Why is an invariant vector subspace sometimes called a representation? For example in Lie algebras, say su(3), the subspace characterized by the highest weight (1,0) is an irreducible representation of dimension 3 of su(3).
However, a representation of a Lie algebra is a Lie algebra homomorphism from the algebra to a subspace of the so called general linear algebra of some vector space. Or in other words, the representation is a map that assigns elements of the algebra to elements of the set of linear endomorphisms of some vector space.
In the previous example, the subspace (1,0) is a subspace in which the action of the endomorphisms maps its elements into themselves. So by the definition, the irreducible representation should be the mapping that associates the endomorphisms to the elements of the algebra and not the space in which they act.