Abstract view
A BottBorelWeil Theorem for Diagonal Indgroups


Published:20110517
Printed: Dec 2011
Ivan Dimitrov,
Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6
Ivan Penkov,
Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany
Abstract
A diagonal indgroup is a direct limit of classical affine algebraic
groups of growing rank under a class of
inclusions that contains the inclusion
$$
SL(n)\to SL(2n), \quad
M\mapsto \begin{pmatrix}M & 0 \\ 0 & M \end{pmatrix}
$$
as a typical special case. If $G$ is a diagonal indgroup and
$B\subset G$ is a Borel indsubgroup,
we consider the indvariety $G/B$ and compute the cohomology
$H^\ell(G/B,\mathcal{O}_{\lambda})$
of any $G$equivariant line bundle $\mathcal{O}_{\lambda}$ on
$G/B$. It has been known that, for a generic $\lambda$,
all cohomology groups of $\mathcal{O}_{\lambda}$ vanish, and that a
nongeneric equivariant
line bundle $\mathcal{O}_{\lambda}$ has at most one
nonzero cohomology group. The new result of this paper is a
precise description of when
$H^j(G/B,\mathcal{O}_{\lambda})$ is nonzero and the proof of the fact
that, whenever nonzero,
$H^j(G/B, \mathcal{O}_{\lambda})$ 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 $H^j(G/B,
\mathcal{O}_{\lambda})$ is then computed
by a procedure similar to that in the classical BottBorelWeil theorem.