Evaluation birepresentations of affine type A Soergel bimodules

Miemietz, Vanessa;Dos Santos Santana Forte Vaz, Pedro;Mackaay, Marco
(2024) Advances in mathematics — Vol. 436 (2024)

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In this paper, we use Soergel calculus to define a monoidal functor, called the evaluation functor, from extended affine type A Soergel bimodules to the homotopy category of bounded complexes in finite type A Soergel bimodules. This functor categorifies the well-known evaluation homomorphism from the extended affine type A Hecke algebra to the finite type A Hecke algebra. Through it, one can pull back the triangulated birepresentation induced by any finitary birepresentation of finite type A Soergel bimodules to obtain a triangulated birepresentation of extended affine type A Soergel bimodules. We show that if the initial finitary birepresentation in finite type A is a cell birepresentation, the evaluation birepresentation in extended affine type A has a finitary cover, which we illustrate by working out the case of cell birepresentations with subregular apex in detail.
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Miemietz, V., Dos Santos Santana Forte Vaz, P., & Mackaay, M. (2024). Evaluation birepresentations of affine type A Soergel bimodules. Advances in mathematics, 436. https://doi.org/10.1016/j.aim.2023.109401 (Original work published 2024)