Generalized Preservation Principle in Finite Theta Correspondence

It is known that irreducible cuspidal characters satisfy the preservation principle in the Howe correspondences of finite reductive dual pairs. In this article, we generalize the preservation principle to any irreducible characters of classical groups.Comment: The results in this paper replace Section 8-10 of old version of arXiv:1906.0115

Finite Theta Correspondence of Almost Characters

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.Comment: 39 page