- Source: 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
]
{\displaystyle k[X]}
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[X]_{(\lambda )}}
be the sum of all G-submodules of
k
[
X
]
{\displaystyle k[X]}
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[X]}
. Then there is a direct sum decomposition:
k
[
X
]
=
⨁
λ
k
[
X
]
(
λ
)
{\displaystyle k[X]=\bigoplus _{\lambda }k[X]_{(\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[X]_{(\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[G/H]_{(\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[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
×
N
{\displaystyle G\times N}
-submodules of
k
[
G
/
H
]
(
λ
)
{\displaystyle k[G/H]_{(\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[G/H]_{(\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
See also
Algebra representation
Notes
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.
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