On a family of distributions obtained from Eisenstein series. I: Application of the Paley-Wiener theorem.

*(English)*Zbl 0541.22010This paper represents a step towards the higher rank trace formula for an adelic reductive algebraic group. References for the classical result include, for example, A. Selberg [J. Indian Math. Soc., New Ser. 20, 47–87 (1956; Zbl 0072.08201)], D. Hejhal [The Selberg trace formula for \(\text{PSL}(2,\mathbb R)\), Vols. I, II (Lect. Notes Math. 548 and 1001) (1976; Zbl 0347.10018), (1983; Zbl 0543.10020)], the reviewer’s [Harmonic analysis on symmetric spaces and applications, New York: Springer-Verlag (1985; Zbl 0574.10029)]. In the classical case of \(\text{SL}(2,\mathbb Z)\), say, a truncation of the Eisenstein series term was cancelled against a truncated term coming from the parabolic elements of \(\text{SL}(2,\mathbb Z)\). In the higher rank situation this process will certainly be more complicated. The reviewer found it useful to read the author’s paper “The trace formula for noncompact quotient” in [Proc. Int. Congr. Math., Warszawa 1983, Vol. 2, 849–859 (1984; Zbl 0563.10024)]. Here the author considered the case \(\text{SL}(2,\mathbb Z)\) in some detail.

The paper under review uses the asymptotic formula from the author’s paper [Duke Math. J. 49, 35–70 (1982; Zbl 0518.22012)] and introduces some cutoff functions \(B^{\varepsilon}_{\pi}\) associated to \(\varepsilon>0\) and \(\pi\), an irreducible unitary representation of \(M_ p({\mathbb A})^ 1\). The proof of the main theorem involving these cutoff functions is “indirect”. The main ingredients are two rather deep results from other papers. The first result is that the terms from truncated Eisenstein series are polynomial in the truncation variable \(T\), for suitably regular \(T\). The second ingredient is a multiplier theorem for the Hecke algebra on \(G({\mathbb R})^ 1=G({\mathbb R})\cap G({\mathbb A})^ 1\). The second ingredient was proved as a corollary of the Paley-Wiener theorem in an earlier paper of the author.

The paper under review uses the asymptotic formula from the author’s paper [Duke Math. J. 49, 35–70 (1982; Zbl 0518.22012)] and introduces some cutoff functions \(B^{\varepsilon}_{\pi}\) associated to \(\varepsilon>0\) and \(\pi\), an irreducible unitary representation of \(M_ p({\mathbb A})^ 1\). The proof of the main theorem involving these cutoff functions is “indirect”. The main ingredients are two rather deep results from other papers. The first result is that the terms from truncated Eisenstein series are polynomial in the truncation variable \(T\), for suitably regular \(T\). The second ingredient is a multiplier theorem for the Hecke algebra on \(G({\mathbb R})^ 1=G({\mathbb R})\cap G({\mathbb A})^ 1\). The second ingredient was proved as a corollary of the Paley-Wiener theorem in an earlier paper of the author.

Reviewer: A.Terras

##### MSC:

22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |

43A80 | Analysis on other specific Lie groups |