On the concept of the rank of a spherical homogeneous space

Abstract

Let G be a connected reductive complex linear algebraic group, H a closed subgroup of it, and B c G a Borel subgroup. We consider the multiplicative subgroup of C(G) consisting of the non-zero rational functions f on G such that f{gh) = f(g) and f(b~1g) = x(b)-f(g), where g e G, h e H, b e B, and χ: B -> C* is a character depending on /. Let ;/'(G, H) be the factor group of it by the subgroup of constant functions. The group ;fi(G, H) was introduced by Luna and Vust in (1) in connection with the problem of a combinatorial classification of open equivariant embeddings of the homogeneous space G/H. The homogeneous space G/H is said to be spherical if B has an open orbit on G/H. It is easy to see that rank J'(G, H) C* is a character. The homogeneous space G/H is again spherical, and is equivariantly embedded in C for a suitable choice of φ. It follows easily from Lemma 1 that the rank of G/H is greater by 1 than the rank of G/H. On the other hand, if A' = A' X {2 £ C* | | ζ | = 1 }, then dim K\G/H = dim K\G/H+ 1. Therefore, we assume henceforth without loss of generality that G/H is a quasi-affine variety equivariantly embedded in C. We decompose the algebra of regular functions on G/H into a sum of irreducible G-modules Kλ, where λ is a highest weight and runs through some semigroup © of dominating weights (since G/H is spherical, there are no multiplicitie s in the decompositon; see (3)).