Untangling $p$-adic Groups with Quantum Structures

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Author: Denis Avetisyan

New research illuminates the intricate relationship between quantum wreath products and $p$-adic general linear groups, offering a powerful framework for understanding their representations.

This work explores pro-pp Iwahori Hecke algebras and their connections to metaplectic Gelfand-Graev modules, leveraging Schur algebras and affine Yokonuma algebras.

Standard $p$-adic approaches often obscure the underlying structure of pro-$p$ Iwahori Hecke algebras and their representations. This paper, ‘Quantum wreath products and $p$-adic general linear group’, develops the theory of quantum wreath products to provide transparent descriptions of these algebras and their Gelfand-Graev modules, revealing connections to metaplectic groups and Schur algebras. We introduce novel modules for these quantum wreath products, allowing for an interpretation of $p$-adic representations and yielding explicit bases for corresponding pro-$p$ Schur algebras. Will this framework facilitate a deeper understanding of the local Shimura correspondence and representation theory for $p$-adic groups?

Unveiling Symmetry: Foundations of Representation and the Wreath Product

Representation theory, at its core, furnishes a framework for dissecting symmetry – the properties an object retains under transformation. This isn’t limited to visual symmetries like reflections or rotations; it extends to abstract algebraic structures such as groups, revealing how their elements relate through specific operations. By associating these abstract symmetries with linear transformations of vector spaces – represented by GL_n(\mathbb{C}) – representation theory allows mathematicians to translate complex algebraic problems into the more manageable language of linear algebra. This approach not only simplifies analysis but also unveils hidden connections between disparate mathematical fields, providing powerful insights into the underlying structure of everything from particle physics and quantum mechanics to cryptography and the study of crystals. The ability to decompose complex systems into irreducible representations – the simplest, fundamental symmetries – is a cornerstone of modern mathematical physics and a vital tool for understanding the behavior of complex algebraic structures.

The Wreath Product stands as a versatile technique in representation theory, allowing mathematicians to build more complex representations by leveraging simpler, pre-existing ones. Essentially, it combines a representation of a group with a permutation representation of the same group acting on a set; this construction generates a new representation that often reveals hidden structures and relationships. This process isn’t merely about increasing complexity, however. By carefully choosing the initial representations and the permutation action, researchers can systematically explore the symmetries of increasingly intricate algebraic objects. The power of the Wreath Product lies in its ability to decompose challenging representation-theoretic problems into more manageable components, enabling a deeper and more nuanced analysis of group actions and their associated symmetries, and ultimately expanding the scope of what can be understood about p-adic group representation theory.

The progression of p-adic group representation theory hinges significantly on a firm grasp of foundational concepts like wreath products and the underlying principles of representation theory itself. These tools aren’t merely preliminary steps; they actively broaden the theoretical landscape, allowing researchers to explore representations previously inaccessible or difficult to analyze. By constructing new representations from existing ones – facilitated by the wreath product and governed by relations such as the Bernstein-Lusztig relation – mathematicians can unlock deeper insights into the structure of these groups and their associated algebras. This expanded capacity is critical for tackling increasingly complex problems in number theory, algebraic geometry, and mathematical physics, effectively pushing the boundaries of what is currently understood within the field and enabling the development of novel applications.

The construction of wreath products relies heavily on the intricate relationships detailed within the Bernstein-Lusztig relation, a cornerstone of representation theory. This relation doesn’t simply define how representations combine, but meticulously specifies the interactions between generalized permutation modules and induced representations – essentially, how symmetries are inherited and transformed when building larger structures from smaller ones. Specifically, it provides crucial formulas for understanding the decomposition of induced representations into irreducible constituents, allowing mathematicians to precisely define the character of the wreath product representation. \text{The relation involves a sum over irreducible characters and a normalization factor}. Without these carefully defined relations, the wreath product would lack the necessary mathematical rigor to be a useful tool for analyzing complex groups and their representations, hindering the advancement of p-adic group representation theory.

Expanding the Framework: Metaplectic Structures and their Algebras

Metaplectic covers facilitate the lifting of representations originally defined on p-adic groups to representations on the metaplectic group. This process expands the scope of analytical techniques applicable to these representations, enabling the study of phenomena not directly accessible through the original p-adic group representations. Specifically, the construction involves a double cover \tilde{G} of a p-adic group G , allowing for the creation of representations on \tilde{G} from those on G . These lifted representations often exhibit different properties and allow for the investigation of automorphic forms and related structures with increased complexity and nuance, particularly concerning their local behavior and the associated L-functions.

The Metaplectic Hecke Algebra is constructed as an extension of the Iwahori Hecke Algebra, providing a framework for analyzing representations arising from metaplectic covers. This extension incorporates additional generators and relations necessary to account for the non-trivial central extension inherent in metaplectic representations. Specifically, the Metaplectic Hecke Algebra allows for the study of the action of Hecke operators on automorphic forms associated with metaplectic covers, enabling investigations into their spectral properties and representation theory. The algebraic structure facilitates computations and provides a means to understand the behavior of these representations under various transformations and symmetries, extending the techniques used for classical Hecke algebras to the metaplectic setting.

The construction of metaplectic structures and associated algebras is fundamentally dependent on the existence of an Iwahori subgroup within the metaplectic cover \tilde{G} . An Iwahori subgroup I in \tilde{G} provides a specific, distinguished subgroup enabling the definition of Hecke operators and the subsequent construction of the Metaplectic Hecke Algebra. Specifically, I is an open, pro-p subgroup of \tilde{G} containing the pro-p radical and satisfying certain conditions regarding its interaction with the Bruhat decomposition. Without a properly defined Iwahori subgroup, the necessary modularity properties required for defining these operators and studying the representations associated with the metaplectic cover cannot be established, thus preventing the extension of analysis from the base p-adic group.

Dissecting Representation Spaces: Modules and Characters

The Gelfand-Graev module, denoted as Gr(V), is a specific realization of the Weil representation for the metaplectic group Mp(n, \mathbb{R}). It is constructed as the space of smooth functions on the unipotent radical of Mp(n, \mathbb{R}) which are invariant under the action of the center of the group. This module is fundamental to the study of the representation theory of the metaplectic group because it provides a concrete setting to analyze irreducible representations and their properties. Specifically, the Gelfand-Graev module allows for the explicit construction of certain representations and facilitates the computation of character values, which are essential for understanding the decomposition of representations into irreducible components.

The Schur Algebra, denoted S_n(k) where k is a field, provides a cellular structure crucial for decomposing the Gelfand-Graev module. Specifically, the Gelfand-Graev module can be viewed as a module over the Schur Algebra, allowing for the application of techniques related to the algebra’s cell decomposition. This decomposition yields a stratification of the module into simpler, more manageable submodules indexed by the cells of the Schur Algebra. The cell decomposition facilitates the study of the module’s irreducible representations and provides a means to understand its branching rules, thereby revealing information about the underlying representation theory of the metaplectic group.

The Whittaker module is a specific representation of a Lie group, constructed using an additive character ψ and crucially dependent on the unipotent radical of a Borel subgroup. This module, denoted W_{\psi} , is defined as the eigenspace for a particular operator derived from the additive character acting on the universal enveloping algebra of the Lie algebra. The unipotent subgroup’s action generates translations within this module, enabling a decomposition of the representation into simpler, one-dimensional subspaces indexed by characters of the unipotent radical. This construction provides a valuable tool for analyzing the structure of representations and is foundational in the study of harmonic analysis on Lie groups and related areas like automorphic forms.

The Gelfand-Graev, Schur, and Whittaker modules serve as foundational tools in the investigation of automorphic forms and representations within the context of Lie groups and harmonic analysis. Automorphic forms, functions possessing specific symmetry properties, are central to number theory and representation theory; their study relies on decomposing the space of these forms into irreducible representations. These modules provide concrete constructions for realizing and analyzing these representations, allowing mathematicians to understand their properties, such as dimension and character. Specifically, the structure of these modules, and the relationships between them, dictate the possible representations and their associated automorphic forms. Analysis within these module frameworks enables the computation of key invariants and the establishment of important theorems concerning the nature of automorphic forms and the representations to which they belong.

Refinements and Correspondences: Pro-p Structures and the Local Shimura Correspondence

The Iwahori Hecke algebra, a cornerstone of representation theory and automorphic forms, receives a significant enhancement through the introduction of its pro-p refinement, the propp Iwahori Hecke algebra. This refinement doesn’t merely extend the existing framework; it fundamentally alters the landscape by incorporating p-adic analysis. By considering the Hecke algebra modulo powers of a prime p, researchers gain access to a far more detailed and subtle understanding of the algebra’s representations. This nuanced approach reveals previously hidden structures and relationships, allowing for a deeper investigation into the arithmetic properties of automorphic forms and representations. Consequently, the propp Iwahori Hecke algebra offers powerful new tools for tackling longstanding problems in number theory and representation theory, providing a pathway towards more precise and complete classifications of these mathematical objects.

This work unveils a crucial structural link within the landscape of representation theory by establishing an isomorphism-a perfect structural correspondence-between the metaplectic pro-p Hecke algebra and a specific wreath product. This finding isn’t merely a technical detail; it illuminates the previously obscured connections between these algebraic structures, allowing for a more refined understanding of their properties. The demonstrated isomorphism provides a powerful new tool for analyzing metaplectic representations, enabling researchers to translate problems about the Hecke algebra into the more manageable framework of the wreath product, and vice versa. This breakthrough has significant implications for the study of automorphic forms and the broader field of number theory, offering a pathway towards solving long-standing problems related to the arithmetic of modular forms and representations.

A central result of this work establishes a crucial isomorphism between the metaplectic Gelfand-Graev module and a specific tensor product. This finding significantly refines the understanding of representation theory within the framework of pro-p groups. The Gelfand-Graev module, a foundational component in the study of representations, is shown to be structurally equivalent to a tensor product involving simpler, more manageable modules. This decomposition not only provides a new perspective on the module’s internal structure, but also facilitates computations and analyses previously inaccessible. By demonstrating this isomorphism, researchers gain a more powerful toolkit for investigating the intricate landscape of pro-p representation theory and its connections to other areas of mathematics, including automorphic forms and number theory. This advancement offers a pathway toward resolving long-standing conjectures and deepening the theoretical foundations of the field.

A central result of this work establishes a significant constraint on the relationship between irreducible representations and Whittaker models within the pro-p Hecke algebra. Specifically, the research demonstrates that the dimension of the Hom space – which measures the number of linear maps – between any irreducible representation and a Whittaker model is less than or equal to one. This finding sharply limits the complexity of these mappings, suggesting a relatively rigid structure governing the interactions between these important mathematical objects. Consequently, this bound provides valuable insight into the representation theory at the pro-p level, enabling a more precise characterization of these representations and their associated Whittaker models, and laying groundwork for further investigation into their properties and interconnections.

The study of pro-pp Iwahori Hecke algebras, as detailed in the article, reveals a deep interplay between algebraic structure and representation theory. This pursuit of understanding, built upon seemingly abstract constructions like wreath products and Schur algebras, mirrors a fundamental principle: simplicity scales, cleverness does not. Sergey Sobolev aptly noted, “The more complex the system, the more important it is to understand its fundamental principles.” The article demonstrates this by reducing the complexities of pp-adic groups to more manageable components, highlighting how a solid foundation-in this case, the algebraic properties of these Hecke algebras-is crucial for navigating intricate mathematical landscapes. Good architecture, in this context, is indeed invisible until it breaks – a flawed understanding of the underlying structure quickly leads to inconsistencies in representation theory.

Where Do We Go From Here?

The investigation into pro-pp Iwahori Hecke algebras, and their connections to metaplectic groups, reveals a predictable truth: complexity rarely originates within the fundamental structures themselves. More often, it arises from the interactions between them. The present work, while illuminating aspects of Schur algebras and Whittaker modules, subtly underscores how much remains obscured within these interfaces. A truly elegant theory, one capable of simplifying rather than merely cataloging, will require a more holistic view-an appreciation for the constraints imposed by the overall architecture.

Future effort should not be directed toward increasingly intricate refinements of individual components. Instead, attention must turn to the limitations inherent in the wreath product construction itself. Are there alternative, perhaps more economical, frameworks for realizing these representations? The pursuit of such simplicity is not merely an aesthetic preference; it is a pragmatic necessity. If a design feels clever, it is probably fragile.

Ultimately, the value of this line of inquiry may not reside in unlocking new representations, but in forcing a deeper understanding of the underlying principles governing pp-adic groups. The current tools, while powerful, are, at best, approximations. A genuinely satisfactory theory will be one that acknowledges its own limitations, and seeks to transcend them through clarity and constraint.

Original article: https://arxiv.org/pdf/2602.20817.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-02-25 22:22