A note on toric periods in unramified families

TL;DR

This work constructs an explicit, $A$-valued normalization of toric periods for unramified parabolically induced representations over both $Q(A)$ and its specialization $A$. By realizing the representation in a Whittaker model and employing a formal zeta-integral, the author defines a linear form $l_A$ uniquely up to $A^{\times}$, satisfying $l_A(f^{\mathrm{sph}})=1-q^{-1}X_1X_2^{-1}$. The main finding identifies $l_A(i_A(\chi))$ as the non-free $A$-submodule of $A$ generated by $(1-q^{-1/2}X_1)$ and $(1-q^{-1}X_1X_2^{-1})$, equivalently $(1-q^{1/2}X_2)$ and $(1-q^{-1}X_1X_2^{-1})$, thereby embedding the $A$-structure precisely into $A$. This clarifies how the $Q(A)$-level normalization specializes to $A$ and suggests broader unramified Gan–Gross–Prasad generalizations; it also notes a subtle irreducibility phenomenon for $i_A(\chi)$.

Abstract

Let $A$ be the algebra $\mathbb{C}[X_1^{\pm 1},X_2^{\pm 1}]$ and $Q(A)$ its quotient field. In this short article, we exhibit the correct normalization for the toric period on the parabolically induced unramified family over $Q(A)$, so that it behaves optimally under restriction to the parabolically induced unramified family over $A$. This answers a question raised by D. Prasad, and points towards potential generalizations to a broader unramified Gan-Gross-Prasad setting.

Theorems & Definitions (9)

• Theorem A
• Remark 1.3.1
• Remark 1.3.2
• Definition 2.2.1
• Definition 2.2.2
• Lemma 2.2.3
• proof
• Theorem 3.1.1
• Theorem 3.2.1