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 of X as a left regular representation: . 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.
Let be the sum of all G-submodules of that are isomorphic to the simple module ; it is called the -isotypic component of . Then there is a direct sum decomposition:
where the sum runs over all simple G-modules . 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., . For example, is multiplicity-free as -module. More precisely, given a closed subgroup H of G, define
by setting and then extending by linearity. The functions in the image of are usually called matrix coefficients. Then there is a direct sum decomposition of -modules (N the normalizer of H)
which is an algebraic version of the Peter–Weyl theorem (and in fact the analytic version is an immediate consequence.) Proof: let W be a simple -submodules of . We can assume . Let be the linear functional of W such that . Then . That is, the image of contains and the opposite inclusion holds since is equivariant.
- Let be a B-eigenvector and X the closure of the orbit . It is an affine variety called the highest weight vector variety by Vinberg–Popov. It is multiplicity-free.
The Kostant–Rallis situation
- G is not assumed to be connected so that the results apply to finite groups.
- Goodman–Wallach 2009, Remark 12.2.2.
- Roe Goodman, Nolan R. Wallach, Symmetry, Representations, and Invariants (2009)