On a question of Drinfeld on the Weil representation I: the finite field case

TL;DR

This paper decomposes the Weil representation of GL2 over finite fields and cubic algebras into irreducible components, addressing a question posed by Drinfeld about the structure of these representations.

Contribution

It provides the first explicit decomposition of the Weil representation for GL2 over finite fields and cubic algebras, answering Drinfeld's question.

Findings

Decomposition of Weil representation into irreducibles for finite fields

Explicit analysis of Weil representation over cubic algebras

Addresses a longstanding question by Drinfeld

Abstract

Let F be a finite field of odd cardinality, and let G= GL2(F). The group G \times G \times G acts on F^2 \otimes F^2 \otimes F^2 via symplectic similitudes, and has a natural Weil representation. Answering a question rasised by V. Drinfeld, we decompose that representation into irreducibles. We also decompose the analogous representation of GL2(A), where A is a cubic algebra over F.