## Representation Theory and Mathematical Physics Seminar: Spectra of quantum integrable systems, Langlands duality and category O

Seminar | December 6 | 4-5 p.m. | 939 Evans Hall

David Hernandez, Paris Diderot (7)

Department of Mathematics

The spectrum of a quantum integrable system is crucial to understand its properties. R-matrices give power tools to study such spectra. A better understanding of transfer-matrices obtained from R-matrices led us to the proof of several results for the corresponding quantum integrable systems. In particular, their spectra can be described in terms of "Baxter polynomials" as conjectured by Frenkel-Reshetikhin. They appear naturally in the study of a category O of representations of a Borel subalgebra of a quantum affine algebra (in the $sl_2$-case, this is due to Bazhanov-Lukyanov-Zamolochikov). The properties of geometric objects attached to the Langlands dual Lie algebra (the affine opers) led us to establish new relations in the Grothendieck ring of this category O, from which one can derive the generic Bethe Ansatz equations between the roots of the Baxter polynomials. They are also related to a cluster algebra structure on the Grothendieck ring.

​(Based on joint works with M. Jimbo, E. Frenkel and B. Leclerc). Supported by the European Research Council under the European Union's Framework Programme H2020 with ERC Grant Agreement number 647353 Qaffine.

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