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 $$k[X]$$ of X as a left regular representation: $$(g \cdot f)(x) = f(g^{-1} x)$$. 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 $$k[X]_{(\lambda)}$$ be the sum of all G-submodules of $$k[X]$$ that are isomorphic to the simple module $$V^{\lambda}$$; it is called the $$\lambda$$-isotypic component of $$k[X]$$. Then there is a direct sum decomposition:
 * $$k[X] = \bigoplus_{\lambda} k[X]_{(\lambda)}$$

where the sum runs over all simple G-modules $$V^{\lambda}$$. 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., $$\operatorname{dim} k[X]_{(\lambda)} \le \operatorname{dim} V^{\lambda}$$. For example, $$G$$ is multiplicity-free as $$G \times G$$-module. More precisely, given a closed subgroup H of G, define
 * $$\phi_{\lambda}: V^{{\lambda}*} \otimes (V^{\lambda})^H \to k[G/H]_{(\lambda)}$$

by setting $$\phi_{\lambda}(\alpha \otimes v)(gH) = \langle \alpha, g \cdot v \rangle$$ and then extending $$\phi_{\lambda}$$ by linearity. The functions in the image of $$\phi_{\lambda}$$ are usually called matrix coefficients. Then there is a direct sum decomposition of $$G \times N$$-modules (N the normalizer of H)
 * $$k[G/H] = \bigoplus_{\lambda} \phi_{\lambda}(V^{{\lambda}*} \otimes (V^{\lambda})^H)$$,

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 $$G \times N$$-submodules of $$k[G/H]_{(\lambda)}$$. We can assume $$V^{\lambda} = W$$. Let $$\delta_1$$ be the linear functional of W such that $$\delta_1(w) = w(1)$$. Then $$w(gH) = \phi_{\lambda}(\delta_1 \otimes w)(gH)$$. That is, the image of $$\phi_{\lambda}$$ contains $$k[G/H]_{(\lambda)}$$ and the opposite inclusion holds since $$\phi_{\lambda}$$ is equivariant.

Examples

 * Let $$v_{\lambda} \in V^{\lambda}$$ be a B-eigenvector and X the closure of the orbit $$G \cdot v_\lambda$$. It is an affine variety called the highest weight vector variety by Vinberg–Popov. It is multiplicity-free.