Optimal eigenfunction bounds on compact Lie groups

Let $M$ be a compact Lie group of dimension $d$ and rank $r\geq 2$, and let $f$ be an eigenfunction at frequency scale $N$. The relevant estimate is

$$\|f\|_{L^p(M)}\leq C N^{\frac{d-2}{2}-\frac{d}{p}}\|f\|_{L^2(M)}.$$

The optimal eigenfunction-bound conjecture. This estimate should hold for every $p>2+\frac{4}{d-2}$, with an $\varepsilon$-loss if $2\leq r\leq 4$. The paper establishes evidence for the claim by reducing its class-function case to conjectured torus eigenfunction bounds.