Subconvex bounds for orthogonal groups and triple products.

In 2023, Yueke Hu and Paul Nelson, in their paper ``Subconvex bounds for $U_{n + 1} \times U_n$ in horizontal aspects'', established a subconvex bound valid in certain horizontal aspects for $L$-functions attached to automorphic representations of unitary groups $U_{n + 1} \times U_n$. My current work concerns establishing analogous bounds for the orthogonal family $\SO_{n + 1} \times \SO_n$. I will first explain how the unitary and orthogonal cases fundamentally differ, and how previous Lie algebraic methods break down in the orthogonal setting. To bypass this, our strategy is to adapt Hu and Nelson's group-level techniques to the orthogonal geometry. This talk focuses on the $\GL_2$ triple product; in particular, I will show how working directly at the group level resolves the geometric transversality problem, providing the key structural ingredient needed to extract subconvexity bounds for the triple product $L$-function.