Eisenstein series twisted with non-expanding cusp monodromies

Abstract

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.

Appendix A

In the proof of Theorem 5.1 the integral

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

(17)

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

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

Proof

Throughout let

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

and

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

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

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

$\square$