The smallest representations of nonlinear covers of odd orthogonal groups

We construct the smallest genuine representations of a nonlinear cover of the group ${\rm SO}^0(p,q)$ where $p+q$ is odd. We determine correspondences of infinitesimal characters arising from restricting the smallest representations to dual pairs ${{\frak s}{\frak o}}(p,a) \oplus {{\frak s}{\frak o}}(b)$ where $a+b=q$.