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Poles of Siegel Eisenstein Series on U(n, n)

Published online by Cambridge University Press:  20 November 2018

Victor Tan*
Affiliation:
National University of Singapore, Singapore 119260 email: [email protected]
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Abstract

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Let $U\left( n,\,n \right)$ be the rank $n$ quasi-split unitary group over a number field. We show that the normalized Siegel Eisenstein series of $U\left( n,\,n \right)$ has at most simple poles at the integers or half integers in certain strip of the complex plane.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1999

References

[Art] Arthur, J., Eisenstein series and the Trace formula. Proc. Sympos. Pure Math. Part 1 XXXIII, Amer. Math. Soc., Providence, RI, 1979, 253274.Google Scholar
[GPS] Gelbart, S., Piatetski-Shapiro, I. and Rallis, S., Explicit constructions of automorphic L-functions. Lecture Notes in Math. 1254 , Springer-Verlag, New York, 1987.Google Scholar
[How] Howe, R., A notion of rank for unitary representations of classical groups. Cortona, C. I. M. E., ed. (1980).Google Scholar
[Kar] Karel, M., Values of certain Whittaker functions on p-adic groups. Illinois J. Math. 26 (1982), 552575.Google Scholar
[KR1] Kudla, S. and Rallis, S., Poles of Eisenstein series and L-functions. In: Festschrift in honor of I. Pietetski- Shapiro, Israel Mathematical Conference Proceedings 3 (1990), 81110.Google Scholar
[KR2] Kudla, S. and Rallis, S., A regularized Siegel-Weil formula: the first term identity. Ann. of Math. 140 (1994), 180.Google Scholar
[KS] Kudla, S. and Sweet, W. J., Degenerate principal series representation for U(n; n). Israel J. Math. 98 (1997), 253306.Google Scholar
[Lai] Lai, K. F., Tamagawa number of reductive algebraic groups. Compositio Math. 41 (1980), 153188.Google Scholar
[Lee] Lee, S., On some degenerate principal series representations of U(n; n). J. Funct. Anal. 126 (1994), 305366.Google Scholar
[Shi] Shimura, G., On Eisenstein Series. Duke Math. J. 50 (1983), 417476.Google Scholar
[Sie] Siegel, C. L., Uber die analytische theorie der quadratischen formen. Ann. of Math. 36 (1935), 527606.Google Scholar
[Tan] Tan, V., A regularized Siegel-Weil formula on U(2; 2) and U(3). Duke Math. J. (2) 94 (1998), 341378.Google Scholar
[Wal] Wallach, N., Lie algebra cohomology and holomorphic continuation of generalised Jacquet integrals. Representation of Lie groups, Kyoto, Adv. Stud. Pure Math. 14 (1988), 123151.Google Scholar
[Wei] Weil, A., Sur la formule de Siegel dans la theorie des groupes classiques. Acta Math. 113 (1965), 187.Google Scholar