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A linear representation (cf. [[Representation of a Lie algebra|Representation of a Lie algebra]])  $  \rho $
of a finite-dimensional semi-simple split Lie algebra  $  \mathfrak g $
over a field  $  k $
of characteristic zero with a split [[Cartan subalgebra|Cartan subalgebra]]  $  \mathfrak t $, 
having the following properties.

1) In the space  $  V $
of  $  \rho $
there is a cyclic vector  $  v $(
i.e.  $  V $
is the smallest  $  \mathfrak g $-
invariant subspace containing  $  v $).

2)  $  \rho ( h) v = \lambda ( h) v $
for all  $  h \in \mathfrak t $, 
where  $  \lambda $
is some fixed linear form on  $  \mathfrak t $
with values in  $  k $.

3) If  $  \alpha _ {1} \dots \alpha _ {r} $
is a system of simple roots, defined by a lexicographical order on the set  $  \Delta $
of all roots of  $  \mathfrak g $
relative to  $  \mathfrak t $(
cf. [[Root system|Root system]]), and if  $  e _ {\alpha _ {i}  } , \mathfrak t _ {\alpha _ {i}  } , h _ {\alpha _ {i}  } $
are the vectors from the Chevalley basis of  $  \mathfrak g $
corresponding to  $  \alpha _ {i} $, 
$  i = 1 \dots r $, 
then  $  \rho ( e _ {\alpha _ {i}  } ) ( v) = 0 $
for all  $  i = 1 \dots r $. 
Thus,  $  \lambda $
is a weight relative to the restriction of  $  \rho $
to  $  \mathfrak t $(
cf. [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]]); it is called a highest weight. The space  $  V $
is called a cyclic  $  \mathfrak g $-
module with highest weight  $  \lambda $
and generator  $  v $, 
and  $  v $
is called a highest weight vector.

There exists for every linear form  $  \lambda $
on  $  \mathfrak t $
a unique, up to equivalence, irreducible representation  $  \rho _  \lambda  $
of  $  \mathfrak g $
with highest weight  $  \lambda $. 
The  $  \mathfrak g $-
module  $  V ( \lambda ) $
determined by  $  \rho _  \lambda  $
is a direct sum of weight subspaces relative to the restriction of  $  \rho _  \lambda  $
to  $  \mathfrak t $. 
Their weights have the form

$$ 
\lambda -
\sum _ {i = 1 } ^ { r } 
n _ {i} \alpha _ {i} ,
$$

where the  $  n _ {i} $
are non-negative integers. The weight subspace  $  V _  \mu  ( \lambda ) $
of weight  $  \mu $
is finite-dimensional, spanned over  $  k $
by vectors of the form

$$ 
( \rho _  \lambda  ( f _ {\alpha _ {i _ {1}  } } ) \dots
\rho _  \lambda  ( f _ {\alpha _ {i _ {s}  } } ) ) ( v ) ,
$$

and for any  $  h \in \mathfrak t $
the restriction of  $  \rho _  \lambda  ( h) $
to  $  V _  \mu  ( \lambda ) $
is the operator of scalar multiplication by  $  \mu ( h) $. 
The space  $  V _  \lambda  ( \lambda ) $
is one-dimensional; the weight  $  \lambda $
is the only highest weight of  $  \rho _  \lambda  $
and can be characterized as the unique weight of the  $  \mathfrak t $-
module  $  V ( \lambda ) $
such that any other weight has the form

$$ 
\lambda -
\sum _ {i = 1 } ^ { r } 
n _ {i} \alpha _ {i} ,
$$

where the  $  n _ {i} $
are non-negative integers.

A representation  $  \rho _  \lambda  $
is finite-dimensional if and only if  $  \lambda $
is a dominant linear form on  $  \mathfrak t $, 
i.e.  $  \lambda ( h _ {\alpha _ {i}  } ) $
is a non-negative integer for  $  i = 1 \dots r $. 
Every irreducible finite-dimensional linear representation of  $  \mathfrak g $
has the form  $  \rho _  \lambda  $
for some dominant linear form  $  \lambda $
on  $  \mathfrak t $(
hence all such representations are classified, up to equivalence, by the dominant linear forms on  $  \mathfrak t $). 
The set of all weights of a finite-dimensional representation  $  \rho _  \lambda  $
relative to  $  \mathfrak t $
is invariant relative to the [[Weyl group|Weyl group]] of  $  \mathfrak g $(
regarded as a group of linear transformations of  $  \mathfrak t $), 
and if weights  $  \mu $
and  $  \gamma $
belong to one orbit of the Weyl group, then the dimensions of the spaces  $  V _  \mu  ( \lambda ) $
and  $  V _  \gamma  ( \lambda ) $
are equal. For every weight  $  \mu $
and every root  $  \alpha \in \Delta $
the number  $  \mu ( h _  \alpha  ) $
is an integer; if, moreover,  $  \mu + \alpha $
is also a weight, then

$$ 
\rho ( e _  \alpha  )
( V _  \mu  ( \lambda ))  \neq  0
$$

(here  $  h _  \alpha  $
is the element in  $  \mathfrak t $
corresponding to  $  \alpha $
and  $  e _  \alpha  $
is the root vector of  $  \alpha $).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,   "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> , ''Theórie des algèbres de Lie. Topologie des groupes de Lie'' , ''Sem. S. Lie'' , '''Ie année 1954–1955''' , Secr. Math. Univ. Paris  (1955)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D.P. Zhelobenko,   "Compact Lie groups and their representations" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Cartan,   "Les tenseurs irréductibles et les groupes linéaires simples et semi-simples"  ''Bull. Sci. Math.'' , '''49'''  (1925)  pp. 130–152</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  Harish-Chandra,   "On some applications of the universal enveloping algebra of a semisimple Lie algebra"  ''Trans. Amer. Math. Soc.'' , '''70'''  (1951)  pp. 28–96</TD></TR></table>