Throughout this post, we let $G$ denote the Lie group $\mathrm{SL}(2,\mathbb{R})$. For $t,\theta \in \mathbb{R}$ we define the following elements of $G$: \begin{align*} A(t) &= \begin{bmatrix}e^t & 0 \\ 0 & e^{-t} \end{bmatrix} \\ B(t) &= \begin{bmatrix} \cosh(t) & \sinh(t) \\ \sinh(t) & \cosh(t) \end{bmatrix} \\ K(\theta) &= \begin{bmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{bmatrix}. \end{align*}

We define corresponding left-invariant differential operators given for $\psi \in C^\infty(G)$ by:

\begin{equation*} (\partial_A \psi)(g) = \frac{\mathrm{d}}{\mathrm{d}t} \psi(g A(t)) \qquad (\partial_B \psi)(g) = \frac{\mathrm{d}}{\mathrm{d}t} \psi(g B(t)) \qquad (\partial_K \psi)(g) = \frac{\mathrm{d}}{\mathrm{d}\theta} \psi(g K(\theta)). \end{equation*}

The Casimir operator $\Omega_G$ of $G$ is given by: \begin{equation*} \Omega_G = \partial_A^2 + \partial_B^2 - \partial_K^2. \end{equation*} Then (up to a scalar multiple) we have that $\Omega_G$ is the unique second-order differential operator on $G$ which commutes with both left and right translation.

In contrast, the Laplace–Beltrami operator $\Delta_G$ of $G$ as a Riemannian manifold is given by: \begin{equation*} \Delta_G = \partial_A^2 + \partial_B^2 + \partial_K^2 = \Omega_G + 2 \partial_K. \end{equation*} It follows from abstract Riemannian geometry that $\Delta_G$ is the infinitesimal generator of a 'Riemannian' heat evolution on $G$. In particular, all $c > 0$ the operator $\exp(c\Delta_G)$ is given by convolution with a smooth heat kernel that has good decay properties at infinity. An explicit formula for this heat kernel appears in the (apparently unpublished) article at Mori - The heat kernel on $\operatorname{SL}(2,\mathbb R)$.

The article Sarkar - A complete analogue of Hardy's theorem on $\operatorname{SL}_2(\mathbb R)$ and characterization of the heat kernel investigates the heat evolution on $G$ associated with the Casimir operator, as opposed to the Laplace–Beltrami operator. More explicitly, the author of that article shows that for any compactly supported smooth function $\psi \in C^\infty_0(G)$ there exists a unique family of functions $(\psi_c :c>0)$ such that $\frac{\mathrm{d}}{\mathrm{d}c} \psi_c = \Omega_G \psi_c$ for all $c>0$ and such that we have $\lim_{c \downarrow 0} \psi_c = \psi$ in various of natural modes of convergence. Unfortunately (for me), that article also proves that there is no heat kernel for this evolution, in the sense that there is no one-parameter family of functions $\exp(c \Omega_G)$ such that $\psi_c$ would be given by convolution with $\exp(c \Omega_G)$. Ultimately, the last fact is a consequence of the fact that $\Omega_G$ fails to be a negative-definite operator.

My purpose in this question is to obtain any known information about the heat evolution associated with the Casimir operator on $\operatorname{SL}(2,\mathbb{R})$ beyond what appears in the above article of Rudra Sarkar. In particular, I would like to obtain any known information about $L^p$ bounds or decay estimates for $\psi_c$ given such estimates for the initial data $\psi$. I would also be interested in anything about a 'wave-type evolution' for the Casimir operator, which would presumably satisfy the differential equation $\frac{\mathrm{d}^2}{\mathrm{d}^2c} \psi_c(g) = \Omega_G \psi_c$.