Schubert Eisenstein Series and the Poisson Summation Conjecture: Meromorphic Continuation and Applications to Analytic Number Theory

Core Concepts

This mathematics research paper establishes a connection between Schubert Eisenstein series and the Poisson summation conjecture, proving the meromorphic continuation of these series in many cases and advancing the understanding of fundamental concepts in analytic number theory.

Abstract

Bibliographic Information: Choie, Y., & Getz, J. R. (2024). Schubert Eisenstein Series and Poisson Summation for Schubert Varieties. arXiv preprint arXiv:2107.01874v5.

Research Objective: This paper investigates the properties of Schubert Eisenstein series, a novel modification of degenerate Eisenstein series, and their relationship to the Poisson summation conjecture in the field of analytic number theory.

Methodology: The authors employ techniques from algebraic geometry and representation theory, focusing on Braverman-Kazhdan spaces and their relation to Schubert varieties. They define Schwartz spaces and a Fourier transform for these spaces, ultimately proving a Poisson summation formula for a specific family of varieties related to Schubert varieties.

Key Findings: The paper proves that Schubert Eisenstein series admit meromorphic continuation in a significant number of cases, confirming a conjecture by Bump and Choie. This is achieved by proving the Poisson summation conjecture for a particular family of varieties related to Schubert varieties.

Main Conclusions: The established link between Schubert Eisenstein series and the Poisson summation conjecture provides a powerful tool for studying these series and their applications in number theory. The meromorphic continuation result has significant implications for understanding the analytic behavior of these functions.

Significance: This research significantly advances the understanding of Schubert Eisenstein series and their connection to broader concepts in number theory, particularly the Poisson summation conjecture. The results have potential implications for the study of automorphic forms and L-functions.

Limitations and Future Research: The meromorphic continuation of Schubert Eisenstein series is established under specific conditions, leaving room for further exploration of more general cases. The paper also suggests investigating the potential of generalized Schubert Eisenstein series in constructing integral representations of automorphic L-functions, opening avenues for future research in this direction.

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Schubert Eisenstein series and Poisson summation for Schubert varieties

by YoungJu Choi... at arxiv.org 11-19-2024

https://arxiv.org/pdf/2107.01874.pdf

Deeper Inquiries

How might the results of this paper be extended to address the analytic properties of Schubert Eisenstein series in the context of Kac-Moody groups, as hinted at by the authors?

Extending the results of this paper to the realm of Kac-Moody groups presents a formidable yet enticing challenge. Here's a breakdown of potential pathways and inherent difficulties:
Potential Approaches:

Finite-Dimensional Approximations: Kac-Moody groups can often be studied through their finite-dimensional representations and quotients. One might attempt to define Schubert Eisenstein series on these finite-dimensional objects and relate their analytic properties back to the Kac-Moody setting. This would require carefully analyzing how the analytic behavior of the series changes as one takes larger and larger finite-dimensional approximations.

Geometric Methods: The paper heavily relies on the geometry of Schubert varieties and Braverman-Kazhdan spaces. Kac-Moody flag varieties, while infinite-dimensional, possess rich geometric structures. Adapting tools from algebraic geometry, such as Borel-Weil-Bott theory and the theory of D-modules, to this infinite-dimensional setting could provide insights into the analytic continuation of Schubert Eisenstein series.

Representation Theory of Kac-Moody Algebras: The representation theory of Kac-Moody algebras plays a crucial role in understanding the structure of Kac-Moody groups. Investigating how Schubert Eisenstein series intertwine with representations of these algebras might offer a route to proving their analytic properties. This would likely involve exploring connections with Verma modules, character formulas, and other key aspects of the representation theory.

Difficulties:

Infinite Dimensionality: The primary obstacle is the infinite-dimensional nature of Kac-Moody groups. Many of the geometric and analytic techniques used in the finite-dimensional setting do not directly generalize.

Convergence Issues: Defining and establishing the convergence of infinite sums and integrals becomes significantly more delicate in the Kac-Moody context. New methods for regularizing and analyzing these infinite expressions would be needed.

Lack of Explicit Formulas: Explicit formulas and decompositions, which are often available for finite-dimensional groups, are generally lacking in the Kac-Moody setting. This makes it challenging to perform concrete calculations and establish analytic properties.

Summary:
While extending the results to Kac-Moody groups is a highly speculative endeavor, the potential rewards are significant. It would establish a fascinating link between the theory of automorphic forms and the representation theory of infinite-dimensional Lie algebras, potentially leading to new insights in both fields.

Could there be alternative approaches, not relying on the Poisson summation conjecture, to prove the meromorphic continuation of Schubert Eisenstein series in full generality?

Yes, there could be alternative approaches to proving the meromorphic continuation of Schubert Eisenstein series without directly invoking the Poisson summation conjecture. Here are a few possibilities:

Direct Analytic Continuation: One could attempt a direct analytic continuation of the series by expressing them as integrals over suitable contours and then deforming the contours to demonstrate meromorphic continuation. This approach would require a deep understanding of the asymptotic behavior of the series' summands and the geometry of the integration domains.

Differential Equations: Schubert Eisenstein series might satisfy certain differential equations arising from the action of Casimir operators or other differential operators on the underlying representation spaces. If one could establish the existence of such differential equations and analyze their solutions, it might lead to a proof of meromorphic continuation.

Relation to Other Eisenstein Series: It might be possible to express Schubert Eisenstein series as linear combinations or integrals of more familiar Eisenstein series, whose meromorphic continuation is already established. This approach would involve carefully analyzing the intertwining operators relating different Eisenstein series.

Spectral Theory: The meromorphic continuation of Eisenstein series is closely related to the spectral decomposition of $L^2(G(F)\backslash G(\mathbb{A}))$. One could try to analyze the contribution of Schubert Eisenstein series to this spectral decomposition and deduce their meromorphic continuation from the general theory of Eisenstein series.

Challenges:
Each of these alternative approaches presents its own set of challenges. Direct analytic continuation can be computationally demanding, while establishing the existence of suitable differential equations or relations to other Eisenstein series might require significant ingenuity. The spectral theory approach, while powerful, often involves sophisticated machinery.
Summary:
While the Poisson summation conjecture provides a conceptually elegant approach, exploring alternative methods for proving the meromorphic continuation of Schubert Eisenstein series could lead to new insights and techniques in the theory of automorphic forms.

What are the implications of this research for the Langlands program, particularly in light of the connection between the Poisson summation conjecture and the Langlands functoriality?

This research holds significant implications for the Langlands program, particularly in its pursuit of Langlands functoriality. Here's a breakdown of the connections:
Langlands Functoriality:
The Langlands program predicts deep connections between automorphic representations of different reductive algebraic groups. Functoriality, a central conjecture within this program, asserts that a homomorphism between L-groups of two reductive groups should give rise to a transfer of automorphic representations.
Poisson Summation and L-functions:
The Poisson summation conjecture, as explored in this paper, provides a powerful tool for establishing the meromorphic continuation and functional equations of L-functions. These L-functions are central objects in the Langlands program, encoding deep arithmetic and geometric information.
Implications of the Research:

New Cases of Functoriality: Proving the meromorphic continuation of Schubert Eisenstein series, as achieved in this paper, potentially leads to new cases of Langlands functoriality. The functional equations satisfied by these Eisenstein series can often be interpreted as instances of functoriality, relating automorphic representations of different groups.

Evidence for the Poisson Summation Conjecture: The successful application of the Poisson summation conjecture in this context provides further evidence for its validity. As the conjecture is closely linked to Langlands functoriality, this strengthens the belief in the overall Langlands program.

New Tools for Studying L-functions: The techniques developed in this research, particularly those involving the geometry of Schubert varieties and Braverman-Kazhdan spaces, offer new tools for studying L-functions. These tools could potentially be applied to other families of L-functions, leading to further progress in the Langlands program.

Summary:
This research represents a significant step forward in the Langlands program. By establishing the meromorphic continuation of Schubert Eisenstein series using the Poisson summation conjecture, the authors provide new evidence for Langlands functoriality and develop powerful techniques for studying L-functions. This work opens up exciting new avenues for future research in this vibrant area of mathematics.