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