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.^[1] 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^[2]) if every irreducible representation of G appears at most one time in the coordinate ring; i.e., $\operatorname {dim}k[X]_{{(\lambda )}}\leq \operatorname {dim}V^{{\lambda }}$. 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.

See also

• Algebra representation

Notes

1. G is not assumed to be connected so that the results apply to finite groups.
2. Goodman–Wallach 2009, Remark 12.2.2.

References

• Roe Goodman, Nolan R. Wallach, Symmetry, Representations, and Invariants (2009)