A note on toric periods in unramified families

TL;DR

This paper determines the proper normalization of toric periods in unramified families of algebraic groups, ensuring optimal behavior under restriction, and addresses a question by D. Prasad with implications for broader unramified Gan-Gross-Prasad problems.

Contribution

It provides the correct normalization for toric periods in unramified families, facilitating better understanding of their behavior and potential generalizations.

Findings

Correct normalization for toric periods established

Optimal behavior under restriction demonstrated

Addresses a question by D. Prasad

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.