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Leticia Barchini

Publications and source records attributed to Leticia Barchini.

7 recordsLinked to original sources
arXiv2025

Some unipotent Arthur packets for p-adic split F4

Let $G(k)$ be the split form of the simple exceptional p-adic group of type $F_4$, and let $\mathcal O = F_4(a_3)$ be the minimal distinguished nilpotent orbit. Our main result concerns the class of unipotent representations with cuspidal support at infinitesimal character $Λ$ determined by $\mathcal O$. These representations are parameterized by local systems, $\{(S, \mathcal L)\}$. We compute the characteristic cycles of the perverse sheaves $\text{IC}(S, \mathcal L)$ and determine all micro-packets in the sense of [Vo93]. In [CMBO24], the authors introduced a notion of weak Arthur packets in the p-adic setting. They conjectured that weak Arthur packets are unions of Arthur packets, in an appropriate sense. We verify that weak Arthur packets are unions of micro-packets.

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arXiv2025

Micro-packets for real groups of type $G_2$

In their study of Arthur's conjectures for real groups, Adams, Barbasch, and Vogan introduced the notion of micro-packets. Micro-packets are finite sets of irreducible representations defined using microlocal geometric methods and characteristic cycles. We explore an action of the Weyl group on characteristic cycles to compute all micro-packets of real groups of type $G_2$.

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arXiv2024

Relating Real and P-adic Kazhdan-Lusztig Polynomials

Fix an integral semisimple element $λ$ in the Lie algebra $\mathfrak{g}$ of a complex reductive algebraic group $G$. Let $L$ denote the centralizer of $λ$ in $G$ and let $\mathfrak{g}(-1)$ denote the $-1$ eigenspace of $\mathrm{ad}(λ)$ in $\mathfrak{g}$. Under a natural hypothesis (which is always satisfied for classical subgroups of $\mathrm{GL}(n)$), we embed the closure of each $L$ orbit on $\mathfrak{g}(-1)$ into the closure of an orbit of a symmetric subgroup $K$ containing $L$ on a partial flag variety for $G$. We use this to relate the local intersection homology of the later orbit closures to the former orbit closures. This, in turn, relates multiplicity matrices for split real and $p$-adic groups. We also describe relationships between "microlocal packets'' of representations of these groups.

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arXiv2024

A Note On the Orbits of a Symmetric Subgroup in the Flag Variety

Motivated by relating the representation theory of the split real and $p$-adic forms of a connected reductive algebraic group $G$, we describe a subset of $2^r$ orbits on the complex flag variety for a certain symmetric subgroup. (Here $r$ is the semisimple rank of $G$.) This set of orbits has the property that, while the closure of individual orbits are generally singular, they are always smooth along other orbits in the set. This, in turn, implies consequences for the representation theory of the split real group.

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arXiv2018

Reducible characteristic cycles of Harish-Chandra modules for $\mathrm{U}(p,q)$ and the Kashiwara-Saito singularity

We give examples of reducible characteristic cycles for irreducible Harish-Chandra modules for $\mathrm{U}(p,q)$ by analyzing a four-dimensional singular subvariety of $\mathbb{C}^8$. We relate this singularity to the Kashiwara-Saito singularity arising for Schubert varieties for ${\mathrm{GL}}(8,\mathbb{C})$ with reducible characteristic cycles, as well as recent related examples of Williamson. In particular, this gives another explanation of the simple highest weight modules for ${\mathfrak{g}}{\mathfrak{l}}(12,\mathbb{C})$ with reducible associated varieties that Williamson discovered.

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