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Representation on coordinate rings

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

k [ X ] = ⨁ λ k [ X ] ( λ ) {\displaystyle k=\bigoplus _{\lambda }k_{(\lambda )}}

where the sum runs over all simple G-modules V λ {\displaystyle 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 ] ( λ ) ≤ dim ⁡ V λ {\displaystyle \operatorname {dim} k_{(\lambda )}\leq \operatorname {dim} V^{\lambda }} . For example, G {\displaystyle G} is multiplicity-free as G × G {\displaystyle G\times G} -module. More precisely, given a closed subgroup H of G, define

ϕ λ : V λ ∗ ⊗ ( V λ ) H → k [ G / H ] ( λ ) {\displaystyle \phi _{\lambda }:V^{{\lambda }*}\otimes (V^{\lambda })^{H}\to k_{(\lambda )}}

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

k [ G / H ] = ⨁ λ ϕ λ ( V λ ∗ ⊗ ( V λ ) H ) {\displaystyle k=\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 × N {\displaystyle G\times N} -submodules of k [ G / H ] ( λ ) {\displaystyle k_{(\lambda )}} . We can assume V λ = W {\displaystyle V^{\lambda }=W} . Let δ 1 {\displaystyle \delta _{1}} be the linear functional of W such that δ 1 ( w ) = w ( 1 ) {\displaystyle \delta _{1}(w)=w(1)} . Then w ( g H ) = ϕ λ ( δ 1 ⊗ w ) ( g H ) {\displaystyle w(gH)=\phi _{\lambda }(\delta _{1}\otimes w)(gH)} . That is, the image of ϕ λ {\displaystyle \phi _{\lambda }} contains k [ G / H ] ( λ ) {\displaystyle k_{(\lambda )}} and the opposite inclusion holds since ϕ λ {\displaystyle \phi _{\lambda }} is equivariant.

Examples

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

The Kostant–Rallis situation

This section needs expansion. You can help by adding to it. (June 2014)

See also

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

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