Let $\Delta(z)$ be the modular form associated with Ramanujan $\tau$-function.

For any $k=2,3,...$, $Sym^k\Delta$ is conjectured to be an automorphic form on GL($k+1$) and $L(s, Sym^k\Delta)$ is conjectured to be holomorphic and satisfies a functional equation.

Question 1: For which $k$, has $Sym^k\Delta$ been proved to be automorphic form on GL($k+1$)?

Question 2: For which $k$, has $L(s,Sym^k\Delta)$ been proved to be homorphic and satisfies a functional equation?

I know certainly that both questions is YES for $k=2,3,4$ (Gelbart-Jacquet, Kim-Shahidi, Shahidi). But I heard there is progress beyond that because $\Delta$ is a modular form rather than arbitrary automorphic representation on GL(2).

Let $\Delta(z)$ be the modular form associated with Ramanujan $\tau$-function.

For any $k=2,3,...$, $Sym^k\Delta$ is conjectured to be an automorphic form on $\mathrm{GL}(k+1)$ and $L(s, Sym^k\Delta)$ is conjectured to be holomorphic and satisfies a functional equation.

Question 1: For which $k$, has $Sym^k\Delta$ been proved to be automorphic form on $\mathrm{GL}(k+1)$?

Question 2: For which $k$, has $L(s,Sym^k\Delta)$ been proved to be homorphic and satisfies a functional equation?

I know certainly that both questions is YES for $k=2,3,4$ (Gelbart-Jacquet, Kim-Shahidi, Shahidi). But I heard there is progress beyond that because $\Delta$ is a modular form rather than arbitrary automorphic representation on $\mathrm{GL}(2)$.