Invariant Valuations

This chapter plays a significant, but auxiliary, rôle in the general context of our survey. We investigate the set of

G

-invariant valuations of the function field of a

G

-variety. We have seen in Chap. 3 that

G

-valuations are of importance in the embedding theory, because they provide a material for constructing combinatorial objects (colored data) that describe equivariant embeddings.

Remarkably, a

G

-valuation of a given

G

-field is uniquely determined by its restriction to the multiplicative group of

B

-eigenfunctions, the latter being a direct product of the weight lattice and of the multiplicative group of

B

-invariant functions. Thus a

G

-valuation is essentially a pair composed by a linear functional on the weight lattice and by a valuation of the field of

B

-invariants. Under these identifications, we prove in §20 that the set of

G

-valuations is a union of convex polyhedral cones in certain half-spaces.

The common face of these valuation cones is formed by those valuations, called central, that vanish on

B

-invariant functions. The central valuation cone controls the situation “over the field of

B

-invariant functions”. For instance, its linear part determines the unity component of the group of

G

-automorphisms acting identically on

B

-invariants.

This cone has another remarkable property: it is a fundamental chamber of a crystallographic reflection group called the little Weyl group of a

G

-variety. This group is defined in §22 as the Galois group of a certain symplectic covering of the cotangent bundle constructed in terms of the moment map. The little Weyl group is linked with the central valuation cone via the invariant collective motion on the cotangent variety, which is studied in §23.

For practical applications, we must be able to compute the set of

G

-valuations. For central valuations, it suffices to know the little Weyl group. In §24 we describe the “method of formal curves” for computing

G

-valuations on a homogeneous space. Informally, one computes the order of functions at infinity along a formal curve approaching a boundary

G

-divisor.

Most of the results of this chapter are due to D. Luna and Th. Vust, M. Brion, F. Pauer, and F. Knop. We follow [D. Luna, Th. Vust,

Plongements d’espaces homogènes

, Comment. Math. Helv.

58

(

1983

), no. 2, 186–245], [F. Knop,

Über Bewertungen, welche unter einer reductiven Gruppe invariant sind

, Math. Ann.

295

(

1993

), no. 2, 333–363], [F. Knop,

The asymptotic behavior of invariant collective motion

, Invent. Math.

116

(

1994

), 309–328] in our exposition.