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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
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)

##### 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
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