Unipotent Representations with Cuspidal Support

• The paper unifies Arthur parameters, nilpotent orbits, and perverse sheaves to parameterize irreducible unipotent representations with cuspidal support.
• It outlines a detailed construction of Arthur packets and micro-packets, emphasizing the role of geometric and combinatorial criteria in p-adic groups.
• It demonstrates how invariants like wavefront sets and special pieces precisely determine the structure and automorphic implications of these representations.

A unipotent representation with cuspidal support is an irreducible admissible representation of a reductive $p$-adic group $G$ whose construction and parameterization are tightly connected to the theory of Arthur parameters, nilpotent orbits in the Langlands dual group $G^\vee$, and the microlocal geometry of perverse sheaves. The paper of unipotent Arthur packets and their weak variants addresses the intricate combinatorial and geometric relationships among these representations, their wavefront sets, and their occurrence within the space of automorphic forms.

1. Arthur Parameters, Unipotent Representations, and Cuspidal Support

Let $G$ be a connected reductive group over a non-Archimedean local field $k$. An Arthur parameter is a continuous homomorphism $\psi: W_k' \times SL_2(\mathbb{C}) \to G^\vee$ where $W_k'$ is the Weil-Deligne group and $G^\vee$ is the complex Langlands dual group. A unipotent Arthur parameter is one which is trivial on $W_k'$—these are classified by nilpotent orbits $\mathcal{O}^\vee \subset \mathfrak{g}^\vee$. The corresponding unipotent representations with given infinitesimal character (or cuspidal support) are labeled by purely geometric data associated to that orbit.

Following Lusztig's framework, representations with unipotent cuspidal support are parametrized via the classification of pairs $(S,\mathcal{L})$, where $S$ is a $G^\vee(h)$-orbit in a graded piece of $\mathfrak{g}^\vee$ defined by an $\mathfrak{sl}_2$-triple $\{e,h,f\}$ associated to $\mathcal{O}^\vee$, and $\mathcal{L}$ is an irreducible $G^\vee(h)$-equivariant local system on $S$. Each such pair corresponds to a simple perverse sheaf $IC(S,\mathcal{L})$ and an irreducible representation $X(q^{h/2},S,\mathcal{L})$ of $G(k)$ at infinitesimal character determined by $\mathcal{O}^\vee$ (Barchini et al., 7 Dec 2025, Ciubotaru et al., 2022).

2. Construction of Arthur Packets and Micro-Packets

The construction of (genuine) Arthur packets for unipotent representations exploits the geometry of the nilpotent cone and the structure of perverse sheaves on associated graded spaces. Basic Arthur packets are defined as

$$\Pi^{\mathrm{Art}}_{\psi_{\mathcal{O}}}(G(k)) = \{ X \in \Pi^{\mathrm{Lus}}_{\Lambda}(G(k)) : \mathrm{AZ}(X) \text{ is tempered} \}$$

where $\mathrm{AZ}$ denotes the Aubert-Zelevinsky dual. Each representation $X$ corresponds uniquely to $IC(S,\mathcal{L})$, and the temperedness of its dual reflects the position of the orbit $S$ relative to $\mathcal{O}^\vee$ (Barchini et al., 7 Dec 2025, Ciubotaru et al., 2022).

Micro-packets are finer objects capturing the local geometric behavior: each perverse sheaf $IC(S,\mathcal{L})$ has a characteristic cycle

$$CC(IC(S,\mathcal{L})) = \sum_{S'} \chi^{\mathrm{mic}}_{S'}(IC(S,\mathcal{L})) [T^*_{S'} V]$$

with $\chi^{\mathrm{mic}}_{S'}$ the microlocal multiplicity along the conormal bundle to each $S'$. The micro-packet attached to an orbit $S'$ is

$$\Pi^{\mathrm{mic}}_{S'}(G(k)) = \{ X(q^{h/2},S,\mathcal{L}) : \chi^{\mathrm{mic}}_{S'}(IC(S,\mathcal{L})) > 0 \}$$

This partition reflects the intrinsic singularity structure of the unipotent representations in terms of their geometric supports (Barchini et al., 7 Dec 2025).

3. Weak Arthur Packets: Definition and Decomposition

Weak Arthur packets generalize the notion of Arthur packets, motivated by analogous constructions in real groups (Adams-Barbasch-Vogan, Barbasch-Vogan). For a nilpotent orbit $\mathcal{O}^\vee$, denote by $sp(\mathcal{O}^\vee)$ its Spaltenstein special piece. The weak Arthur packet is defined by

$$\Pi^{\mathrm{weak}}_{\psi_{\mathcal{O}^\vee}}(G(k)) = \{ \mathrm{AZ}(X(q^{h/2},S,\mathcal{L})) : G^\vee \cdot S \in sp(\mathcal{O}^\vee) \}$$

In other words, the weak Arthur packet is obtained by applying the Aubert-Zelevinsky duality to all representations parameterized by local systems on orbits within the special piece of $\mathcal{O}^\vee$ (Barchini et al., 7 Dec 2025, Ciubotaru et al., 2022). The central conjecture, proved for split classical groups and in several explicit exceptional cases (e.g., split $F_4$, orbit $F_4(a_3)$), is that every weak Arthur packet is a finite union of (genuine) Arthur packets, equivalently a union of micro-packets corresponding to those special orbits (Liu et al., 2023, Barchini et al., 7 Dec 2025).

The union-of-packets theorem can be formalized as

$$\Pi^{\mathrm{weak}}_{\psi} = \bigcup_{\psi' : d_{\mathrm{BV}}(\mathcal{O}_A(\psi')) = d_{\mathrm{BV}}(\mathcal{O}_A(\psi))} \Pi^{\mathrm{Art}}_{\psi'}$$

where $d_{\mathrm{BV}}$ denotes the Barbasch–Vogan duality (Liu et al., 2023).

4. Geometric and Combinatorial Criteria: Wavefront Sets and Special Pieces

A key characterization of (weak) unipotent Arthur packets is via wavefront sets. For $X \in \Pi^{\mathrm{Lus}}_{\Lambda}(G(k))$, the canonical unramified wavefront set $\operatorname{CUWF}(X)$ (in Okada's sense) and the geometric wavefront set ${}^{\bar{k}}\operatorname{WF}(X)$ control the inclusions:

• $X \in \Pi^{\mathrm{Art}}_{\psi_{\mathcal{O}^\vee}}$ if and only if $\operatorname{CUWF}(X) \le D(\mathcal{O}^\vee,1)$,
• $$X \in \Pi^{\mathrm{weak}}_{\psi_{\mathcal{O}^\vee}}$$ if and only if $${}^{\bar{k}}\operatorname{WF}(X) \le d(\mathcal{O}^\vee)$$, where $D$ is Achar's duality and $d$ is the Barbasch–Vogan–Spaltenstein duality (Ciubotaru et al., 2022, Liu et al., 2023). For split classical groups, the structure of wavefront sets inside Arthur packets is tightly controlled and can be read off combinatorially from the associated nilpotent partitions.

The special piece $sp(\mathcal{O}^\vee)$, as defined by Lusztig–Spaltenstein, partitions the unipotent locus into minimal sets closed under closure and relevant for both ramification and geometric multiplicity (Gurevich et al., 4 Apr 2024). Each weak Arthur packet is correspondingly partitioned as the union of genuine packets indexed by the orbits in $sp(\mathcal{O}^\vee)$.

5. Structural Properties and Ramification

The decomposition of weak Arthur packets into Arthur packets can be characterized using both representation-theoretic and geometric invariants:

• Minimal Gelfand–Kirillov dimension among representations with fixed infinitesimal character selects the anti-tempered part, corresponding to Arthur packets with the same minimal nilpotent support (Gurevich et al., 4 Apr 2024).
• Sphericity and ramification: The property of an irreducible representation containing vectors fixed by a maximal compact subgroup ("weak sphericity") precisely determines its membership in weak Arthur packets. Weak sphericity matches with belonging to Lusztig's canonical quotient in the representation theory of Weyl groups (Gurevich et al., 4 Apr 2024).
• For each special piece, only representations labeling primitive characters of the component group (in Lusztig's sense) correspond to weakly spherical constituents.

6. Explicit Examples and Case Studies

The split exceptional group $F_4$ with distinguished nilpotent orbit $F_4(a_3)$ provides a concrete test case. Here, the perverse-sheaf construction gives 20 unipotent irreducibles at the relevant infinitesimal character, organized into micro-packets. The basic Arthur packet is the micro-packet for the closed orbit, while the weak Arthur packet is the union of the micro-packets corresponding (via Fourier transform and duality) to the five orbits in $sp(F_4(a_3))$ (Barchini et al., 7 Dec 2025, Ciubotaru et al., 2022).

For split classical groups, the general union-of-packets theorem is established: each weak Arthur packet (for basic parameters) decomposes as the union of Arthur packets indexed by parameters with dual Barbasch–Vogan image matching that of the original (Liu et al., 2023). All members of weak packets so constructed are unitary.

7. Generalizations, Conjectures, and Connections

The theory extends beyond unipotent representations: for any real infinitesimal parameter and any nilpotent orbit in the dual Lie algebra, one can define generalized weak packets as maximal Arthur-stable subsets of irreducible representations with wavefront sets bounded by the given orbit. The full union-of-packets conjecture is expected to hold in this broader scope, subject to additional constraints (not all commuting pairs of nilpotents with the same infinitesimal character need appear) (Liu et al., 2023, Ciubotaru et al., 2022).

There are close analogies to the description of special unipotent Arthur packets for real groups in terms of associated varieties, closure orderings, and the duality structure on nilpotent orbits (Barchini et al., 7 Dec 2025, Ciubotaru et al., 2022). The microlocal and endoscopic frameworks provide a robust toolkit for decomposing the unitary dual and automorphic spectrum via unipotent data.

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References:

• "Some unipotent Arthur packets for p-adic split F4" (Barchini et al., 7 Dec 2025)
• "Some Unipotent Arthur Packets for Reductive $p$-adic Groups" (Ciubotaru et al., 2022)
• "On the weak local Arthur packets conjecture for split classical groups" (Liu et al., 2023)
• "Ramification of weak Arthur packets for p-adic groups" (Gurevich et al., 4 Apr 2024)