A Bott-Borel-Weil Theorem for Diagonal Ind-groups

作者：Dimitrov Ivan^*; Penkov Ivan

来源：Canadian Journal of Mathematics, 2011, 63(6): 1307-1327.

DOI：10.4153/CJM-2011-032-6

A diagonal id-group is a direct limit of classical affine algebraic groups of growing rank under a class of inclusions that contains the inclusion
SL(n) -> SL(2n), M -> ((M)(0) (0)(M))
as a typical special case. If G is a diagonal id-group and B subset of G is a Borel ind-subgroup, we consider the ind-variety G/B and compute the cohomology H(l)(G/B, O(-lambda)) of any G-equivariant line bundle O(-lambda) on G/B. It has been known that, for a generic lambda, all cohomology groups of O(-lambda) vanish, and that a non-generic equivariant line bundle O(-lambda) has at most one nonzero cohomology group. The new result of this paper is a precise description of when H(j)(G/B, O(-lambda)) is nonzero and the proof of the fact that, whenever nonzero, H(j)(G/B, O(-lambda)) is a G-module dual to a highest weight module. The main difficulty is in defining an appropriate analog W(B) of the Weyl group, so that the action of W(B) on weights of G is compatible with the analog of the Demazure "action" of the Weyl group on the cohomology of line bundles. The highest weight corresponding to H(j)(G/B, O(-lambda)) is then computed by a procedure similar to that in the classical Bott-Borel-Weil theorem.

• 出版日期2011-12