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Duality theorems for current groups

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Abstract

The classical Peter—Weyl theorem describes the structure of the space of functions on a semi-simple algebraic group. On the level of characters (in type A) this boils down to the Cauchy identity for the products of Schur polynomials. We formulate and prove an analogue of the Peter—Weyl theorem for current groups. In particular, in type A the corresponding characters identity is governed by the Cauchy identity for the products of q-Whittaker functions. We also formulate and prove a version of the Schur—Weyl theorem for current groups, which can be seen as a specialization of a theorem due to Drinfeld and Chari—Pressley. The link between the Peter—Weyl and Schur—Weyl theorems is provided by the (current version of) Howe duality.

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Acknowledgments

We are grateful to Alexei Borodin, Alexander Braverman, Boris Feigin, Michael Finkelberg, Ryo Fujita, Syu Kato and Shrawan Kumar for useful discussions and correspondence. The research of E. F. was supported by the grant RSF 19-11-00056. The study was partially supported by the HSE University Basic Research Program. A. Kh. is a Young Russian Mathematics award winner and would like to thank its sponsors and jury. The work of Ie. M. was supported by the Japan Society for the Promotion of Science.

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Authors and Affiliations

  1. Department of Mathematics, HSE University, Usacheva str. 6, Moscow, 119048, Russia

    Evgeny Feigin & Anton Khoroshkin

  2. Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Building 3, Moscow, 143026, Russia

    Evgeny Feigin & Ievgen Makedonskyi

  3. Institute for Theoretical and Experimental Physics, Moscow, 117259, Russia

    Anton Khoroshkin

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  1. Evgeny Feigin
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Correspondence to Evgeny Feigin.

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Feigin, E., Khoroshkin, A. & Makedonskyi, I. Duality theorems for current groups. Isr. J. Math. 248, 441–479 (2022). https://doi.org/10.1007/s11856-022-2306-6

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  • DOI: https://doi.org/10.1007/s11856-022-2306-6

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