Weil representations over finite fields and Shintani lift

TL;DR

This paper studies Weil representations of symplectic groups over finite fields, demonstrating their compatibility with Shintani base change and extending results to similitude groups.

Contribution

It establishes the compatibility of Weil representations with Shintani base change for symplectic and similitude groups over finite fields.

Findings

Weil representations are compatible with Shintani base change.

Compatibility extends to groups of similitudes.

Results hold for finite fields of odd cardinality.

Abstract

Let Sp_V(F) be the group of isometries of a symplectic vector space V over a finite field F of odd cardinality. The group Sp_V(F) possesses distinguished representations--- the Weil representations. We know that they are compatible with base change in the sense of Shintani for a finite extension F'/F. The result is also true for the group of similitudes of V.