Fixed-vector subspace

Definition

For a representation $\rho:G\to\operatorname{GL}(V)$ , the fixed-vector subspace, or space of invariants, is

V^G=\{v\in V:\rho(g)v=v\text{ for every }g\in G\}.

For a Lie-algebra representation $d\rho:\mathfrak g\to\mathfrak{gl}(V)$ , it is

V^{\mathfrak g}=\{v\in V:d\rho(X)v=0\text{ for every }X\in\mathfrak g\}.

Both are linear subspaces and subrepresentations carrying the trivial action.

Universal description

The fixed space can be written as an intersection of kernels,

V^G=\bigcap_{g\in G}\ker(\rho(g)-I), \qquad V^{\mathfrak g}=\bigcap_{X\in\mathfrak g}\ker(d\rho(X)).

Equivalently, $V^G$ is naturally the space $\operatorname{Hom}_G(\mathbf1,V)$ of equivariant maps from the trivial one-dimensional representation. Thus $\dim V^G$ is the multiplicity of the trivial representation whenever $V$ is completely reducible.

Compact-group averaging

If $G$ is compact and $V$ is a continuous finite-dimensional representation, normalized Haar measure defines the Reynolds projection

P(v)=\int_G\rho(g)v\,dg.

It satisfies $P^2=P$ and $\operatorname{im}P=V^G$ . This turns the abstract invariant subspace into a computable direct summand.

Group versus Lie algebra

Every group-fixed vector is fixed infinitesimally. If $G$ is connected, then

V^G=V^{\mathfrak g}.

For disconnected $G$ , the equality can fail because $V^{\mathfrak g}$ only records invariance under the identity component. The remaining component group may act nontrivially on $V^{\mathfrak g}$ .

The term “invariant vector” means a fixed vector. It should not be confused with an invariant subspace, whose individual vectors may move within that subspace.

References

Brian C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015, Chapters 4 and 11. Publisher record.
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002, Chapter IV. Publisher record.