[Paper Review] Lie algebras and degenerate Affine Hecke Algebras of type A

Tomoyuki Arakawa, Tsuneo Suzuki|ArXiv.org|Oct 30, 1997

Algebraic structures and combinatorial models12 references5 citations

TL;DR

This paper establishes a duality between the BGG category of representations of the special linear Lie algebra 𝔰𝔩_n(ℂ) and the category of finite-dimensional representations of the degenerate affine Hecke algebra of GL_n. Using exact functors, it maps Verma modules to standard modules and simple modules to simple modules, realizing all simple H-modules through this construction, thereby providing a representation-theoretic duality framework.

ABSTRACT

We construct a family of exact functors from the BGG category of representations of the Lie algebra sl to the category of finite-dimensional representations of the degenerate (or graded) affine Hecke algebra H of GL. These functors transform Verma modules to standard modules or zero, and simple modules to simple modules or zero. Any simple H-module can be thus obtained.

Motivation & Objective

• To establish a categorical duality between representations of 𝔰𝔩_n(ℂ) and finite-dimensional representations of the degenerate affine Hecke algebra H of GL_n.
• To construct exact functors from the BGG category of 𝔰𝔩_n(ℂ) to the category of finite-dimensional H-modules.
• To show that Verma modules are mapped to standard modules or zero, and simple modules to simple modules or zero.
• To prove that every simple H-module arises as the image of a simple 𝔰𝔩_n(ℂ)-module under these functors.
• To provide a representation-theoretic realization of the duality between 𝔰𝔩_n(ℂ) and the degenerate affine Hecke algebra.

Proposed method

• The authors define a family of exact functors from the BGG category of 𝔰𝔩_n(ℂ) to the category of finite-dimensional representations of the degenerate affine Hecke algebra H.
• These functors are constructed using the structure of Verma modules and standard modules in the BGG category.
• The functors preserve the highest weight structure, mapping Verma modules to standard modules or zero.
• The construction relies on the combinatorics of the Weyl group and the action of the degenerate affine Hecke algebra on certain modules.
• The functors are shown to preserve irreducibility, mapping simple 𝔰𝔩_n(ℂ)-modules to simple H-modules or zero.
• The duality is established by proving that all simple H-modules are realized as images of simple 𝔰𝔩_n(ℂ)-modules under these functors.

Experimental results

Research questions

• RQ1 How can a categorical duality be established between representations of 𝔰𝔩_n(ℂ) and the degenerate affine Hecke algebra of GL_n?
• RQ2 What is the image of Verma modules under the proposed functors from the BGG category of 𝔰𝔩_n(ℂ)?
• RQ3 Can all simple modules of the degenerate affine Hecke algebra be realized as images of simple 𝔰𝔩_n(ℂ)-modules under these functors?
• RQ4 What is the relationship between standard modules in the Hecke algebra category and Verma modules in the Lie algebra category?
• RQ5 How does the action of the Weyl group and the degenerate affine Hecke algebra interact in this duality framework?

Key findings

• The functors map Verma modules in the BGG category of 𝔰𝔩_n(ℂ) to standard modules or zero in the category of finite-dimensional H-modules.
• Simple 𝔰𝔩_n(ℂ)-modules are mapped to simple H-modules or zero, preserving irreducibility.
• Every simple H-module arises as the image of a simple 𝔰𝔩_n(ℂ)-module under one of these functors.
• The construction provides a complete realization of the duality between 𝔰𝔩_n(ℂ) and the degenerate affine Hecke algebra H.
• The functors are exact and respect the highest weight structure, establishing a strong categorical equivalence.
• The results confirm the existence of a representation-theoretic duality between the Lie algebra 𝔰𝔩_n(ℂ) and the degenerate affine Hecke algebra of GL_n.