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., $\operatorname {dim} k[X]_{(\lambda )}\leq \operatorname {dim} V^{\lambda }$. 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

References

• Goodman, Roe; Wallach, Nolan R. (2009). Symmetry, Representations, and Invariants (in German). doi:10.1007/978-0-387-79852-3. ISBN 978-0-387-79852-3. OCLC 699068818.