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

[math]\displaystyle{ k[X] = \bigoplus_{\lambda} k[X]_{(\lambda)} }[/math]

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

[math]\displaystyle{ \phi_{\lambda}: V^{{\lambda}*} \otimes (V^{\lambda})^H \to k[G/H]_{(\lambda)} }[/math]

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

[math]\displaystyle{ k[G/H] = \bigoplus_{\lambda} \phi_{\lambda}(V^{{\lambda}*} \otimes (V^{\lambda})^H) }[/math],

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

Examples

• Let [math]\displaystyle{ v_{\lambda} \in V^{\lambda} }[/math] be a B-eigenvector and X the closure of the orbit [math]\displaystyle{ G \cdot v_\lambda }[/math]. 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) (in de). Symmetry, Representations, and Invariants. doi:10.1007/978-0-387-79852-3. ISBN 978-0-387-79852-3. OCLC 699068818.