A regularized Siegel-Weil formula on \(U(1,1)\) and application to theta lifting from \(U(1)\) to \(U(1,1)\)
This paper extends to the unitary dual pair \((U(1,1), U(1,1))\) the work of \textit{S. S. Kudla} and \textit{S. Rallis} on the regularized Siegel-Weil formula for a symplectic-orthogonal dual pair and its application to the study of theta liftings of cuspidal automorphic forms [Ann. Math. (2) 140, No. 1, 1--80 (1994; Zbl 0818.11024)]. In first instance, the author establishes a regularized Siegel-Weil formula following the way of \textit{V. Tan} [Duke Math. J. 94, No. 2, 341--378 (1998 ; Zbl 0989.11025)] and \textit{A. Ichino} [Math. Z. 247, No. 2, 241--277 (2004 ; Zbl 1071.11022)]. He obtains a second term identity, namely a relation between the residue of the regularized theta integral and the constant term of the Siegel-Eisenstein series (Theorem 3.5). Then he derives a Rallis inner product formula which expresses the Petersson product of theta liftings of two characters of \(U(1)\) to \(U(1,1)\) in terms of special values of Hecke \(L\)-functions (Theorem 4.1). Finally, he deduces by choosing a good Schwartz function that the norm of the corresponding theta lifting of a character of \(U(1)\) is a product of a nonzero explicit constant by the value of a Hecke \(L\)-function at \(s=1\) (Theorem 4.2). The methods essentially follow the outline of the papers mentioned earlier.
scientific article; zbMATH DE number 6138259
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