## Duke Mathematical Journal

### Absolute convergence of Eisenstein series on loop groups

Howard Garland

#### Abstract

In earlier work, we established the a.e. convergence of certain Eisenstein series on arithmetic quotients of loop groups. In this article we prove that these series converge everywhere, and uniformly on certain bounded sets

#### Article information

Source
Duke Math. J. Volume 135, Number 2 (2006), 203-260.

Dates
First available in Project Euclid: 17 October 2006

https://projecteuclid.org/euclid.dmj/1161093264

Digital Object Identifier
doi:10.1215/S0012-7094-06-13521-4

Mathematical Reviews number (MathSciNet)
MR2267283

Zentralblatt MATH identifier
1160.11025

#### Citation

Garland, Howard. Absolute convergence of Eisenstein series on loop groups. Duke Math. J. 135 (2006), no. 2, 203--260. doi:10.1215/S0012-7094-06-13521-4. https://projecteuclid.org/euclid.dmj/1161093264.

#### References

• H. Garland, The arithmetic theory of loop algebras, J. Algebra 53 (1978), 480--551.; Erratum, J. Algebra 63 (1980), 285. $\!$;
• —, The arithmetic theory of loop groups, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 5--136.
• —, The arithmetic theory of loop groups, II: The Hilbert-modular case, J. Algebra 209 (1998), 446--532.
• —, Certain Eisenstein series on loop groups: Convergence and the constant term'' in Algebraic Groups and Arithmetic (Mumbai, 2001), Tata Inst. Fund. Res., Mumbai, 2004.
• —, Eisenstein series on loop groups: Maass-Selberg relations 1'' to appear in Algebraic Groups and Homogeneous Spaces (Mumbai, 2004), Tata Inst. Fund. Res., Mumbai.
• —, Eisenstein series on loop groups: Maass-Selberg relations 2, to appear in Amer. J. Math.
• —, Eisenstein series on loop groups: Maass-Selberg relations 3, to appear in Amer. J. Math.
• —, Eisenstein series on loop groups: Maass-Selberg relations 4, in preparation.
• N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of $p$-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5--48.
• F. Shahidi, Infinite dimensional groups and automorphic $L$-functions, Pure Appl. Math. Q. 1 (2005), 683--699.