Representation on coordinate rings

In mathematics, a representation on coordinate rings is a representation of a group on coordinate rings of affine varieties.
Let "X" be an affine algebraic variety over an algebraically closed field "k" of characteristic zero with the action of a reductive algebraic group "G". "G" then acts on the coordinate ring formula_1 of "X" as a left regular representation: formula_2. This is a representation of "G" on the coordinate ring of "X".
The most basic case is when "X" is an affine space (that is, "X" is a finite-dimensional representation of "G") and the coordinate ring is a polynomial ring. The most important case is when "X" is a symmetric variety; i.e., the quotient of "G" by a fixed-point subgroup of an involution.
Isotypic decomposition.
Let formula_3 be the sum of all "G"-submodules of formula_1 that are isomorphic to the simple module formula_5; it is called the formula_6-isotypic component of formula_1. Then there is a direct sum decomposition:
where the sum runs over all simple "G"-modules formula_5. The existence of the decomposition follows, for example, from the fact that the group algebra of "G" is semisimple since "G" is reductive.
"X" is called "multiplicity-free" (or spherical variety) if every irreducible representation of "G" appears at most one time in the coordinate ring; i.e., formula_10.
For example, formula_11 is multiplicity-free as formula_12-module. More precisely, given a closed subgroup "H" of "G", define