= Weight-k hyperbolic Laplacian
{c}
{title2=$\Delta_k$}
The <Weight-k hyperbolic Laplacian> is a differential operator adapted to weight-$k$ transformation laws on the <upper half-plane> $z=x+iy$, $y>0$. Using the sign consistent with the nonnegative weight-zero <hyperbolic Laplacian>, and the holomorphic-weight convention, define
$$
\Delta_k=-y^2(\partial_x^2+\partial_y^2)+iky\partial_x-ky\partial_y
=-4y^2\partial_z\partial_{\bar z}+2iky\partial_{\bar z}.
$$
Here $\partial_z=\tfrac12(\partial_x-i\partial_y)$ and $\partial_{\bar z}=\tfrac12(\partial_x+i\partial_y)$ are <Wirtinger derivatives>. The cited paper uses the opposite overall sign, so its <eigenvalues> must also have their signs reversed.<Polyharmonic Maass forms for PSL(2,Z) by Jeffrey C. Lagarias and Robert C. Rhoades>[Introduction, printed p. 1, weight-k modularity, displayed Laplacian and explicit sign-convention warning]{ref}
The correction terms let the operator preserve the transformation law $f((az+b)/(cz+d))=(cz+d)^k f(z)$ for determinant-one integer matrices, as used for <modular forms> and <Maass forms>. At $k=0$ it reduces to the ordinary <hyperbolic Laplacian>. Every <holomorphic function> is locally annihilated because $\partial_{\bar z}f=0$; being holomorphic alone does not supply modularity or growth conditions.
For a direct calculation, take $f(z)=y^s$. Its $x$ derivatives vanish, giving
$$
\Delta_k y^s=[-s(s-1)-ks]y^s=s(1-k-s)y^s.
$$
Thus both $1$ and $y^{1-k}$ are annihilated; when $k=1$ these coincide and a second independent solution depending only on $y$ is $\log y$. These are local operator examples, not claims that all satisfy a modular transformation law. Conventions using a unit-modulus weight multiplier have a different first-derivative correction, so the displayed formula matters as much as the name.
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