Finite Theta Correspondence of Almost Characters

TL;DR

This paper establishes that the decomposition of Weil characters with respect to almost characters matches that with irreducible characters for certain finite reductive dual pairs, leading to a finite theta correspondence on almost characters.

Contribution

It demonstrates the equivalence of decompositions of Weil characters for specific dual pairs with respect to almost characters and irreducible characters, establishing a finite theta correspondence.

Findings

Decomposition of Weil characters matches for almost and irreducible characters.

Finite theta correspondence on almost characters is established.

Results apply to specific dual pairs like $( ext{Sp}_{2n}, ext{O}^ ext{±}_{2n'})$ and $( ext{Sp}_{2n}, ext{SO}_{2n'+1})$.

Abstract

The theory of almost characters which is closely related to character sheaves is proposed by Lusztig to study the representation theory of finite reductive groups. In this article we show that the decomposition of the Weil character for finite reductive dual pairs $(\Sp_{2n},\rmO^\pm_{2n'})$ or $(\Sp_{2n},\SO_{2n'+1})$ with respect to the almost characters is exactly the same as the decomposition with respect to the irreducible characters. As a consequence, the finite theta correspondence on almost characters is established.