[Paper Review] Raising nilpotent orbits in wave-front sets

Dihua Jiang, Baiying Liu+1 more|arXiv (Cornell University)|Dec 30, 2014

Advanced Algebra and Geometry12 references3 citations

TL;DR

This paper investigates the structure of wave-front sets of smooth representations of reductive groups over local or global fields, showing that if a nilpotent orbit is in the wave-front set, then certain larger orbits—obtained by a specific partition transformation—are also included. The key result is that maximal orbits in the wave-front set under closure ordering are always special nilpotent orbits, extending prior work of Mœglin and supporting a conjecture on the exclusivity of special orbits in wave-front sets.

ABSTRACT

We study wave-front sets of representations of reductive groups over global or non-archimedean local fields.

Motivation & Objective

• To understand the structure of wave-front sets of smooth representations of reductive groups over local or global fields.
• To determine which nilpotent orbits can appear as maximal elements in wave-front sets under the closure ordering.
• To extend Mœglin’s earlier results on irreducible representations to general smooth representations.
• To investigate whether non-special orbits can be maximal in wave-front sets, particularly in exceptional groups.
• To provide a uniform framework for analyzing wave-front sets via raising nilpotent orbits through partition transformations.

Proposed method

• Define the wave-front set of a smooth representation π as the set of nilpotent orbits O such that the twisted Jacquet module π_{N_u, ψ_u} is nontrivial for u ∈ O.
• Use the Jacobson-Morozov theorem to associate a sl₂-triple to each nilpotent element u, enabling the construction of unipotent subgroups N_u and characters ψ_u.
• Apply a raising procedure: if a partition p corresponds to a non-special orbit, replace a pair (i,i) with (i−1,i+1) to obtain a larger orbit p′, and show p′ is also in the wave-front set under suitable conditions.
• Analyze the structure of wave-front sets in classical and exceptional groups using representation-theoretic techniques, including degenerate Whittaker models and character expansions.
• Use the Schrödinger model and Heisenberg group representations to study the behavior of smooth vectors and twisted co-invariants.
• For global fields, define the global wave-front set via nonvanishing of functionals ∫_{N_u(k)\N_u(𝔸)} f(n)ψ̅_u(n) dn on automorphic forms, and extend the raising argument to this setting.

Experimental results

Research questions

• RQ1 Can non-special nilpotent orbits appear as maximal elements in the wave-front set of a smooth representation?
• RQ2 Under what conditions does the presence of a nilpotent orbit in the wave-front set imply the presence of a larger orbit?
• RQ3 To what extent do the raising procedures based on partition transformations preserve membership in the wave-front set?
• RQ4 Are there global automorphic representations whose wave-front sets contain non-special maximal orbits?
• RQ5 Do the results for classical groups extend to exceptional groups, particularly in light of the existence of completely odd, non-special orbits?

Key findings

• For classical groups, if a nilpotent orbit corresponding to a partition p is in the wave-front set, then so is the orbit corresponding to the partition p′ obtained by replacing a pair (i,i) with (i−1,i+1), provided p is not special.
• The process of raising orbits continues until reaching the special expansion p^G of the original partition, implying that all maximal orbits in the wave-front set are special.
• In the global setting, the same raising mechanism applies: if an orbit O is in the global wave-front set, then so is a larger orbit O′ under the same transformation, provided the conditions hold.
• For split exceptional groups, there exist non-special orbits (e.g., the minimal orbit in G₂) that resist elimination via the raising method, but these are ruled out as maximal elements by deeper arguments, supporting the conjecture that only special orbits can be maximal.
• The maximal orbits in the wave-front set of any smooth representation of a reductive group over a local or global field are always special nilpotent orbits, under the closure ordering.
• The method of raising orbits is effective in classical groups and extends to global automorphic forms, with the global wave-front set also containing only special orbits as maximal elements.