The name is absent

SOME OBSERVATIONS ON THE SIEGEL FORMULA

2. In general, let G denote a reductive group, X a vector space, and
p: G → Aut(JV) a rational representation of G in X, all defined over k.
Then, the coordinate ring Ω[X] of X over the universal domain Ω is a graded
ring, and G operates on Ω[X] via p as a group of degree-preserving auto-
morphisms. It is well known (cf. [P]) that every ideal J of the ring of in-
variants Ω[X]^σ has the following property:

J∙Ω[X] ∩Ω[X]^ff = J.

Consequently, the ring Ω[X]^g is finitely generated over Ω and, if I(X)
denotes the affine variety with Ω[X]° as its coordinate ring and
f∙.X → I(X) the corresponding morphism,/is surjective. We observe that
Ω[X]^g is generated over Ω by G-invariant polynomials in k[X]. Conse-
quently, we may assume that I(X) and f are defined over k. We observe
also that, if Ω[X]^g is generated over Ω by algebraically independent,
G-invariant polynomials, we may even assume that they are homogeneous.
In the following definition, we shall assume that G is connected and semi-
simple :

Definition. We say that G operates very nicely on X if (1) I(X) is
an affine space, (2) every fiber f~ ^l(i) contains a G-orbit U(i) of dimension
equal to dim(X) — dim(I(X)) such that f~^^l(i) — U(i) is of codimension
≥ 2, (3) the morphism f is “submersive” at every point of the Zariski open
subset

χ¹ = U ɑ(θ
i s I (X)

of X, (4) the stabilizer subgroup of G at every point x of X' is a semi-
direct product of a connected, semisimple group and a unipotent group.

Theorem 2. Let G denote a connected, semisimple group defined over
к such that the absolutely simple factors of its universal covering group
are of type А-D. Then, for any absolutely admissible representation
p:G → Aut(X) defined over k, G operates very nicely on X.

The proof of this theorem goes as follows: First of all, if G* → G is the
universal covering of G and if p* is the product of p and G* → G, the
stabilizer subgroup of G at any point x of X is the image by G* → G of
the stabilizer subgroup of G* at x. Therefore, we may replace G by G*
and p by p*. We observe also that we are concerned with “absolute prop-
erties.” Therefore, we may assume that G is simply connected and fc-split.
Then, using the lemma in the previous section, we can reduce this case
to the case where G is simple. The rest depends on the case-by-case exami-

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