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Parshin A.N., Shafarevich I.R., (eds.) Springer, Popov, Vinberg. Algebraic geometry IV. Linear algebraic groups. Invariant theory (Enc.Math.55, Springer)(T)(287s).djvu |
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rational invariants separates the orbits &t, <S2 if it contains an invariant that
separates these orbits...
The specialness of SLn and Spn then follows from the lack of nontrivial
"forms" of the objects whose groups of automorphisms they are (Serre [1975])...
In geometric language this means that a birational description of the action
of G on X reduces to a birational description of the action of N(H)/H on the
union of the irreducible components of XH that intersect Xo (see 2.8)...
It therefore follows from
Katsylo's theorem that the field of invariants of any projective representation
of SL2 is rational...
(In the context of Theorem 3.5, there corre-
corresponds to the homomorphism n a G-equivariant closed embedding of a variety
X into the space of a linear representation of G; see Theorem 1.5.)
To keep the notation simple we will assume the algebra A itself has a G-
invariant grading A = (J)"=o An and Ao = k...
Such subgroups of (not necessarily reductive) algebraic
groups are called observable (Bialynicki-Birula, Hochschild, and Mostow
[1963])...
A finitely generated graded algebra A without zero-divisors is called a Cohen-
Macaulay algebra if either of the following equivalent conditions is satisfied:
1) There exists a system of parameters {tly..., td} such that A is a free
k[tu..., rj-module...
If
G is connected, integration over K by means of Weyl's integration formula
(Weyl [1925-26], Adams [1969], Zhelobenko [1970]) can be replaced by inte-
integration with a suitable weight over a maximal torus T of K...
Since this action is transitive, a covariant G/H ->¦ W is uniquely defined by the
image of the point eH...
A more visual necessary
condition is that all of the orbits must be closed and, if X is irreducible, have the
same dimension (cf...
We denote it by X/G, and the morphism X -* X/G defined by the
embedding /c[X]G ci> fe[X] by nXIG...
¦* It suffices to prove this for the principal open subsets U = (X/G)f =
{y e X/G\f(y) # 0}(/e fc[X]G) constituting a base for the topology of X/G...
The pair
(Y, n) is a categorical quotient for the action G:X if and only if conditions A) and
B') are satisfied...
We can try to apply the same idea used in the above proof to the action of
any reductive group...
Since Q = Xf, where / is the
discriminant of a binary form, there exists a geometric quotient Q/G...
A semisimple element he q will be called rational if for some faithful (and
therefore for any) linear representation jR of G the eigenvalues of the linear
operator dR(h) are rational...
This property can be reformulated as follows:
for any E, e g and u e V+{h)°,
?,u € V+(h) implies ? e p(Ji)...
It can be shown that if the group G is simple, then all irreducible components
of S are mapped onto the same subvariety of codimension 2 in 91...
Suppose Y is another affine variety on which G acts and nY/G: Y -> Y/G is the
corresponding categorical quotient...
Since cp is excellent, it follows
from the definition of change of base that q> effects an equivariant analytic
isomorphism of the space n^QtS)/G(U) with the invariant complex neighborhood
Gx
nxjo(Y) °f ^e orbit Gx...
In 5.3 we proved the Hilbert-Mumford criterion,
which says that any point of a finite-dimensional G -module whose orbit con-
contains zero in its closure can be "steered" to zero by means of a suitable one-
dimensional torus of G...
We will see in 6.13 that certain unions of the remaining strata
can be described in a similar way...
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