Cyclotomic Gaudin models: construction and Bethe ansatz

This is a pre-copyedited author produced PDF of an article accepted for publication in Communications in Mathematical Physics, Benoit, V and Young, C, 'Cyclotomic Gaudin models: construction and Bethe ansatz', Commun. Math. Phys. (2016) 343:971, first published on line March 24, 2016. The final publication is available at Springer via http://dx.doi.org/10.1007/s00220-016-2601-3 © Springer-Verlag Berlin Heidelberg 2016To any simple Lie algebra $\mathfrak g$ and automorphism $\sigma:\mathfrak g\to \mathfrak g$ we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of $U(\mathfrak g)^{\otimes N}$ generated by a hierarchy of cyclotomic Gaudin Hamiltonians. It reduces to the Gaudin algebra in the special case $\sigma = \text{id}$. We go on to construct joint eigenvectors and their eigenvalues for this hierarchy of cyclotomic Gaudin Hamiltonians, in the case of a spin chain consisting of a tensor product of Verma modules. To do so we generalize an approach to the Bethe ansatz due to Feigin, Frenkel and Reshetikhin involving vertex algebras and the Wakimoto construction. As part of this construction, we make use of a theorem concerning cyclotomic coinvariants, which we prove in a companion paper. As a byproduct, we obtain a cyclotomic generalization of the Schechtman-Varchenko formula for the weight function.Peer reviewe

Cyclotomic Gaudin models with irregular singularities

Generalizing the construction of the cyclotomic Gaudin algebra from arXiv:1409.6937, we define the universal cyclotomic Gaudin algebra. It is a cyclotomic generalization of the Gaudin models with irregular singularities defined in arXiv:math/0612798. We go on to solve, by Bethe ansatz, the special case in which the Lax matrix has simple poles at the origin and arbitrarily many finite points, and a double pole at infinity

Representation theory on the open Bruhat cell

AbstractThe action of a connected reductive algebraic group G on G/P−, where P− is a parabolic subgroup, differentiates to a representation of the Lie algebra g of G by vector fields on U+, the unipotent radical of a parabolic opposite to P−. The classical instances of this setting that we study in detail are the actions of GLn on the Grassmannian of k-planes (1≤k≤n), of SOn on the quadric of isotropic lines, and of SO2n or SP2n on their respective Grassmannians of maximal isotropic spaces; in each instance, U+ is one of the usual affine charts.We show that both the polynomials on U+ and the polynomial vector fields on U+ formg-modules dual to parabolically induced modules, construct an explicit composition chain of the former module in the case where G is classical simple and U+ is Abelian—these are exactly the cases above—and indicate how this chain can be used to analyse the module of vector fields, as well.We present two proofs of our main theorems: one uses the results of Enright and Shelton on classical Hermitian pairs, and the other is independent of their work. The latter proof mixes classical (and briefly reviewed) facts of representation theory with combinatorial and computational arguments, and is accessible to readers unfamiliar with the vast modern literature on category O