Eisenstein integral solutions to boundary KZB equations

Jasper Stokman (Universiteit van Amsterdam)

Tuesday 28th January, 2020 16:00-17:00 Maths 311B


Let G be a split real connected semisimple Lie group with finite center, and A a maximal split torus. Let K be a maximal compact Lie subgroup of G compatible with A, and V a finite dimensional KxK-representation. It goes all the way back to Harish-Chandra that the restriction to A of an elementary V-valued spherical functions on G satisfies a system of differential equations. This system of equations can be reinterpreted as the eigenvalue equations for the quantum Hamiltonians of a spin generalisation of the quantum hyperbolic Calogero-Moser (CM) system, with V playing the role of total spin space. The elementary V-spherical functions provide a representation theoretic construction of their eigenstates. In this talk I will mix this classical harmonic analytic approach to quantum CM systems with more recent ideas from integrable quantum field theory.

Concretely, I will show that the quantum spin CM Hamiltonians are supplemented with first order quantum integrals for an appropriate class of tensor product representations V. These additional first order differential operators, which arise from the presence of vertex operators, are boundary analogs of Knizhnik-Zamolodchikov-Bernard (KZB) operators. A refinement of the representation theoretic construction of the eigenstates of the quantum spin CM Hamiltonians leads to common eigenfunctions of the boundary KZB operators, given in terms of Eisenstein integrals.

This is joint work with N. Reshetikhin.

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