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We study Whittaker functions on nonlinear coverings of simple algebraic groups over a nonarchimedean local field. We produce a recipe for expressing such a Whittaker function as a weighted sum over a crystal graph and show that in type A, these expressions agree with known formulae for the prime-power-supported coefficients of multiple Dirichlet series.
We prove that the Poisson deformation functor of an affine (singular) symplectic variety is unobstructed. As a corollary, we prove the following result. For an affine symplectic variety with a good -action (where its natural Poisson structure is positively weighted), the following are equivalent.
(1) has a crepant projective resolution.
(2) has a smoothing by a Poisson deformation.
A typical example is (the normalization) of a nilpotent orbit closure in a complex simple Lie algebra. By the theorem, one can see which orbit closure has a smoothing by a Poisson deformation.
We consider the Kähler-Ricci flow on complete finite-volume metrics that live on the complement of a divisor in a compact Kähler manifold . Assuming certain spatial asymptotics on the initial metric, we compute the singularity time in terms of cohomological data on . We also give a sufficient condition for the singularity, if there is one, to be type II.
The twistor space of the sphere is an isotropic Grassmannian that fibers over . An orthogonal complex structure (OCS) on a subdomain of (a complex structure compatible with the round metric) determines a section of this fibration with holomorphic image. In this article, we use this correspondence to prove that any finite energy OCS on must be of a special warped product form, and we also prove that any OCS on that is asymptotically constant must itself be constant. We give examples defined on which have infinite energy and examples of nonstandard OCSs on flat tori in complex dimension and greater.
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