this case, the isotropy representation is faithful (cf. [[Faithful representation|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|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Sulanke,   P. Wintgen,   "Differentialgeometrie und Faserbündel" , Deutsch. Verlag Wissenschaft.  (1972)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Helgason,   "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Cartan,   "La théorie des groupes finis et continus et l'analyse situs" , Gauthier-Villars  (1930)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''2''' , Interscience  (1969)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Helgason,   "Groups and geometric analysis" , Acad. Press  (1984)  pp. Chapt. II, Sect. 4</TD></TR></table>