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:

$k[X]=\underset{\lambda }{⨁}k[X{]}_{(\lambda )}$

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., $\dim k[X{]}_{(\lambda )}\le \dim V^{\lambda }$. For example, $G$ is multiplicity-free as $G\times G$-module. More precisely, given a closed subgroup H of G, define

${\varphi }_{\lambda }:V^{\lambda *}\otimes (V^{\lambda }{)}^H\to k[G/H{]}_{(\lambda )}$

by setting ${\varphi }_{\lambda }(\alpha \otimes v)(gH)=⟨\alpha ,g\cdot v⟩$ and then extending ${\varphi }_{\lambda }$ by linearity. The functions in the image of ${\varphi }_{\lambda }$ are usually called matrix coefficients. Then there is a direct sum decomposition of $G\times N$-modules (N the normalizer of H)

$k[G/H]=\underset{\lambda }{⨁}{\varphi }_{\lambda }(V^{\lambda *}\otimes (V^{\lambda }{)}^H)$,

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 $w(gH)={\varphi }_{\lambda }({\delta }_1\otimes w)(gH)$. That is, the image of ${\varphi }_{\lambda }$ contains $k[G/H{]}_{(\lambda )}$ and the opposite inclusion holds since ${\varphi }_{\lambda }$ 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

References

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