Gross--Prasad periods for reducible representations

TL;DR

This paper extends the understanding of GL_2-invariant periods from irreducible to certain reducible representations over nonarchimedean local fields, providing new dimension bounds and epsilon-factor conditions.

Contribution

It generalizes Prasad's theorem to reducible representations of Whittaker type, expanding the class of representations for which period dimensions are understood.

Findings

Dimension of period space is at most 1 for the studied reducible representations.

Non-zero periods occur under specific epsilon-factor conditions.

Extends Harris--Scholl results to non-split algebra cases.

Abstract

We study GL_2(F)-invariant periods on representations of GL_2(A), where F is a nonarchimedean local field and A/F a product of field extensions of total degree 3. For irreducible representations, a theorem of Prasad shows that the space of such periods has dimension at most 1, and is non-zero when a certain epsilon-factor condition holds. We give an extension of this result to a certain class of reducible representations (of Whittaker type), extending results of Harris--Scholl when A is the split algebra F x F x F.