A regularized Siegel–Weil formula on U(1,1) and application to theta lifting from U(1) to U(1,1)

In this paper, we obtain a second term identity of the regularized Siegel–Weil formula for the unitary dual pair (U(1,1),U(1,1)), and we use this to derive a Rallis inner product formula for theta lifting from U(1) to U(1,1). By choosing a good Schwartz function in the inner product formula, we obtain a special value formula for some Hecke L-function at s=1 in terms of the norm of a theta function on U(1,1).