Integrable structures arising from split symmetric pairs.

Tuesday 13th March, 2018 16:00-17:00 Maths 311B

Harmonic analysis on symmetric spaces is known to be intimately related

to solving quantum Calogero-Moser systems. The reason is that the action of Casimirs on spherical

functions is described by the action of commuting differential operators on the radial components

of the spherical functions. In the first half of the talk I will extend this result to vector-valued

spherical functions. It leads to a new class of quantum spin Calogero-Moser systems.

In the second half of the talk I will attach boundary conformal blocks to

split symmetric pairs. I will show that they solve boundary versions of

Knizhnik-Zamolodchikov-Bernard type equations. These are compatible systems of

equations involving a new universal solution of the classical dynamical reflection

equation with respect to Felder's trigonometric dynamical r-matrix.

The first part of the talk naturally ties in because it provides compatible (higher) Laplace equations for

the boundary conformal blocks.

If time permits I will say something about the extension of the results to affine symmetric pairs.

The talk is based on joint work with Nicolai Reshetikhin.

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