[Paper Review] Metaplectic representations of affine Hecke algebras and Weyl groups

Siddhartha Sahi, Jasper V. Stokman+1 more|arXiv (Cornell University)|Aug 3, 2018

Advanced Algebra and Geometry20 references3 citations

TL;DR

This paper provides a uniform, elementary proof that the Chinta-Gunnells Weyl group action on the fraction field of a group algebra arises from a quotient of a parabolically induced affine Hecke algebra module. By realizing this action through metaplectic generalizations of Demazure-Lusztig operators, the authors establish a concrete algebraic framework that confirms the action's consistency and structure in the context of Weyl group multiple Dirichlet series.

ABSTRACT

Chinta and Gunnells expressed the functional equations of Weyl group multiple Dirichlet series in terms of an explicit, but rather intricate, multi-parameter linear Weyl group action on the fraction field of the group algebra $\mathbb{C}[\Lambda]$ of some lattice $\Lambda$. In this paper we realize the Chinta-Gunnells Weyl group action by localizing a suitable quotient of a parabolically induced affine Hecke algebra module. This gives an elementary and uniform proof that the Chinta-Gunnells formulas indeed define an action of the Weyl group. The key step in the paper is to realize the quotient affine Hecke algebra module in terms of Demazure-Lusztig type operators involving metaplectic generalizations of divided-difference operators.

Motivation & Objective

• To provide a uniform and elementary proof of the Chinta-Gunnells Weyl group action on the fraction field of a group algebra.
• To realize the intricate multi-parameter Weyl group action as arising from a quotient of a parabolically induced affine Hecke algebra module.
• To establish a connection between Weyl group multiple Dirichlet series and representation-theoretic structures via metaplectic generalizations of divided-difference operators.
• To clarify the algebraic origin of the functional equations of multiple Dirichlet series through a systematic module-theoretic construction.

Proposed method

• Construct a quotient of a parabolically induced affine Hecke algebra module as the underlying representation space.
• Define metaplectic generalizations of divided-difference operators that act on this module.
• Use Demazure-Lusztig type operators built from these generalized operators to realize the Weyl group action.
• Demonstrate that the resulting action matches the Chinta-Gunnells formulas via explicit computation in the localized fraction field.
• Leverage the algebraic structure of the affine Hecke algebra to ensure consistency and uniformity across root systems.
• Establish the action's compatibility with the multi-parameter Weyl group action by verifying the braid and quadratic relations.

Experimental results

Research questions

• RQ1 How can the Chinta-Gunnells Weyl group action on the fraction field of a group algebra be systematically realized from a representation-theoretic construction?
• RQ2 What role do metaplectic generalizations of divided-difference operators play in constructing the Weyl group action?
• RQ3 Can the Chinta-Gunnells formulas be derived from a quotient of an affine Hecke algebra module in a uniform and elementary way?
• RQ4 How does the structure of the parabolically induced affine Hecke algebra module support the Weyl group action in the context of multiple Dirichlet series?
• RQ5 What is the precise relationship between Demazure-Lusztig operators and the functional equations of Weyl group multiple Dirichlet series?

Key findings

• The Chinta-Gunnells Weyl group action is realized as an action on a quotient of a parabolically induced affine Hecke algebra module.
• The action is constructed uniformly across root systems using metaplectic generalizations of divided-difference operators.
• The Demazure-Lusztig type operators built from these generalized operators reproduce the Chinta-Gunnells formulas exactly.
• The construction provides a new, elementary, and uniform proof that the Chinta-Gunnells formulas define a genuine Weyl group action.
• The algebraic framework ensures the action satisfies the required braid and quadratic relations of the Weyl group.
• The method establishes a direct link between the representation theory of affine Hecke algebras and the functional equations of multiple Dirichlet series.