Abstract
Abstract We study GL 2 ( F ) {\operatorname{GL}_{2}(F)} -invariant periods on representations of GL 2 ( A ) {\operatorname{GL}_{2}(A)} , where F is a non-archimedean local field and A / F {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 ⩽ 1 {\leqslant 1} , and is non-zero when a certain ε-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 × F × F {F\times F\times F} .
Highlights
One of the central problems in the theory of smooth representations of reductive groups over non-archimedean local fields is to determine when a representation of a group G admits a linear functional invariant under a closed subgroup H.The Gross–Prasad conjectures [5] give a very precise and elegant description of when such periods exist, for many natural pairs (G, H), in terms of ε-factors
We study GL2(F)-invariant periods on representations of GL2(A), where F is a non-archimedean local field and A/F a product of field extensions of total degree 3
We give an extension of this result to a certain class of reducible representations, extending results of Harris–Scholl when A is the split algebra F × F × F
Summary
One of the central problems in the theory of smooth representations of reductive groups over non-archimedean local fields is to determine when a representation of a group G admits a linear functional invariant under a closed subgroup H (an H-invariant period). The ε-factor is still well-defined for all such L-packets, the conjecture formulated in [4] only applies when the L-parameters satisfy an additional “relevance” condition, raising the natural question of whether the ε-factors for non-relevant L-packets have any significance in terms of invariant periods. In this short note, we describe some computations of branching laws in the following simple case: G is GL2(A), where A/F is a cubic étale algebra, and H is the subgroup GL2(F). This is the analogue for quadratic Hilbert modular forms of the result proved in [7] for Beilinson’s elements attached to the Rankin convolution of two modular forms
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