I saw two different definitions for the weight $k$ non-Euclidean Laplacian. First, in Daniel Bump's book Automorphic Forms and Representations, the following definitions are given for smooth $\mathbb C$-valued functions on the upper half plane (Chapter 2.1):

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I was not able to verify (1.4) by the way. Maybe I made a mistake, but from what I calculated, the first equality was not correct. I was wondering whether one of the definitions (1.1), (1.2), (1.3) was not written correctly.

Edit: I have checked again, and (1.4) is actually correct.

Wikipedia defines the weight $k$ non-Euclidean Laplacian by

$$\Delta_k = -y^2( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}) + iky (\frac{\partial}{\partial x} + i \frac{\partial}{\partial y})$$

Which definition is correct?

  • $\begingroup$ Just as a comment, there are some errata on that page (129): sporadic.stanford.edu/bump/match/errata.html $\endgroup$ – user608030 Dec 22 '18 at 4:33
  • $\begingroup$ Those errors seem to have been fixed in my edition $\endgroup$ – D_S Dec 22 '18 at 4:39
  • $\begingroup$ Won't the first one lead to if $f(z)$ is smooth and $1$-periodic then $f(z) = \sum_n a_n(y) e^{2i \pi n x}$ and if it is in the kernel of $\Delta_k$ then $f(z) = c_0+\sum_{n\ne 0} c_n h_k(2\pi |n|y) e^{2i \pi n x}$ where $\partial_y^2 h_k(y) = h_k(y)+ h_k(y) C_k/ y$ $\endgroup$ – reuns Dec 22 '18 at 5:51
  • $\begingroup$ I don't know..what is $(2\pi|n|y)$? $\endgroup$ – D_S Dec 22 '18 at 17:19
  • $\begingroup$ $h_k(t)$ evaluated at $t = 2 \pi N y$ with $N = |n|$. And $f(z) = c_0+\sum_{n\ne 0} c_n h_k(2\pi |n|y) e^{2i \pi n x}$ is what we need for the Mellin transform of $f(x+i.)-c_0$ being a Dirichlet series $\endgroup$ – reuns Dec 22 '18 at 23:38

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