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 this talk, I will discuss recent work with collaborators that, if $G$ is a $p$-adic group for which the $p$-adic Kazhdan-Lusztig hypothesis holds, then the forward direction of the Gross-Prasad conjecture holds: An $L$-packet $\Pi_\phi(G)$ contains a generic representation if and only if $L(s, \phi, \text{Ad})$ is regular at $s=1$. I will also discuss an analogous result for ABV-packets, and implications for Shahidi's enhanced genericity. Finally, I will discuss some results towards showing that Arthur parameters of classical groups and $\mathrm{GL}(n)$ are associated to orbits of singular closure in their Vogan variety, and counter-examples for non-classical groups.

We will see that these conditions naturally appear from the following result: Two totally real number fields $K_1$ and $K_2$ have the same Dedekind zeta functions $\zeta_{K_1}(s)$ and $\zeta_{K_2}(s)$ if and only if the real numbers \[\lim_{s\rightarrow 1}(s-1)\zeta_{K_1}(s)\quad\text{and}\quad \lim_{s\rightarrow1}(s-1)\zeta_{K_2}(s)\] are linearly dependent over the field of algebraic numbers $\overline{\mathbb{Q}}$. I will conclude with a representation-theoretic reformulation of this result.

In this talk, we discuss a strategy to canonicalize Hiraga and Saito's result, effectively removing the indeterminate constant in their work. We use the example of $n=2$ to illustrate this strategy.

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 this talk, we will discuss how to extend this result to small residue fields using Shintani lifting of the associated parahoric representations. This gives us a correspondence between these representations of $G(F)$ and those of $G(F')$ coming from the pair $(S(F'), \phi \circ \mathrm{Nm}_{F'/F})$, where $F'/F$ is a finite unramified extension.

The log canonical threshold (LCT) is a numerical invariant that measures the severity of singularities of a divisor relative to a fixed ambient log structure and plays a central role in the minimal model program.

In this setting, the LCT of the Vinberg enveloping monoid is studied via its colored cone. The main result relates the LCT in terms of root-theoretic and combinatorial data extracted from the colored cone of the Vinberg enveloping monoid.

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 $\mathrm{SL}(2,\mathbb{Q}_2)$.

The linear case of Rankin--Selberg Flicker--Rallis periods is historically well-studied, and the modern approach on unitary groups suggested by Gan--Gross--Prasad is to reduce to these two cases by means of the relative trace formula.

The problem is solved for unramified local unitary groups but remains open even in low-dimensional ramified local fields. In this talk I will present what is known and how this relates to orbital integrals and the Arithmetic Transfer Conjecture for Shimura varieties. I will then formulate a conjecture establishing an explicit transfer in the ramified setting, together with a proof for rank one groups. This is joint work with Wei Zhang.