[Paper Review] Discrete series of odd general spin groups

Yeansu Kim, Ivan Matić|arXiv (Cornell University)|Jun 4, 2017

Advanced Algebra and Geometry17 references3 citations

TL;DR

This paper establishes a simplified, uniform classification of non-cuspidal discrete series representations for odd general spin groups over non-archimedean local fields of characteristic zero, using a modified version of the Mœglin-Tadić admissible triple framework. By replacing intricate intertwining operator theory with systematic Jacquet module calculations, the authors provide a more concrete and directly applicable classification that extends to other reductive $p$-adic groups.

ABSTRACT

We obtain a classification of discrete series representations of odd general spin groups, generalizing the Mœglin-Tadić classification for classical groups. Using mostly algebraic methods, available in both classical and the odd general spin case, we provide a simplified and shorter version of the classification of non-cuspidal discrete series in terms of the admissible triples. The presented classification can potentially be directly applied to many other reductive $p$-adic groups.

Motivation & Objective

• To extend the Mœglin-Tadić classification of discrete series representations to odd general spin groups over non-archimedean local fields of characteristic zero.
• To simplify and unify the classification proof by replacing subtle intertwining operator theory with algebraic Jacquet module techniques.
• To define the $\epsilon$-function exclusively via Jacquet modules, enabling more concrete characterizations and avoiding analytic continuation.
• To establish a bijective correspondence between isomorphism classes of non-cuspidal discrete series and admissible triples, with a new, more direct construction.
• To demonstrate that the classification is directly applicable to other reductive $p$-adic groups, including even general spin and similitude classical groups.

Proposed method

• Construct a bijective correspondence between non-cuspidal discrete series representations and admissible triples, defined via Jordan blocks, partial cuspidal support, and an $\epsilon$-function.
• Use Jacquet module techniques to compute and characterize the $\epsilon$-function on consecutive pairs in Jordan blocks, replacing the need for standard intertwining operators.
• Define the $\epsilon$-function only on consecutive pairs $((a_-,\rho),(a,\rho))$, simplifying technical challenges in injectivity proofs.
• Apply an inductive construction based on methods from [24], [26], and [16] to prove surjectivity, significantly shortening the original Mœglin-Tadić arguments.
• Leverage the square-integrability criterion and uniqueness of partial cuspidal support to ensure the classification is both injective and surjective.
• Use results from [22, Section 4] for Jacquet module calculations in the classical group case to avoid redundant computations in the GSpin setting.

Experimental results

Research questions

• RQ1 Can the Mœglin-Tadić classification of discrete series for classical groups be simplified and extended to odd general spin groups using algebraic methods?
• RQ2 How can the $\epsilon$-function be redefined using only Jacquet module data, avoiding the need for analytic continuation of intertwining operators?
• RQ3 What structural advantages does defining the $\epsilon$-function on consecutive Jordan block pairs provide for proving injectivity?
• RQ4 Can the surjectivity of the classification map be established via an inductive construction that avoids the original lengthy arguments of Mœglin and Tadić?
• RQ5 To what extent is this classification method directly applicable to other reductive $p$-adic groups beyond classical and odd general spin groups?

Key findings

• The paper establishes a natural bijection between isomorphism classes of non-cuspidal discrete series representations and admissible triples for odd general spin groups.
• The $\epsilon$-function is fully defined via Jacquet modules, yielding concrete characterizations (i), (ii), and (iii) in Theorem 1.1(3), which are not available in the original Mœglin-Tadić framework.
• The injectivity of the classification map is proven using a refined definition of the $\epsilon$-function on consecutive pairs, avoiding technical difficulties from earlier approaches.
• Surjectivity is established via an inductive procedure based on [24], [26], and [16], significantly shortening the original proofs in [22].
• The classification is shown to be directly applicable to other reductive $p$-adic groups, including even general spin groups and similitude classical groups.
• For each admissible triple, there exists a unique discrete series representation realizing it, as proven by constructing such representations via induced representations and subrepresentation uniqueness.