31 research outputs found
Global branching laws by global Okounkov bodies
Let be a complex semisimple group, and let be a
semisimple subgroup. We show that the branching cone of the pair ,
which (asymptotically) parametrizes all pairs of irreducible
finite-dimensional -representations which occur as subrepresentations of
a finite-dimensional irreducible -representation , can be identified
with the pseudo-effective cone, \overline{\mbox{Eff}}(Y), of some GIT
quotient of the flag variety of the group . Moreover, we prove
that the quotient is a Mori dream space.
As a consequence, the global Okounkov body of , with respect
to some admissible flag of subvarieties of , is fibred over the branching
cone of , and the fibre over a point
carries information about (the asymptotics of) the multiplicity of in .
Using the global Okounkov body , we easily derive a
multi-dimensional generalization of Okounkov's result about the log-concavity
of asymptotic multiplicities
Okounkov bodies for ample line bundles
Let be an ample line bundle over a nonsingular
complex projective variety . We construct an admissable flag of subvarieties for which the associated
Okounkov body for is a rational polytope
Okounkov bodies for ample line bundles with applications to multiplicities for group representations
Let be an ample line bundle over a complex normal
projective variety . We construct a flag of subvarieties for which the associated Okounkov body for
is a rational polytope. In the case when is a homogeneous
surface, and the pseudoeffective cone of is rational polyhedral, we also
show that the global Okounkov body is a rational polyhedral cone if the flag of
subvarieties is suitably chosen. Finally, we provide an application to the
asymptotic study of group representations.Comment: supersedes the preprint arXiv:1007.191
Global Okounkov bodies for Bott-Samelson varieties
We use the theory of Mori dream spaces to prove that the global Okounkov body
of a Bott-Samelson variety with respect to a natural flag of subvarieties is
rational polyhedral. In fact, we prove more generally that this holds for any
Mori dream space which admits a flag of Mori dream spaces satisfying a certain
regularity condition. As a corollary, Okounkov bodies of effective line bundles
over Schubert varieties are shown to be rational polyhedral. In particular, it
follows that the global Okounkov body of a flag variety is rational
polyhedral.
As an application we show that the asymptotic behaviour of dimensions of
weight spaces in section spaces of line bundles is given by the counting of
lattice points in polytopes.Comment: A new and simpler definition of a good flag is introduced, and
Bott-Samelson varieties are shown to admit such flag