4 problems

Let $G$ be a connected reductive group over a local field $F$, let $\rho:{}^LG\to\operatorname{GL}_{V_\rho}(\mathbb{C})$ be as above, and let $\mathcal{S}_\rho$ be a Schwartz space…

Local theta-lift nonvanishing conjecture. The local theta lift $\Theta_{n,r}(\pi_v)$ is nonzero if and only if $Z_v^*(s_{m,n})$ is nonzero on…

Weak holomorphy conjecture for normalized local zeta integrals. The following assertions hold:

Holomorphy conjecture for normalized local zeta integrals. The function $Z_v^*(s,f_{1,v},f_{2,v},Phi_v,pi_v)$ is holomorphic for all $s\in\mathbb{C}$ and for every choice of…