p-adic interpolation for ring class twists of cohomological automorphic representations of GL(n) over a CM field

Let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_K)$ for $K$ a CM field. I will explain two approaches to constructing new $p$-adic interpolation series for critical values of the standard L-function of $\pi$ twisted by ring class characters of $K$. One approaches uses Eulerian integral presentations, and the other work-in-progress on the Ichino-Ikeda conjecture for unitary groups $\mathrm{U}_n(\mathbb{A}_K) \times \mathrm{U}_1(\mathbb{A}_K)$. Both approaches reduce to choosing vectors in the corresponding representation spaces, and identifying some “trace” Hecke operator to show relevant distribution relations. Moreover, as I will explain, both constructions suggest a natural extension of the nonvanishing theorems of Cornut-Vatsal via reductions to $p$-adic unipotent flows in this setting.