An explicit formula of the unramified Shintani functions for \((\mathbf{GSp}_4, \mathbf{GL}_2 \times_{\mathbf{GL}_1} \mathbf{GL}_2)\)
For a non-archimedean local field \(F\) of arbitrary characteristics, let \(G\) be \(\mathbf{GSp}_4(F)\) and \(G_0\) a subgroup of \(G\) isomorphic to \((\mathbf{GL_2\times_{\mathbf{GL_1}}\mathbf{GL}_2})(F)\). The article under review studies the unramified Shintani functions for the pair \((G,G_0)\). Let \(X_{nr}(T)\) (resp. \(X_{nr}(T_0)\)) be the group of unramified characters of the diagonal subgroup \(T\) of \(G\) (resp. \(T_0\) of \(G_0\)). For \((\xi,\Xi)\in X_{nr}(T)\times X_{nr}(T_0)\), let \(\mathcal{S}(\xi,\Xi)\) denote the space of Shintani functions of type \((\xi,\Xi)\). The author establishes the uniqueness of the Shintani functions, namely, \(\dim_{\mathbb{C}}\mathcal{S}(\xi,\Xi)=1\) for \((\xi\Xi)|_Z\equiv 1\), and \(=0\) otherwise; here \(Z\) is the center of \(G\); and when \(\mathcal{S}(\xi,\Xi)\) is non-zero, the author gives an explicit formula for the Shintani function \(W_{\xi,\Xi}\in\mathcal{S}(\xi,\Xi)\) satisfying \(W_{\xi,\Xi}(1_4)=1\) (see Theorem A). This generalizes the result of [\textit{S. Kato} et al., Tohoku Math. J., II. Ser. 55, No. 1, 1--564 (2003; Zbl 1037.22034)] for the split special orthogonal group \(\mathbf{SO}_5\).
scientific article; zbMATH DE number 6802527
scientific article