Local vs. global conjugacy

Let $H$ and $H'$ be two subgroups of a connected reductive group $G$, and assume that they are element-wise conjugate (or locally conjugate). One asks whether they are conjugate. When $G=\mathrm{GL}(n,{\mathbb C})$, the answer is yes and the reason is exactly the character theory. How about the other group? Moreover, the discrepancy between local and global conjugacy reflects somewhat one important feature of the automorphic representations of $G$, namely the multiplicity.
In this talk, we will review the local-global issue, and its relation to the multiplicities. We will address some recent results on local-global conjugacy issue and multiplicities for $\mathrm{SO}(2N)$.