Stade, Eric [Bump, Daniel; Friedberg, Solomon; Hoffstein, Jeffrey] On explicit integral formulas for \(\text{GL}(n, \mathbb R)\)-Whittaker functions. With an appendix by Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein. (English) Zbl 0731.11027 Duke Math. J. 60, No. 2, 313-362 (1990). The author considers the Whittaker functions \(W_{(n,\nu)}(y)\) associated with nonramified principal series representations of the group \(\text{GL}(n, \mathbb R)\). The main result is: Theorem. Let \(n\geq 2\). If \(\nu \in {\mathbb C}^{n-1}\), put \(\lambda_{j-1}=n\nu_ j/(n-2)\) for \(2\leq j\leq n- 2\) and \(\lambda =(\lambda_ 1,\lambda_ 2,...,\lambda_{n-3})\). Also define \(u_ 0=1/u_{n-1}=0\) and \(u^ 0_{n-1}=1\). Then \[ W^*_{(n,\nu)}(y)=2^{n-1}\int_{({\mathbb R}_+)^{n-2}}\prod^{n- 1}_{i=1}u_ i^{r_{i,1}-r_{i,n-i}}K_{\mu_ 1}(2\pi y_ i\sqrt{(1+u^ 2_{i-1})(1+1/u_ i^ 2)}) \]\[ \times W^*_{(n- 2,\lambda)}(\frac{y_ 2}{u_ 2}u_ 1,\frac{y_ 3}{u_ 3}u_ 2,...,\frac{y_{n-2}}{u_{n-2}}u_{n-3})\prod^{n- 2}_{i=1}\frac{du_ i}{u_ i} \] where \(K\) denotes the \(K\)-Bessel function \(K_{\mu}(2\pi y)=\frac{1}{2}\int^{\infty}_{0}t^{\mu}\exp (-\pi y(t+1/t))\frac{dt}{t}\) \((y>0)\). Here \(W^*_{(n,\nu)}\) is \(W_{(n,\nu)}\) divided by some powers of the \(y_ i\)’s. As applications of the above theorem, he expresses the local factors at the archimedean places for the exterior square automorphic \(L\)-function on \(\text{GL}(n)\) and for automorphic functions for \(\text{GL}(2,\mathbb R)\times \text{GL}(3,\mathbb R)\) as products of Gamma functions. He also gives growth estimates for \(W_{(n,\nu)}\). Reviewer: I.K.Ohta (Tokyo) Cited in 3 ReviewsCited in 26 Documents MSC: 11F55 Other groups and their modular and automorphic forms (several variables) 11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods Keywords:automorphic L-functions; local factors; Whittaker functions; principal series representations; growth estimates PDFBibTeX XMLCite \textit{E. Stade}, Duke Math. J. 60, No. 2, 313--362 (1990; Zbl 0731.11027) Full Text: DOI References: [1] D. Bump, Automorphic Forms on \(\mathrm GL(3,\mathbbR)\) , Springer Lecture Notes in Mathematics, vol. 1083, Springer-Verlag, Berlin, 1984. · Zbl 0543.22005 [2] D. Bump, The Rankin-Selberg method: A survey , to appear in the proceedings of the Selberg Symposium, Oslo, 1987. · Zbl 0668.10034 [3] D. Bump, Barnes’ second lemma and its application to Rankin-Selberg convolutions , to appear in Amer. J. Math. JSTOR: · Zbl 0639.10021 · doi:10.2307/2374544 [4] D. Bump and S. Friedberg, The exterior square automorphic \(L\)-functions on \(GL(n)\) , · Zbl 0712.11030 [5] D. Bump and J. Huntley, in preparation. [6] I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products , Academic Press, New York, 1980, corrected and enlarged edition. · Zbl 0521.33001 [7] H. Jacquet, Fonctions de Whittaker associées aux groupes de Chevalley , Bull. Soc. Math. France 95 (1967), 243-309. · Zbl 0155.05901 [8] B. Kostant, On Whittaker vectors and representation theory , Invent. Math. 48 (1978), no. 2, 101-184. · Zbl 0405.22013 · doi:10.1007/BF01390249 [9] I. I. Pjateckij-Šapiro, Euler subgroups , Lie Groups and their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), Halsted, New York, 1975, pp. 597-620. · Zbl 0329.20028 [10] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series , J. Indian Math. Soc. 20 (1956), 47-87. · Zbl 0072.08201 [11] J. Shalika, The multiplicity one theorem for \(\mathrm GL\sbn\) , Ann. of Math. (2) 100 (1974), 171-193. JSTOR: · Zbl 0316.12010 · doi:10.2307/1971071 [12] E. Stade, Poincaré series for \(\mathrm GL(3,\mathbf R)\)-Whittaker functions , Duke Math. J. 58 (1989), no. 3, 695-729. · Zbl 0699.10041 · doi:10.1215/S0012-7094-89-05833-X [13] I. Vinogradov and L. Takhtadzhyan, Theory of Eisenstein Series for the group \(\mathrmSL(3,\mathbbR)\) and its application to a binary problem , J. Soviet Math. 18 (1982), 293-324. · Zbl 0476.10024 · doi:10.1007/BF01084842 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.