Archimedean Rankin-Selberg integrals.

*(English)*Zbl 1250.11051
Ginzburg, David (ed.) et al., Automorphic forms and \(L\)-functions II. Local aspects. A workshop in honor of Steve Gelbart on the occasion of his sixtieth birthday, Rehovot and Tel Aviv, Israel, May 15–19, 2006. Providence, RI: American Mathematical Society (AMS); Ramat Gan: Bar-Ilan University. (ISBN 978-0-8218-4708-4/pbk). Contemporary Mathematics 489. Israel Mathematical Conference Proceedings, 57-172 (2009).

This excellent and voluminous (120 p.) paper is best described by its introduction:

The goal of these notes is to give a definitive exposition of the local Archimedean theory of the Rankin-Selberg integrals for the group \(\text{GL}(n)\). Accordingly, the ground field \(F\) is either \(\mathbb R\) or \(\mathbb C\). The integrals at hand are attached to pairs of irreducible representations \((\pi, V)\) and \((\pi', V')\) of \(\text{GL}(n, F)\) and \(\text{GL}(n', F)\) respectively. More precisely, each integral is attached to a pair of functions \(W\) and \(W'\) in the Whittaker models of \(\pi\) and \(\pi'\), respectively and, in the case \(n = n'\), a Schwartz function in \(n\) variables. More generally, it is necessary to consider instead of a pair \((W,W')\) a function in the Whittaker model of the completed tensor product \(V\otimes V'\). The integrals depend on a complex parameter \(s\). They converge absolutely for \(\text{Re}\,s \gg 0\). The goal is to prove that they extend to holomorphic multiples of the appropriate Langlands \(L\)-factor, are bounded at infinity in vertical strips, and satisfy a functional equation where the Langlands \(\varepsilon\) factor appears. This is what is needed to have a complete theory of the converse theorems (J. W. Cogdell and I. I. Piatetski-Shapiro [Publ. Math., Inst. Hautes Étud. Sci. 79, 157–214 (1994; Zbl 0814.11033)], [J. Reine Angew. Math. 507, 165–188 (1999; Zbl 0912.11022)], [(*) Contributions to automorphic forms, geometry, and number theory. Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birthday, Johns Hopkins University, Baltimore, MD, USA, 2002. Baltimore, MD: Johns Hopkins University Press, 255–278 (2004; Zbl 1080.11038)]). An alternate approach may be found in [S. D. Miller and W. Schmid, The Rankin-Selberg method for automorphic distributions. Representation theory and automorphic forms, Prog. Math. 255, 111–150 (2008; Zbl 1124.11004)].

More is proved. Namely, it is proved that the \(L\)-factor itself is a sum of such integrals. At this point in time, this result is not needed. Nonetheless, it has esthetic appeal. Indeed, it shows that the factors \(L\) and \(\varepsilon\) are determined by the representations \(\pi\) and \(\pi'\). Anyway, by using this general result and by following Cogdell and Piatetski-Shapiro (*), it is shown that for the case \((n,n-1)\) and \((n, n)\) the relevant \(L\)-factor is obtained in terms of vectors which are finite under the appropriate maximal compact subgroups. The result is especially simple in the unramified situation, a result proved by E. Stade [Duke Math. J. 60, No. 2, 313–362 (1990; Zbl 0731.11027)], [Am. J. Math. 123, No. 1, 121–161 (2001; Zbl 1017.11022)] with a different proof.

A first version of these notes was published earlier (cf. the author and J. Shalika [Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Pt. I: Papers in representation theory, Pap. Workshop L-Functions, Number Theory, Harmonic Anal., Tel-Aviv/Isr. 1989, Isr. Math. Conf. Proc. 2, 125–207 (1990; Zbl 0712.22011)]). The present notes are more detailed. Minor mistakes of the previous version have been corrected. More importantly, in contrast to that version, the methods are uniform as all the results are derived from an integral representation of the Whittaker functions, the theory of the Tate integral, and the Fourier inversion formula. The estimates for a Whittaker function are derived from coarse estimates which are then improved by applying the same coarse estimates to the derivatives of the Whittaker function, a method first used by Harish-Chandra. This is simpler than giving an explicit description of the Whittaker functions and then deriving estimates, as was done in the previous version. In [Contributions to automorphic forms, geometry, and number theory. Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birthday, Johns Hopkins University, Baltimore, MD, USA, 2002. Baltimore, MD: Johns Hopkins University Press, 373–419 (2004; Zbl 1084.11022)], the author proposed another approach to the study of the integrals. Again, the approach of the present notes is in fact simpler. Thus I hope that these notes can be indeed regarded as a definitive treatment of the question.

Difficult results on smooth representations and Whittaker vectors due to N. R. Wallach [Real reductive groups II. Pure and Applied Mathematics, 132, Pt. 2. Boston, MA etc.: Academic Press (1992; Zbl 0785.22001)], W. Casselman [Can. J. Math. 41, No. 3, 385–438 (1989; Zbl 0702.22016)], W. Casselman, H. Hecht and D. Miličić [Proc. Symp. Pure Math. 68, 151–190 (2000; Zbl 0959.22010)] are used in an essential way.

Needless to say, these notes owe much to my former collaborators, Piatetski- Shapiro and Shalika. In particular, the ingenious induction step from \((n, n-1)\) to \((n, n)\) is due to Shalika.

For the entire collection see [Zbl 1167.11002].

The goal of these notes is to give a definitive exposition of the local Archimedean theory of the Rankin-Selberg integrals for the group \(\text{GL}(n)\). Accordingly, the ground field \(F\) is either \(\mathbb R\) or \(\mathbb C\). The integrals at hand are attached to pairs of irreducible representations \((\pi, V)\) and \((\pi', V')\) of \(\text{GL}(n, F)\) and \(\text{GL}(n', F)\) respectively. More precisely, each integral is attached to a pair of functions \(W\) and \(W'\) in the Whittaker models of \(\pi\) and \(\pi'\), respectively and, in the case \(n = n'\), a Schwartz function in \(n\) variables. More generally, it is necessary to consider instead of a pair \((W,W')\) a function in the Whittaker model of the completed tensor product \(V\otimes V'\). The integrals depend on a complex parameter \(s\). They converge absolutely for \(\text{Re}\,s \gg 0\). The goal is to prove that they extend to holomorphic multiples of the appropriate Langlands \(L\)-factor, are bounded at infinity in vertical strips, and satisfy a functional equation where the Langlands \(\varepsilon\) factor appears. This is what is needed to have a complete theory of the converse theorems (J. W. Cogdell and I. I. Piatetski-Shapiro [Publ. Math., Inst. Hautes Étud. Sci. 79, 157–214 (1994; Zbl 0814.11033)], [J. Reine Angew. Math. 507, 165–188 (1999; Zbl 0912.11022)], [(*) Contributions to automorphic forms, geometry, and number theory. Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birthday, Johns Hopkins University, Baltimore, MD, USA, 2002. Baltimore, MD: Johns Hopkins University Press, 255–278 (2004; Zbl 1080.11038)]). An alternate approach may be found in [S. D. Miller and W. Schmid, The Rankin-Selberg method for automorphic distributions. Representation theory and automorphic forms, Prog. Math. 255, 111–150 (2008; Zbl 1124.11004)].

More is proved. Namely, it is proved that the \(L\)-factor itself is a sum of such integrals. At this point in time, this result is not needed. Nonetheless, it has esthetic appeal. Indeed, it shows that the factors \(L\) and \(\varepsilon\) are determined by the representations \(\pi\) and \(\pi'\). Anyway, by using this general result and by following Cogdell and Piatetski-Shapiro (*), it is shown that for the case \((n,n-1)\) and \((n, n)\) the relevant \(L\)-factor is obtained in terms of vectors which are finite under the appropriate maximal compact subgroups. The result is especially simple in the unramified situation, a result proved by E. Stade [Duke Math. J. 60, No. 2, 313–362 (1990; Zbl 0731.11027)], [Am. J. Math. 123, No. 1, 121–161 (2001; Zbl 1017.11022)] with a different proof.

A first version of these notes was published earlier (cf. the author and J. Shalika [Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Pt. I: Papers in representation theory, Pap. Workshop L-Functions, Number Theory, Harmonic Anal., Tel-Aviv/Isr. 1989, Isr. Math. Conf. Proc. 2, 125–207 (1990; Zbl 0712.22011)]). The present notes are more detailed. Minor mistakes of the previous version have been corrected. More importantly, in contrast to that version, the methods are uniform as all the results are derived from an integral representation of the Whittaker functions, the theory of the Tate integral, and the Fourier inversion formula. The estimates for a Whittaker function are derived from coarse estimates which are then improved by applying the same coarse estimates to the derivatives of the Whittaker function, a method first used by Harish-Chandra. This is simpler than giving an explicit description of the Whittaker functions and then deriving estimates, as was done in the previous version. In [Contributions to automorphic forms, geometry, and number theory. Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birthday, Johns Hopkins University, Baltimore, MD, USA, 2002. Baltimore, MD: Johns Hopkins University Press, 373–419 (2004; Zbl 1084.11022)], the author proposed another approach to the study of the integrals. Again, the approach of the present notes is in fact simpler. Thus I hope that these notes can be indeed regarded as a definitive treatment of the question.

Difficult results on smooth representations and Whittaker vectors due to N. R. Wallach [Real reductive groups II. Pure and Applied Mathematics, 132, Pt. 2. Boston, MA etc.: Academic Press (1992; Zbl 0785.22001)], W. Casselman [Can. J. Math. 41, No. 3, 385–438 (1989; Zbl 0702.22016)], W. Casselman, H. Hecht and D. Miličić [Proc. Symp. Pure Math. 68, 151–190 (2000; Zbl 0959.22010)] are used in an essential way.

Needless to say, these notes owe much to my former collaborators, Piatetski- Shapiro and Shalika. In particular, the ingenious induction step from \((n, n-1)\) to \((n, n)\) is due to Shalika.

For the entire collection see [Zbl 1167.11002].

Reviewer: Olaf Ninnemann (Berlin)