Wavefront sets of irreducible representations with non-integral infinitesimal characters

Question

Let $G$ be a complex reductive algebraic group (connected, simply connected etc), viewed as a real group. We study the representations of $G$, and we follow the notations in the paper of Barbasch and Vogan:

"Unipotent representations of complex semisimple groups", Annals of Math. Vol.121, 41-110, 1985.

Consider the irreducible representations $\overline{X}(\lambda, \mu)$ of $G$ (following Zhelobenko's notation or the notation in the above paper of Barbasch-Vogan, which means the irreducible constituent of principal series $X(\lambda, \mu)$ containing the extremal weight $\lambda-\mu$) and their wavefront sets, we know their wavefront sets are closures of nilpotent orbits of the Lie algebra.

We have the following results (Definition 1.10 in the above paper):

If $\lambda, \mu$ are integral, and the wavefront set of $\overline{X}(\lambda, \mu)$ is the closure $\overline{O}$ of the nilpotent orbit $O$, then the orbit $O$ is special, i.e. under the Springer correspondence, it correspond to a special representation of the Weyl group $W$ (in the sense of Lusztig).

My question is:

1. Is there any example of irreducible representation, with the closure of a non-special orbit as its wavefront set? For integral infinitesimal characters and special orbits, I have a bunch of examples. But I have never seen any non-special orbits as wavefront sets.

2. In Theorem 3.20 of the above paper of Barbasch-Vogan, the authors show an effective way to calculate wavefront sets of any irreducible representations with integral infinitesimal characters. So for non-integral infinitesimal characters, is there any generalization of the theorem 3.20, that we can use to calculate their wavefront sets?

Thanks!

Answer 1

1) An example is the metapletic representation in the complex symplectic group. They have half-integral infinitesimal characters, and the wavefront set is the minimal orbit which is non-special.

2) Check out the book by Monty McGovern (here). In Section 5. He mentioned how to find out the wavefront set/associated variety (which are the same by a deep theorem of Schmid and Vilonen) for $U(g)/J(\lambda)$, where $\lambda$ can be any infinitesimal character.