[Paper Review] Conjectures about distinction and Asai $L$-functions of generic representations of general linear groups over local fields

Nadir Matringe|ArXiv.org|Nov 10, 2008

Advanced Algebra and Geometry11 references3 citations

TL;DR

This paper establishes a conjectural equivalence between the Rankin-Selberg type Asai L-function and the Asai L-function of the Langlands parameter for generic representations of $GL(n,K)$, where $K/F$ is a quadratic extension of $p$-adic fields. Using a method inspired by Cogdell and Piatetski-Shapiro, it shows that this equality holds if and only if a conjecture on the classification of distinguished generic representations—based on their inducing quasi-square-integrable representations—is true, and proves the result for principal series representations.

ABSTRACT

Let $K/F$ be a quadratic extension of p-adic fields. The Bernstein-Zelevinsky's classification asserts that generic representations are parabolically induced from quasi-square-integrable representations. We show, following a method developed by Cogdell and Piatetski-Shapiro, that the equality of the Rankin-Selberg type Asai $L$-function of generic representations of $GL(n,K)$ and of the Asai $L$-function of the Langlands parameter, is equivalent to the truth of a conjecture about classification of distinguished generic representations in terms of the inducing quasi-square-integrable representations. As the conjecture is true for principal series representations, this gives the expression of the Asai L-function of such representations.

Motivation & Objective

• To establish a precise link between the Asai $L$-function defined via Rankin-Selberg integrals and the Asai $L$-function derived from the Langlands parameter of a generic representation of $GL(n,K)$.
• To investigate the connection between the analytic properties of the Asai $L$-function—specifically poles at zero—and the representation-theoretic notion of distinction (existence of a nonzero $GL(n,F)$-invariant linear form).
• To formulate and analyze a conjecture that characterizes distinguished generic representations in terms of the Galois conjugation and duality of their inducing quasi-square-integrable representations.
• To prove that the equality of the two Asai $L$-functions holds for irreducible principal series representations, assuming the conjecture on distinguished representations.
• To show that the equality of the Asai $L$-functions is equivalent to the truth of the conjecture on the classification of distinguished generic representations.

Proposed method

• Adapts the method of Cogdell and Piatetski-Shapiro for computing $L$-functions of pairs to the Asai $L$-function setting, focusing on the distinction of representations.
• Uses the Bernstein–Zelevinsky classification to express generic representations as normalized parabolic inductions from quasi-square-integrable representations.
• Applies the theory of Whittaker models and intertwining operators to relate the meromorphic properties of Rankin-Selberg integrals to the structure of inducing data.
• Employs the functional equation and meromorphic continuation of Rankin-Selberg integrals to analyze poles of the Asai $L$-function.
• Uses the inductive structure of induced representations to decompose the $L$-function into products involving subrepresentations and their Galois twists.
• Applies the integration formula from [J-P-S] to relate integrals over $GL(r,F)$ to those over $GL(n,F)$, enabling the comparison of $L$-functions via Whittaker functionals.

Experimental results

Research questions

• RQ1 Under what conditions on the inducing quasi-square-integrable representations is a generic representation of $GL(n,K)$ distinguished (i.e., admits a nonzero $GL(n,F)$-invariant linear form)?
• RQ2 Is the Rankin-Selberg type Asai $L$-function of a generic representation equal to the Asai $L$-function of its Langlands parameter?
• RQ3 Does the Asai $L$-function of a generic representation have an exceptional pole at zero if and only if the representation is distinguished?
• RQ4 Can the equality of the Asai $L$-functions be reduced to a conjecture on the structure of inducing data for generic representations?
• RQ5 What is the explicit form of the Asai $L$-function for irreducible principal series representations of $GL(n,K)$?

Key findings

• The equality of the Rankin-Selberg type Asai $L$-function $L_F^K( heta)$ and the Asai $L$-function of the Langlands parameter $L_F^{K,W}( heta)$ for a generic representation $ heta$ of $GL(n,K)$ is equivalent to the truth of Conjecture 1.1 on the classification of distinguished representations.
• For irreducible principal series representations, the conjecture on distinguished representations is true, which implies that $L_F^K( heta) = L_F^{K,W}( heta)$ for such representations.
• The Asai $L$-function of a generic representation $ heta$ has an exceptional pole at zero if and only if $ heta$ is distinguished, under the assumption that the conjecture on inducing data holds.
• If the representation $ heta$ is decomposed as $ heta = heta_d imes heta_u$, with $ heta_d$ being the part induced from representations satisfying $ heta_{i+1}^ au = heta_i^ au$, then the pole order of $L_F^K( heta)$ at zero is determined solely by $L_F^K( heta_d)$, provided $ heta_u$ is nonzero and composed of distinct, non-distinguished representations.
• The function $L_F^K( heta_d)$ divides the Euler factor $L_{(0)}( heta)$, which ensures that the pole at zero of $L_F^K( heta)$ is not exceptional unless $ heta_u = 0$, thus linking the analytic and representation-theoretic properties.