The purpose of this post is to ask for the community's opinion on a conjecture about a certain mean-value property for functions on $\operatorname{SL}(2,\mathbb{R})$. This conjecture appears at the end of the post, after some motivation and preliminaries. I asked about related issues in another MathOverflow question, but did not receive a response.
Let $p,q \in [1,\infty]$ satisfy $1 \leq \frac{1}{p}+\frac{1}{q} \leq 2$ and let $G$ be a locally compact Polish group with left Haar measure $\mu_G$. For $\phi \in L^p(G,\mu_G)$ and $\varphi \in L^q(G,\mu_G)$ we define the convolution $\phi \ast \psi$ by: \begin{equation} [\phi \ast \psi](g) = \int_G \phi(h) \psi(h^{-1}g)\, \mathrm{d} \mu_G(h) \end{equation}
For $d \in \mathbb{N}$ and $t > 0$ we let $\mathbf{1}^t_d: \mathbb{R}^d \to \mathbb{R}$ denote the indicator function of the closed ball of radius $t$ around the origin in $\mathbb{R}^d$. The following is classical:
Laplacian eigenfunction theorem: For all $d \in \mathbb{N}$ and all $y \geq 0$ there exists a function $\alpha_d^y:[0,\infty) \to (0,\infty)$ such that if $\psi \in C^2(\mathbb{R}^d)$ satisfies $\Delta_{\mathbb{R}^d}\psi = -y\psi$ then we have $\mathbf{1}^t_d \ast \psi = \alpha_d^y(t)\psi$ for all $t > 0$.
For all $d \in \mathbb{N}$ and all $t > 0$ we that $\alpha_d^0(t)$ is equal to the $d$-dimensional Lebesgue measure of the ball of radius $t$. This recovers the mean value property for harmonic functions. The functions $\alpha_3^y$ are related to Bessel functions, and more generally to spherical functions on Lie groups. The above theorem is also true for disks in the hyperbolic plane.
Now, adopt the notation $\mathbb{G}$ for the Lie group $\operatorname{SL}(2,\mathbb{R})$. For $r,\theta \in \mathbb{R}$ we define the following elements of $\mathbb{G}$. \begin{align*} \mathcal{A}(r) &= \begin{pmatrix}e^r & 0 \\ 0 & e^{-r} \end{pmatrix} & \mathcal{B}(r) &= \begin{pmatrix} \cosh(r) & \sinh(r) \\ \sinh(r) & \cosh(r) \end{pmatrix} & \mathcal{K}(\theta) &= \begin{pmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{pmatrix} \end{align*}
We define three corresponding left-invariant differential operators given for $\psi \in C^1(\mathbb{G})$ by: \begin{equation*} [\partial_\mathcal{A} \psi](g) = \frac{\mathrm{d}}{\mathrm{d}r} \psi(g\mathcal{A}(r)) \Bigg \vert_{r=0} \qquad [\partial_\mathcal{B} \psi](g) = \frac{\mathrm{d}}{\mathrm{d}r} \psi(g\mathcal{B}(r))\Bigg \vert_{r=0} \qquad [\partial_\mathcal{K} \psi](g) = \frac{\mathrm{d}}{\mathrm{d}\theta} \psi(g\mathcal{K}(\theta))\Bigg \vert_{\theta=0} \end{equation*}
The Casimir operator $\Omega_\mathbb{G}$ of $\mathbb{G}$ is the differential operator defined on $C^2(\mathbb{G})$ by: \begin{equation} \Omega_G = \partial_A^2 + \partial_B^2 - \partial_K^2 \label{eq.defcas} \end{equation}
There are various senses in which the Casimir operator is to $\mathbb{G}$ as the Laplacians are to $\mathbb{R}^d$. Note, however, that $\Omega_\mathbb{G}$ is not the Laplace-Beltrami operator $\Delta_\mathbb{G}$ of the Riemannian metric on $\mathbb{G}$. Instead, we have: \begin{equation*} \Delta_\mathbb{G} = \partial_A^2 + \partial_B^2 + \partial_K^2 = \Omega_\mathbb{G} + 2\partial_K^2 \end{equation*}
Casimir eigenfunction conjecture (weak version): There exists a bounded, compactly supported Borel function $V: \mathbb{G} \to \mathbb{C}$ along with $y \geq 1$ and a nonzero $\beta \in \mathbb{C}$ such that for all $\psi \in C^\infty(\mathbb{G})$ satisfying $\Omega_\mathbb{G}\psi = -y\psi$ we have $V \ast \psi = \beta \psi$.
The intuition here is that if $d \in \mathbb{N}$ and $t > 0$ then making the substitutions below transforms my Casimir eigenfunction conjecture into the classical Laplacian eigenfunction theorem. \begin{equation*} \bigl(\mathbb{G},\Omega_\mathbb{G},V,\beta \bigr) \mapsto \bigl(\mathbb{R}^d,\Delta_{\mathbb{R}^d},\mathbf{1}_d^t, \alpha_d^y(t) \bigr) \end{equation*}
The above conjecture is written to be logically weakest formulation which will suffice for the application we have in mind. However, in accordance with the theorem for $\mathbb{R}^d$ I would intuitively expect that the following stronger version holds.
Casimir eigenfunction conjecture (strong version): There exists a one-parameter family $(V_t:\mathbb{G} \to \mathbb{C})_{t > 0}$ of bounded, compactly supported Borel functions with the following property. For all $y \geq 1$ there exists a nonzero function $\beta_y: (0,\infty) \to \mathbb{R}$ such that for all $\psi \in C^\infty(\mathbb{G})$ satisfying $\Omega_\mathbb{G}\psi = -y\psi$ and all $t > 0$ we have $V_t \ast \psi = \beta_y(t) \psi$.
