Semi-invariant(2)

A common eigenvector of a family of endomorphisms of a vector space or module. If $ G $ is a set of linear mappings of a vector space $ V $ over a field $ K $, a semi-invariant of $ G $ is a vector $ v \in V $, $ v \neq 0 $, such that $$ g v = \chi ( g ) v , g \in G , $$ where $ \chi : \ G \rightarrow K $ is a function, called the weight of the semi-invariant $ v $. A semi-invariant of weight $ 1 $ is also called an invariant. The most frequently considered case is that of a linear group $ G \subset \mathop{\rm GL}\nolimits ( V ) $, in which case $ \chi : \ G \rightarrow K ^{*} $ is a character of $ G $ and may be extended to a polynomial function on $ \mathop{\rm End}\nolimits \ V $. If $ \phi : \ G \rightarrow \mathop{\rm GL}\nolimits (V) $ is a linear representation of a group $ G $ in $ V $, then a semi-invariant of the group $ \phi ( G ) $ is also called a semi-invariant of the representation $ \phi $( cf. also Linear representation, invariant of a). Let $ G $ be a linear algebraic group, $ H $ a closed subgroup of $ G $ and $ \mathfrak h \subset \mathfrak g $ the Lie algebras of these groups. Then there exist a faithful rational linear representation $ \phi : \ G \rightarrow \mathop{\rm GL}\nolimits ( E ) $ and a semi-invariant $ v \in E $ of $ \phi ( H ) $ such that $ H $ and $ \mathfrak h $ are the maximal subsets of $ G $ and $ \mathfrak g $ whose images in $ \mathop{\rm End}\nolimits \ V $ have $ v $ as semi-invariant. This implies that the mapping $ a H \mapsto K \phi ( a ) v $, $ a \in G $, defines an isomorphism of the algebraic homogeneous space $ G/H $ onto the orbit of the straight line $ K v $ in the projective space $ P ( E ) $.

The term semi-invariant of a set $ G \subset \mathop{\rm End}\nolimits \ V $ is sometimes applied to a polynomial function on $ \mathop{\rm End}\nolimits \ V $ which is a semi-invariant of the set of linear mappings $ \eta ( G ) $ of the space $ K [ \mathop{\rm End}\nolimits \ V ] $, where $$ ( \eta ( g ) f \ ) ( X ) = f ( X g ) , $$ $$ g \in G , f \in K [ \mathop{\rm End}\nolimits \ V ] , X \in \mathop{\rm End}\nolimits \ V . $$ If $ G \subset \mathop{\rm GL}\nolimits ( V ) $ is a linear algebraic group and $ \mathfrak g $ is its Lie algebra, then $ G $ has semi-invariants $$ f _{1} \dots f _{n} \in K [ \mathop{\rm End}\nolimits \ V ] $$ of the same weight such that $ G $ and $ \mathfrak g $ are the maximal subsets of $ \mathop{\rm GL}\nolimits (V) $ and $ \mathop{\rm End}\nolimits \ V $ for which $ f _{1} \dots f _{n} $ are semi-invariants (Chevalley's theorem).

References

[1]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2]	J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039
[3]	C. Chevalley, "Théorie des groupes de Lie" , 2 , Hermann (1951) MR0051242 Zbl 0054.01303