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By Dixon E. T.

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This shows that the orbit V ( \ ) ' z is G^-invariant. Let us take a suitable identification of V4' with the affine space C" so that the point z is identified with the origin. 3 the group G^ acts on V4' linearly and has the point z in the closure of any orbit. So the action of G^ on V4' is a good C*-action. , TJ. We assume that each coordinate function T, is homogeneous of some degree q^> 0. Let R = \]{\)'z. Being an orbit of a unipotent group acting on an affine variety, R is closed in V4". Since it is G^-invariant, it can be given by a system of equations F^ == ...

Then the radical of the stabilizer G^ is of dimension k or less. Proof. 18. 3. Theorem. — Let B be a Borel subgroup ofG and let G act on X x G/B diagonally. Then Pic°(X x G/B) is abundant. In particular, when G is a torus, Pic^X) is abundant. Proof. — Notice first that Pic°(G/B) ^ ^(T), where T is a maximal torus contained in B (see [KKV], p. 65). We want to show that for any point (^^[B]) with reductive stabilizer the isotropy representation homomorphism P(^[B]) ^ Pic^X x G/B) ->^(G^,^])) is surjective.

H, where {HJ,gi are the walls that contain / and { H , } , g j are the walls that do not contain /. ) Since the set of walls is finite, we see that the set of cells is finite. Also it is obvious that any wall is a union of cells. 10. Lemma. — Let F be a cell. For any /, /' e F one has X^/) = X^/'). Denote this set by X^F). Then X^F') C X^F) if F n F' + 0, F + F'. Proof. — We already know that X^) === X^/'). 8 (iii)). This implies that X8^) = X8^) and hence X^) = X^'). Now,_if ^eX^F'), then M\x) ^ 0 for any / e F'.

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