On the analytic continuation of Eisenstein series for Siegel's modular group of degreen |
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Authors: | Ulrich Christian |
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Institution: | 1. Mathematisches Institut der Universit?t, Bunsenstrasse 3/5, D-3400, G?ttingen, Federal Republic of Germany
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Abstract: | For \(M = \left( {\begin{array}{*{20}c} {A B} \\ {C D} \\ \end{array} } \right)\) ∈ Γ(n)=Sp(n?) andZ=Z+iY,Y > 0, set $$M\left\langle Z \right\rangle = (AZ + B)(CZ + D)^{ - 1} = X_M + iY_M ;M\{ Z\} = CZ + D.$$ Denote with Γ n (n) the subgroup defined byC=0. Forr∈? and a complex variable ω form the Eisenstein series $$E(n,r,Z,\omega ) = \sum\limits_{M\varepsilon I'_n (n)\backslash \Gamma (n)} {(DetM\{ Z\} )^{ - 2r} (DetY_M )^{\omega - r} } .$$ It is proved thatE(n, r, Z, ω) can be meromorphically continued to the ω-plane and satisfies a functional equation. Forr=1, 2, (n?1)/2], (n+1)/2] the functionE(n, r, Z, ω) is holomorphic at ω-r. For 3≤r≤(n?3)/2] the functionE(n, r, Z, ω) may have poles at ω=r. But the pole-order is for two smaller than known until now. This result says especially that the Eisenstein series has Hecke summation forr=1, 2, (n?1)/2], (n+1)/2]. |
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