Asymptotics for number of indecomposable components of tensor powers of the natural ${\mathrm{S}\mathrm{L}}_2$-module in odd characteristic

Let $K$ be an algebraically closed field of odd characteristic $p$, let $G={\mathrm{S}\mathrm{L}}_2(K)$, and let $V$ be the natural representation of $G$. Let $b_k$ denote the number of $G$-indecomposable factors of $V^{\otimes k}$, counted with multiplicity, and let ${\delta }_p=1-{\log }_{p^2}\text{\hspace{0.17em}⁣}\left(\frac{p+1}{2}\right)$. Then there exists a smooth, strictly positive, multiplicatively $p^2$-periodic function $\omega (t)$ such that $b_k$ is asymptotic to $\omega (k)k^{-{\delta }_p}{2}^k$. We also show that $t^{-\delta }\omega (t)$ arises as the limiting density of renormalized convolutions of rescaled copies of a positive weight $3\mathrm{/}2$ theta function, obtained from the boundary heat flux of the Dirichlet heat kernel on $(0,p)$.