On Fourier restriction type problems on compact Lie groups

In this article, we obtain new results for Fourier restriction type problems on compact Lie groups. We first provide a sharp form of $L^p$ estimates of irreducible characters in terms of their Laplace-Beltrami eigenvalue and as a consequence provide some sharp $L^p$ estimates of joint eigenfunctions for the ring of conjugate-invariant differential operators. Then we improve upon the previous range of exponent for scale-invariant Strichartz estimates for the Schr\"odinger equation, and provide new $L^p$ bounds of Laplace-Beltrami eigenfunctions in terms of their eigenvalue similar to known bounds on tori. A key ingredient in our proof of these results is a barycentric-semiclassical subdivision of the Weyl alcove in a maximal torus. On each component of this subdivision we carry out the analysis of characters and exponential sums, and the circle method of Hardy--Littlewood and Kloosterman.