Micro-Packets in p-adic Groups

• Micro-packets are finite sets of irreducible p-adic representations parameterized via microlocal geometry and perverse sheaves.
• They leverage characteristic cycles and microlocal multiplicities to classify unipotent representations in precise packets.
• They serve as atomic building blocks for weak Arthur packets, facilitating the decomposition of automorphic spectra.

A micro-packet is a finite collection of irreducible representations of a $p$-adic reductive group, parameterized microlocally via characteristic cycles of perverse sheaves on the nilpotent cone, which reflects deep geometric and representation-theoretic structures. The concept, originating with Vogan [Vo93], forms a crucial organizing node in the theory of unipotent representations, weak Arthur packets, and the detailed decomposition of packets for $p$-adic groups, especially in recent approaches at the interface of geometric and harmonic analysis. Micro-packets serve as the atomic blocks for "weak Arthur packets," and their structure and parameterization are now central in the precise description of unipotent automorphic spectra in split reductive and exceptional groups.

1. Microlocal and Geometric Origins

The micro-packet arises from the microlocal study of representation theory, specifically the interaction between representation theory of $p$-adic groups and the geometry of orbits in the dual Lie algebra. Given a split reductive group $G$ (over a non-Archimedean local field $k$) with complex dual group $G^\vee$, and taking a nilpotent $G^\vee$-orbit $\mathcal O\subset\mathfrak g^\vee$, one fixes an $\mathfrak{sl}_2$-triple $\{e,h,f\}$ with $h\in\mathrm{Lie}\,T$ such that $q^{h/2}\in G^\vee$ determines an infinitesimal character.

The irreducible unipotent representations with cuspidal support at infinitesimal character $q^{h/2}$ are classified by pairs $(S,\mathcal L)$, where $S$ is a $G^\vee(h)$-orbit in the associated graded piece $\mathfrak g(2)$ and $\mathcal L$ is a simple $G^\vee(h)$-equivariant local system on $S$.

For each such pair, the geometric counterpart is the intersection cohomology complex $\text{IC}(S,\mathcal L)$ on $\mathfrak g(2)$. The characteristic cycle $\operatorname{CC}(\text{IC}(S,\mathcal L))$ can be expressed as: $$\operatorname{CC}(\text{IC}(S,\mathcal L)) = \sum_{S'} \chi^{\mathrm{mic}}_{S'}(\text{IC}(S, \mathcal L)) \cdot [\overline{T^*_{S'}(\mathfrak g(2))}],$$ where $\chi^{\mathrm{mic}}_{S'}$ is the microlocal multiplicity attached to the conormal bundle of $S'$. Nonvanishing $\chi^{\mathrm{mic}}_{S'}$ signals the importance of the corresponding micro-packet indexed by $S'$ (Barchini et al., 7 Dec 2025).

2. Formal Definition and Construction

Fixing $(O, h)$ as above, define the unipotent Lusztig packet $\Pi^{\mathrm{Lus}}_{q^{h/2}}(G(k))$ as the set of irreducible unipotent representations of $G(k)$ with infinitesimal character $q^{h/2}$ (Barchini et al., 7 Dec 2025). For each $G^\vee(h)$-orbit $S' \subset \mathfrak g(2)$, the $S'$-micro-packet is: $$\Pi^{\mathrm{mic}}_{S'}(G(k)) := \{ X(q^{h/2}, S, \mathcal L) \in \Pi^{\mathrm{Lus}}_{q^{h/2}}(G(k)) \mid \chi^{\mathrm{mic}}_{S'}(\text{IC}(S, \mathcal L)) \neq 0 \}.$$ Equivalently, $X$ lies in $\Pi^{\mathrm{mic}}_{S'}$ exactly if the perverse sheaf $\text{IC}(S, \mathcal L)$ has nonzero microlocal contribution in the conormal bundle over $S'$ in its characteristic cycle.

Micro-packets are always finite, and in the computed examples (e.g., $F_4(a_3)$) provide subsets of Lusztig's packet that can be understood as atomic building blocks for more intricate representation-theoretic constructions (Barchini et al., 7 Dec 2025).

3. Micro-Packets within the Theory of Weak Arthur Packets

Weak Arthur packets are unions of micro-packets. Fixing a nilpotent orbit $O$ and its associated $q^{h/2}$, the weak Arthur packet is defined as

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

where $\operatorname{AZ}$ denotes the normalized Aubert–Zelevinsky involution, and $\mathrm{sp}(O)$ is the special piece containing $O$. The main conjecture (verified for $F_4(a_3)$ in (Barchini et al., 7 Dec 2025)) states that every such weak Arthur packet is a union of micro-packets, each corresponding to "dual" orbits $\widehat{S}$ lying in $\mathrm{sp}(O)$: $$\Pi^{\mathrm{weak}}_{\psi_O}(G(k)) = \bigcup_{G^\vee \cdot S \in \mathrm{sp}(O)} \Pi^{\mathrm{mic}}_{\widehat S}(G(k)).$$ Each $\Pi^{\mathrm{mic}}_{\widehat S}$ behaves analogously to an Arthur packet for a simplified parameter, and $AZ$ interchanges micro-packets for dual orbits (Barchini et al., 7 Dec 2025).

4. Explicit Computations: The Case of $F_4(a_3)$

For $G$ split of type $F_4$ and $\mathcal O = F_4(a_3)$, the unipotent Lusztig packet at $q^{h/2}$ consists of $20$ representations $X_1, ..., X_{20}$ parametrized by pairs $(S_i, \mathcal L)$, $i=0, ..., 11$ (including local system multiplicities). The characteristic cycles $\operatorname{CC}(\text{IC}(S_i, \mathcal L))$ are computed explicitly using Kashiwara’s index formula and Kazhdan–Lusztig polynomials.

Representative micro-packets are:

• $$\Pi^{\mathrm{mic}}_{S_0} = \{ X_5, X_{13}, X_{17}, X_{19}, X_{20} \}$$
• $$\Pi^{\mathrm{mic}}_{S_1} = \{ X_5, X_9, X_{13}, X_{15}, X_{17}, X_{19} \}$$
• $$\Pi^{\mathrm{mic}}_{S_2} = \{ X_5, X_9, X_{11}, X_{13}, X_{17}, X_{18} \}$$
• $$\Pi^{\mathrm{mic}}_{S_3} = \{ X_5, X_8, X_9, X_{13}, X_{15} \}$$
• $$\Pi^{\mathrm{mic}}_{S_7} = \{ X_5, X_7, X_8, X_9, X_{11} \}$$

The weak Arthur packet for $F_4(a_3)$ is then

$$\Pi^{\mathrm{weak}}_{\psi_{F_4(a_3)}}(G(k)) = \Pi^{\mathrm{mic}}_{S_0} \cup \Pi^{\mathrm{mic}}_{S_1} \cup \Pi^{\mathrm{mic}}_{S_2} \cup \Pi^{\mathrm{mic}}_{S_3} \cup \Pi^{\mathrm{mic}}_{S_7}$$

(Barchini et al., 7 Dec 2025).

5. Structural and Theoretical Properties

Each micro-packet is closely related to the geometry of the nilpotent cone and to the asymptotics of character expansions. Micro-packets are finite, typically behave as units under the Aubert–Zelevinsky duality, and are often (conjecturally always) sets of unitary representations whose constituents can be tracked through their microlocal invariants.

Micro-packets fulfill fundamental organizing roles:

• The constituents of a weak Arthur packet can be exactly described as a union of micro-packets corresponding to certain orbits.
• In concrete settings, each micro-packet is naturally associated to an Arthur-type or simplified parameter, explaining and controlling the structure of larger packets (Barchini et al., 7 Dec 2025).

6. Microlocal Multiplicities and Computational Framework

The determination of a micro-packet depends on explicit computation of the microlocal multiplicities $\chi^{\mathrm{mic}}_{S'}(\text{IC}(S, \mathcal L))$. These are governed by formulas: $$\chi^{\mathrm{mic}}_{S'}(P) = \sum_{S''} c(S', S'') \chi^{\mathrm{loc}}_{S''}(P),$$ with $c(S', S'')$ determined from the geometry of conormal bundles, and $\chi^{\mathrm{loc}}_{S''}(P)$ tracking the stalk cohomology at $S''$. Fourier transform identities and D-module techniques are also employed, especially to handle non-spherical cases (Barchini et al., 7 Dec 2025).

7. Broader Implications and Directions

The existence and explicit characterization of micro-packets integrate geometric, combinatorial, and analytic facets of $p$-adic representation theory. Their role is pivotal as the smallest geometric subpackets respecting characteristic cycle data, and their union provides precise control over the intricate overlaps and decomposition of weak Arthur packets. This framework admits generalization, including exceptional types, and suggests new intersections between geometric representation theory, harmonic analysis, and Langlands duality (Barchini et al., 7 Dec 2025).