Suppose $H$ is reductive and contains a Cartan subgroup $T$ of $G$. If $g\in N_G(H)$ then $gTg^{-1}$ is another Cartan subgroup of $H^0$, so $hg\in N_G(T)$ for some $h\in H$. I think this defines a group homomorphism $N_G(H)/H\rightarrow N_G(T)/T=W$. This applies to the $E_7$ and $E_8$ examples. I don't know what to do if this condition doesn't hold. Also I agree that the natural statement involves $N_G(T)\cap N_G(H)$.

Another reference for the same topic is "Langlands Classification and Unitary Dual of SU(2,2)", MR0789291 (Lectures in Applied Math. 21, 1985).

Yes, absolute value, misprint corrected.

Thanks - I've edited my answer accordingly.

There are quite a few papers on the character of the oscillator representation over a local field. In the case of $\mathbb R$ and $\mathbb C$ see [Adams, Israel J. Math 97] or [Torasso, Math. Annalen 1980].

The metaplectic representation of $Sp(2n,\mathbb C)$ does exist. In fact, since $Sp(2n,\mathbb C)$ is simply connected it is an honest representation (rather than a representation of a covering group). In particular one of its two irreducible factors is the spherical representation with infinitesimal character $[n-1/2,n-3/2,...,1/2]$ in the usual coordinates.

See Section 11 of Algorithms for Representation Theory of Real Groups, by myself and Fokko du Cloux. In particular Remark 11.5 says that the fiber of the map from K\G/B to twisted involutions is naturally in bijection with characters of the component group of the dual group. For $GL(n,\mathbb R)$ the dual group is $U(p,q)$ whose Cartans are connected.

Fair enough. I withdraw the objection.