In this talk, we prove an analogous criterion for Adams--Barbasch--Vogan micro-packets: the micro-packet attached to \(\phi\) contains a generic representation precisely when the associated orbit is open. As an application, we establish the Enhanced Shahidi Conjecture for real groups, showing that the Arthur packet attached to an \(A\)-parameter \(\psi\) contains a generic representation if and only if the restriction \(\psi|_{\mathrm{SL}_2}\) is trivial.

Micro-packets are defined in terms of characteristic cycles of $D$-modules. To establish our result, we explicitly compute these characteristic cycles in the case of $D$-modules associated with generic representations. We do this by exploiting the existence of a Weyl group action on both the domain and the codomain of the characteristic cycle map, which makes this map equivariant.

In joint work with Kimball Martin, we obtain a dimension formula that allows for prescribed supercuspidals of any odd-power conductor (the so-called ramified supercuspidals). We are able to get around having to compute the tricky orbital integrals by using the trace of an Atkin-Lehner operator. As a consequence of our formula, we observe a bias in global dimension formulas favoring certain supercuspidals over others with the same conductor exponent, though asymptotically there is balance as first shown in a much more general setting by Kim, Shin and Templier. Interestingly, the magnitude of this bias is independent of the (odd) conductor exponents. The sign of the bias is the global root number.

In ongoing joint work with David Schwein, we connect the two approaches when $p=2$ for wildly ramified irreducible supercuspidal representations of SL(2,F), where the key objects are the fascinating collections of representations whose L-packets have four elements. In this talk, we present our recent progress, focussing particularly on the case of 4-packet representations of $\mathrm{SL}(2,\mathbb{Q}_2)$.