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Representation on coordinate rings ：ウィキペディア英語版

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'' is not assumed to be connected so that the results apply to finite groups.〕 ''G'' then acts on the coordinate ring

k()

of ''X'' as a left regular representation:

(g \cdot f)(x) = f(g^ 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()_

be the sum of all ''G''-submodules of

k()

that are isomorphic to the simple module

V^

; it is called the

\lambda

-isotypic component of

k()

. Then there is a direct sum decomposition:
:

k() = \bigoplus_ k()_

where the sum runs over all simple ''G''-modules

V^

. 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.,

)_ \le \operatorname V^

.
For example,

G

is multiplicity-free as

G \times G

-module. More precisely, given a closed subgroup ''H'' of ''G'', define
:

\phi_: V^*} \otimes (V^)^H \to k()_

by setting

\phi_(\alpha \otimes v)(gH) = \langle \alpha, g \cdot v \rangle

and then extending

\phi_

by linearity. The functions in the image of

\phi_

are usually called matrix coefficients. Then there is a direct sum decomposition of

G \times N

-modules (''N'' the normalizer of ''H'')
:

) = \bigoplus_ \phi_(V^*} \otimes (V^)^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()_

. We can assume

V^ = W

. Let

\delta_1

be the linear functional of ''W'' such that

\delta_1(w) = w(1)

. Then

w(gH) = \phi_(\delta_1 \otimes w)(gH)

.
That is, the image of

\phi_

contains

k()_

and the opposite inclusion holds since

\phi_

is equivariant.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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