Evaluation birepresentations of affine type A Soergel bimodules

TL;DR

This paper constructs a categorified evaluation functor linking extended affine type A Soergel bimodules to finite type A, enabling the transfer of birepresentations and revealing new structural properties.

Contribution

It introduces a monoidal evaluation functor that categorifies the evaluation homomorphism and explores its impact on birepresentations in affine type A Soergel bimodules.

Findings

The evaluation functor categorifies the evaluation homomorphism.

Finitary cell birepresentations induce birepresentations with a finitary cover.

Detailed analysis of birepresentations with subregular apex.

Abstract

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.