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 $({\text{Sp}}_{2n},{\mathrm{O}}_{2n^{\mathrm{\prime }}}^{\pm })$ or $({\text{Sp}}_{2n},{\text{SO}}_{2n^{\mathrm{\prime }}+1})$ 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.