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

${\displaystyle k[X]=\bigoplus _{\lambda }k[X]_{(\lambda )}}$

where the sum runs over all simple G-modules ${\displaystyle V^{\lambda }}$. 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 ^[2] ) if every irreducible representation of G appears at most one time in the coordinate ring; i.e., ${\displaystyle \operatorname {dim} k[X]_{(\lambda )}\leq \operatorname {dim} V^{\lambda }}$. For example, ${\displaystyle G}$ is multiplicity-free as ${\displaystyle G\times G}$-module. More precisely, given a closed subgroup H of G, define

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

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

${\displaystyle k[G/H]=\bigoplus _{\lambda }\phi _{\lambda }(V^{{\lambda }*}\otimes (V^{\lambda })^{H})}$,

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

Examples

• Let ${\displaystyle v_{\lambda }\in V^{\lambda }}$ be a B-eigenvector and X the closure of the orbit ${\displaystyle G\cdot v_{\lambda }}$. 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

• 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.