[Paper Review] Derived Langlands III: PSH algebras and their numerical invariants

Victor Snaith|arXiv (Cornell University)|Jun 12, 2020

Advanced Algebra and Geometry12 references4 citations

TL;DR

This paper introduces a conjectural Hopf-like algebra structure on the hyperHecke algebras of general linear groups over p-adic local fields or finite fields, generalizing PSH algebras and their numerical invariants. It constructs a coproduct via Bruhat decomposition and double coset analysis, showing compatibility with tensor products and suggesting a deeper algebraic framework for derived Langlands correspondence through hyperHecke algebras.

ABSTRACT

This sequel to Derived Langlands II studies some PSH algebras and their numerical invariants, which generalise the epsilon factors of the local Langlands Programme. It also describes a conjectural Hopf algebra structure on the sum of the hyperHecke algebras of products of the general linear groups over a $p$-adic local field or a finite field.

Motivation & Objective

• To generalize the theory of PSH algebras and their numerical invariants to the context of hyperHecke algebras of general linear groups over p-adic local fields or finite fields.
• To investigate the possibility of a Hopf-like algebra structure on the direct sum of hyperHecke algebras of products of GL_n groups.
• To extend the framework of derived Langlands theory by embedding admissible representations into a derived category via monomial categories and bar resolutions.
• To establish a coproduct on the hyperHecke algebra using double coset decompositions and induced representations, motivated by the Segal conjecture and Brauer’s induction theorem.

Proposed method

• Constructs a coproduct on the hyperHecke algebra of GL_n(F) using double coset representatives and conjugation actions on parabolic subgroups.
• Uses the isomorphism P_{a,n-a}/U_{a,n-a} ≅ GL_a(F) × GL_{n-a}(F) to descend representations from parabolic subgroups to the general linear groups.
• Applies the Bruhat decomposition combinatorially to ensure compatibility between multiplication and coproduct maps across tensor products of hyperHecke algebras.
• Imposes a condition that the character ((zg^{-1})^{-1})^*(φ) is trivial on unipotent subgroups to ensure well-definedness of the coproduct components.
• Defines the coproduct as a sum over a = 0 to n, restricted to terms where the character condition holds, ensuring positivity and compatibility with the PSH algebra structure.
• Relies on the structure of PSH algebras over ℤ, with distinguished bases and self-adjointness, to define positivity and duality in the hyperHecke algebra.

Experimental results

Research questions

• RQ1 Can the hyperHecke algebra of GL_n(F) for a non-Archimedean local field F be endowed with a Hopf-like algebra structure?
• RQ2 How does the coproduct on the hyperHecke algebra relate to the tensor product structure of PSH algebras?
• RQ3 What is the role of the Bruhat decomposition in constructing a consistent coproduct on hyperHecke algebras?
• RQ4 Do the numerical invariants of PSH algebras, such as Kondo-Gauss sums, generalize to the hyperHecke algebra setting?
• RQ5 Is there a canonical way to lift induced representations from parabolic subgroups to the full general linear group in this algebraic framework?

Key findings

• The coproduct on the hyperHecke algebra is defined via double coset representatives and conjugation, with components descending to GL_a(F) × GL_{n-a}(F) through quotienting by unipotent radicals.
• The construction ensures that the coproduct is compatible with multiplication, as verified by composition of maps involving m*, m, and the twist T in the tensor product category.
• The coproduct is restricted to terms where ((zg^{-1})^{-1})^*(φ) is trivial on U_{a,n-a} ∩ zg^{-1}Hgz^{-1}, guaranteeing well-definedness and positivity.
• The paper proves that the composition of maps involving m* ⊗ m* followed by T and m × m equals the composition involving m and m*, confirming coassociativity-like properties.
• The construction generalizes Kondo-Gauss sums from GL_n(𝔽_q) to the hyperHecke algebra setting, suggesting a link to epsilon factors in the local Langlands program.
• The author conjectures a full Hopf algebra structure on the direct sum of hyperHecke algebras over all ordered partitions of n, with the coproduct defined via Bruhat decomposition and character conditions.