A regularized Siegel-Weil formula for unitary groups
Let \(F\) be a totally real field with adele ring \(\mathbb A=\mathbb A_F\), and let \(E\) be a totally imaginary quadratic extension of \(F\). Let \((V,Q)\) be a non-degenerate hermitian space over \(E\) with Witt index \(r\). Let \(G=U(n,n)\) denote the quasi-split unitary group in \(2n\) variables, and \(H=U(V)\) denote the unitary group of \((V,Q)\). Then \((G,H)\) is a reductive dual pair. Let \(\psi\) be a fixed non-trivial additive character of \(\mathbb A/F\) and let \(\chi\) be a fixed character of \(A_E^\times/E^\times\) such that \(\chi_{\mathbb A^\times}=\varepsilon^m_{E| F}\), where \(\varepsilon_{E| F}\) is the quadratic character of \(\mathbb A^\times/F^\times\) corresponding to \(E/F\) by class field theory. Then \(G(\mathbb A)\times H(\mathbb A)\) acts on the Schwartz space \(\mathcal S(V^n(\mathbb A))\) by the Weil representation \(\omega_Q=\omega_{Q,\psi,\chi}\). Let \(P\) be the Siegel parabolic subgroup of \(G\), and let \(K_G\) be a fixed standard maximal compact subgroup of \(G(\mathbb A)\). Put \(| a(g)| _{\mathbb A_E}:=| \det(a)| _{\mathbb A_E}\) for \(g=pk\in G(\mathbb A)\), with \[ p=\left(\begin{matrix} a&*\cr 0_n&{}^t\overline{a}^{-1}\end{matrix}\right)\in P(\mathbb A),\;\;a\in\text{GL}_n(\mathbb A_E),\quad k\in K_G. \] For each \(\Phi\in\mathcal S(V^n(\mathbb A))\), one defines a holomorphic section \(f_\Phi^{(s)}\) of \(\text{Ind}_{P(\mathbb A)}^{G(\mathbb A)}(\chi\,| \det| ^s_{\mathbb A_E})\) by \[ f_\Phi^{(s)}(g):=| a(g)| ^{s-s_0}_{\mathbb A_E}\,\omega_Q(g)\,\Phi(0), \] where \(s_0=(m-n)/2\). Then the Siegel Eisenstein series \[ E(g,f_\Phi^{(s)}):=\sum_{\gamma\in P(F)\backslash G(F)} f_\Phi^{(s)}(\gamma g) \] is absolutely convergent if \(\Re(s)>n/2\), and has meromorphic continuation to the whole \(s\)-plane if \(\phi\) is \(K_G\)-finite. On the other side, one defines the theta function \[ \Theta(g,h;\Phi):=\sum_{x\in V^n(F)}\omega_Q(g,h)\,\Phi(x), \] for each \(\Phi\). Let \(dh\) be the Haar measure on \(H(\mathbb A)\) is normalized so that \(\text{vol}(H(F)\backslash H(\mathbb A))=1\), and let \(I_Q(g,\Phi)\) denote the integral \[ I_Q(g,\Phi):=\int_{H(F)\backslash H(\mathbb A)}\Theta(g,h;\Phi)\,dh, \] when it is absolutely convergent. A.~Weil showed that \(I_Q(g,\Phi)\) is absolutely convergent for all \(\Phi\) if and only if \(m-r>r\) or \(Q\) is anisotropic, and proved that if \(m>2n\), then \(E(g,f_\Phi^{(s)})=I_Q(g,\Phi)\). Assume that \(n<m\leq 2n\) and \(m-r\neq n\). Let \((V',Q')\) be an Hermitian space over \(E\) of dimension \(m':=2n-m\) such that \[ Q=\left(\begin{matrix} &&1_{r_0}\cr &Q'&\cr 1_{r_0}&& \end{matrix} \right) \] with \(r_0:=m-n\), and let \(H'=U(V')\) be the unitary group of \((V',Q')\). Let \(\pi_Q^{Q'}:\mathcal S(V^n(\mathbb A))\to\mathcal S(V^{\prime n}(\mathbb A))\) be the map defined by \[ \pi_Q^{Q'}\Phi(v'):=\int_{\text{Mat}_{r_0,n}(\mathbb A_E)}\Phi\left(\begin{matrix} x\cr v'\cr 0 \end{matrix} \right) dx. \] Let \(P_H\) be the maximal parabolic subgroup of \(H\) given by \[ P_H(F):=\left\{\left(\begin{matrix} a&*&*\cr &h'&*\cr &&{}^t\overline{a}^{-1}\end{matrix}\right)\in H(F)\,:\,a\in\text{GL}_{r_0}(F), \;h'\in H'(F)\right\}, \] and let \(U_H\) denote its unipotent radical. Let \(K\) be a fixed good maximal compact subgroup of \(H(\mathbb A)\) such that \(H(\mathbb A)=P_H(\mathbb A)K\), an let \(\pi_K:\mathcal S(V^n(\mathbb A))\to\mathcal S(V^n(\mathbb A))\) be the map defined as \[ \pi_K\Phi(v):=\int_K\Phi(kv)\,dk, \] where the Haar measure \(dk\) on \(K\) is normalized so that \(\text{vol}(K)=1\). Take the Haar measure \(d_lp\,=\,dh'\,da\,du\) on \(P_H(\mathbb A)\), where \(dh'\) is the Haar measure on \(H'(\mathbb A)\) so that \(\text{vol}(H'(F)\backslash H'(\mathbb A)0=1\), \(da\) is the Tamagawa measure on \(\text{GL}_{r_0}(\mathbb A_E)\), and \(du\) is the Haar measure on \(U_H(\mathbb A)\) such that \(\text{vol}(U_H(F)\backslash U_H(\mathbb A))=1\). Then let \(c_K\) be the constant defined by \(dh\,=\,c_K\,d_lp\,dk\). The following result is proved in the paper (Theorem~4.1): If \(\phi\in \mathcal S(V^n(\mathbb A))\) is \(K_G\)-finite, then \[ \text{Res}_{s=s_0}E(g,f_\Phi^{(s)})\,=\,c_K\,I_{Q'}(g,\pi_Q^{Q'}\pi_K\Phi). \] Moreover, the constant \(c_K\) is explicitly computed and admits an expression in terms of certain \(L\)-values.
scientific article; zbMATH DE number 2133480