Poles of Eisenstein series on quaternion groups
This paper determines the poles of the normalized Siegel Eisenstein series of quaternion groups. Let \(k\) be a totally real field and let \(D\) be a quaternion division algebra over \(k\). Let \(\varepsilon=\pm 1\) and let \(W\) be a \(D\)-vector space of dimension \(2n\) equipped with an \(\varepsilon\)-Hermitian form. It defines the group \(O^*(4n)\) if\(\varepsilon= -1\) and \(\text{Sp}(n, n)\) if \(\varepsilon= 1\). We denote this group by \(G\). Let \(P= MN\) be the Siegel parabolic subgroup which stabilizes a maximal isotropic subgroup of \(W\). Let \(\chi: \mathbb A^\times/k^\times\to \mathbb C^\times\) be a (unitary) Hecke character. Let \(v=|\cdot|\) denote the norm character on \(\mathbb A^\times/k^\times\). Let det denote the determinant on \(M\simeq\text{GL}(n, D)\). Then we form the normalized induced degenerate principal series representation \[ I(\chi, s)= \text{Ind}^{G(\mathbb A)}_{P(\mathbb A)}(\chi v^s(\text{det\,}a)), \] where \(s\in\mathbb C\). Let \(\Phi(g, s)\in I_s\). Suppose \(\Phi(g, s)\) is smooth and \(K_v\)-finite at \(v=\infty\) and locally constant and compactly supported at all finite places \(v\). Then we define the Eisenstein series \[ E(g,s,\Phi)= \sum_{\gamma\in P(k)\setminus G(k)} \Phi(\gamma g,s) \] for \(g\in G(\mathbb{A})\). It is well-known that the right-hand side converges for \({\mathfrak R}(s)>\rho_n\) where \(\rho_n:= {1\over 2}(n+\varepsilon)\). The Eisenstein series has a meromorphic continuation to the complex plane and satisfies certain functional equation. Eisenstein series are very important in the theory of automorphic representations. One defines the normalized Eisenstein series \[ E^*(g, s,\Phi)= \Biggl(\prod_{v\not\in S} b_v(s)\Biggr)\, E(g,s,\Phi), \] where \(S\) is a finite set of places which includes all the bad primes. We refer to the paper of the precise definition of \(b_v(s)\) which is product zeta functions. The main result of this paper shows that the poles of \(E^*(g,s,\Phi)\) are at most simple and occurs within the set \[ \Biggl\{-\rho_{4n},{1\over 2}- \rho_{4n},\;1-\rho_{4n},\dots, 0,\dots, \rho_{4n}\Biggr\}. \]
scientific article; zbMATH DE number 5759073
scientific article