A weak second term identity of the regularized Siegel-Weil formula for unitary groups

Abstract

Following W. T. Gan and S. Takeda, we obtain a weak second term identity of the regularized Siegel-Weil formula for the unitary dual pair $$(U(n,n),U(V))$$ , where $$V$$ is a split hermitian space of dimension $$2r$$ with $$r+1\le n \le 2r-1$$ . As an application, we obtain a Rallis inner product formula for theta lifts from $$U(W)$$ to $$U(V)$$ for a skew-hermitian space $$W$$ of dimension $$n$$ .