The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in P7. We compute defining polynomials for three versions of this family, over the Satake hypersurface, over the Göpel variety, and over the reflection representation of type E7. We develop classical themes such as theta functions and Coble’s quartic hypersu...

By the geometric Satake correspondence, the number of components of certain fibres of the affine Grassmannian convolution morphism equals the tensor product multiplicity for representations of the Langlands dual group. On the other hand, in the case of GLn, combinatorial objects called hives also count tensor product multiplicities. The purpose of this paper is to give a simple bijection betwee...

We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight k, subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson’s formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show t...

This paper begins a series studying D-modules on the Feigin-Frenkel semi-infinite flag variety from the perspective of the Beilinson-Drinfeld factorization (or chiral) theory. Here we calculate Whittaker-twisted cohomology groups of Zastava spaces, which are certain finite-dimensional subvarieties of the affine Grassmannian. We show that such cohomology groups realize the nilradical of a Borel ...

• Decomposition by central characters • Square-integrable cuspforms • Smoothness of cuspforms • Eigen-cuspforms and automorphic representation • Dirichlet series versus zeta and L-functions • L-functions defined via local data • Factoring unitary representations of adele groups • Spherical representations and Satake parameters • Local data, L-groups, higher L-functions References and historical...

Pollen from wind-pollinated trees has traditionally been assumed to be abundant and to travel long distances, resulting in extensive gene flow. However, recent empirical work by Knapp et al., genetic analysis by Sork et al., and theoretical models by Satake and Iwasa conclude that short-distance dispersal of limited pollen might be common and play an important role in causing the highly variabl...

Seidel-Smith and Manolescu constructed knot homology theories using symplectic fibrations whose total spaces were certain varieties of matrices. These knot homology theories were associated to SL(n) and tensor products of the standard and dual representations. In this paper, we place their geometric setups in a natural, general framework. For any complex reductive group and any sequence of minu...

1. Spaltenstein’s theorem and Hotta’s construction 1 2. Equivariant cohomology and divided differences 5 3. Review of: Borel subgroups, parabolic subgroups, the Bruhat decomposition 10 4. The Steinberg scheme 12 5. Algebras of constructible correspondences 13 6. Hall algebras 17 7. Geometric construction of Uq(n+) for simply-laced Lie algebras 20 8. Convolution in Borel-Moore homology 23 9. Gro...

We suggest an explanation for the part of Satake Correspondence which relates quantum cohomology complex Grassmannians and projective space, as well their respective Stokes data, based on original physics approach using tt* equations. also use data equations to provide a Lie-theoretic link between particles in affine Toda models solitons certain sigma-models. Along way, we review some old ideas...

The representation theory of reductive groups, such as the group GLn of invertible complex matrices, is an important topic, with applications to number theory, algebraic geometry, mathematical physics, and quantum topology. One way to study this representation theory is through the geometric Satake correspondence (also known as geometric Langlands duality). This correspondence relates the geome...