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Eisenstein series twisted with non-expanding cusp monodromies


Let \(\Gamma \) be a geometrically finite Fuchsian group and suppose that \(\chi :\Gamma \rightarrow {{\,\mathrm{GL}\,}}(V)\) is a finite-dimensional representation with non-expanding cusp monodromy. We show that the parabolic Eisenstein series for \(\Gamma \) with twist \(\chi \) converges on some half-plane. Further, we develop Fourier-type expansions for these Eisenstein series.

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The authors are grateful to Julie Rowlett for generously providing Lemma A.1. They wish to thank the Max Planck Institute for Mathematics in Bonn for great hospitality and excellent working conditions during part of the preparation of this manuscript.

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Correspondence to Anke Pohl.

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AP acknowledges support by the DFG Grant PO 1483/2-1.

Appendix A

Appendix A

In the proof of Theorem 5.1 the integral

$$\begin{aligned} \Phi (r,a,y,s) = \int _{-\infty }^\infty \frac{x^r {{\,\mathrm{{\mathbf {e}}}\,}}(a x)}{ (x^2 + y^2)^s} dx \end{aligned}$$

arose, which we substituted with the expressions (13). For \(a=0\), the equivalence of (17) and (13) is given by [9, 3.251(2), 8.384(1)]. For the remaining cases, the equivalence was kindly shown to us by Julie Rowlett. In the following we provide her proof.

Lemma A.1

(J. Rowlett) For any \(r\in {\mathbb {N}}_0\), \(a\in {\mathbb {R}}\), \(a\not =0\), \(y\in {\mathbb {R}}\) and \(s\in {\mathbb {C}}\) with \({{\,\mathrm{Re}\,}}s> (r+1)/2\) we have

$$\begin{aligned} \Phi (r,a,y,s) = \frac{2\pi ^s}{i^r \Gamma (s)} y^{\frac{1}{2}-s} \frac{\partial ^r}{\partial a^r}\left( |a|^{s-\frac{1}{2}} K_{s-\frac{1}{2}}\big (2\pi |a|y\big )\right) . \end{aligned}$$


Throughout let

$$\begin{aligned} (\mathcal F f)(\xi ) :=\int _{\mathbb {R}}f(x)e^{-2 \pi i\xi x}dx \end{aligned}$$


$$\begin{aligned} (\mathcal F^{-1} f)(x) = \int _{\mathbb {R}}f(\xi ) e^{2 \pi i\xi x}d\xi \end{aligned}$$

denote the Fourier transform and its inverse, respectively. Note that

$$\begin{aligned} \Phi (0,a,y,s) = \mathcal F^{-1} \left[ \frac{1}{ (x^2 + y^2)^s} \right] (a). \end{aligned}$$

By [25, p. 172] (note that \(a\not =0\)), we have

$$\begin{aligned} \Phi (0,a,y,s) = \frac{2\pi ^s|a|^{s-\frac{1}{2}}}{\Gamma (s)} y^{\frac{1}{2}-s} K_{s-\frac{1}{2}}(2\pi |a| y). \end{aligned}$$

Using standard properties of the Fourier transformation it follows that for all \(r\in {\mathbb {N}}_0\) we have

$$\begin{aligned} \Phi (r,a,y,s)&= \int _{-\infty }^\infty \frac{x^r {{\,\mathrm{{\mathbf {e}}}\,}}(a x)}{ (x^2 + y^2)^s} dx\\&= \mathcal F^{-1} \left[ \frac{x^r}{ (x^2 + y^2)^s} \right] (a)\\&= i^{-r} \frac{\partial ^r}{\partial a^r} \mathcal F^{-1} \left[ \frac{1}{ (x^2 + y^2)^s} \right] (a)\\&= i^{-r} \frac{\partial ^r}{\partial a^r} \left( \frac{2\pi ^s|a|^{s-\frac{1}{2}}}{\Gamma (s)} y^{\frac{1}{2}-s} K_{s-\frac{1}{2}}(2\pi |a| y) \right) \\&= \frac{2\pi ^s}{i^r \Gamma (s)} y^{\frac{1}{2}-s} \frac{\partial ^r}{\partial a^r}\left( |a|^{s-\frac{1}{2}} K_{s-\frac{1}{2}}\big (2\pi |a|y\big )\right) . \end{aligned}$$

\(\square \)

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Fedosova, K., Pohl, A. Eisenstein series twisted with non-expanding cusp monodromies. Ramanujan J 51, 649–670 (2020).

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