A nonabelian Fourier transform for tempered unipotent representations, II

Abstract: We extend the construction of the nonabelian Fourier transform proposed in our previous paper from the space of unipotent elliptic characters to the space of unipotent compact characters of a reductive $p$-adic group, and we investigate its relation with the branching to maximal compact open subgroups. We prove that this Fourier transform is compatible with parahoric restriction for certain groups, including $\mathrm{SL}_n, \mathrm{GL}_n$, and $G_2$. We also relate the compact setting to the elliptic setting using parabolic induction.

• The paper extends the nonabelian Fourier transform from elliptic characters to unipotent compact representations for reductive p-adic groups.
• It establishes explicit isomorphisms between representation spaces using parabolic induction and parahoric restriction, validated for classical types.
• The analysis computes root of unity factors and enumerates compact pairs, offering insights into L-packet stability and the local Langlands correspondence.

Extension of the Nonabelian Fourier Transform to Tempered Unipotent Compact Representations

Overview and Motivation

The paper "A nonabelian Fourier transform for tempered unipotent representations, II" (2606.14485) significantly generalizes the theory of nonabelian Fourier transforms initiated in [ACR] to the space of unipotent compact representations for reductive $p$-adic groups. The nonabelian Fourier transform originally constructed for elliptic characters—a quotient by parabolically induced characters—is extended to the rigid quotient corresponding to compact representations, following the perspective of [CH2]. The work investigates compatibility with parahoric restriction and demonstrates how induction and restriction functors enable commutation of the Fourier transform across representations associated to various pure inner twists and maximal compact open subgroups.

Theoretical Foundations

Unipotent Representation Parametrization and the Local Langlands Correspondence

Unipotent representations for $p$-adic groups are classified via the local Langlands correspondence and are organized into $L$-packets labeled by conjugacy classes within the Langlands dual group $G^\vee$. For tempered unipotent representations, this labeling involves the Jordan decomposition $x=su$, where $s$ is semisimple and compact, $u$ is unipotent, and the packet is indexed by irreducible characters of the component group $A_{G^\vee}(x)$.

Compact representations, in the sense considered, arise from restrictions to maximal compact open subgroups $K$ of the $p$-adic group $p$0. These subgroups yield finite reductive quotients $p$1 whose representation theory connects to well-understood Deligne–Lusztig theory for finite groups of Lie type, and whose nonabelian Fourier transforms are handled via generalizations of Lusztig’s original constructions [Lubook], [Lu86].

Nonabelian Fourier Transform Extension

Previously, the nonabelian Fourier transform was constructed as an involution on elliptic representations, tracking isomorphisms between Grothendieck groups of unipotent representations modulo parabolic induction. In this work, the involution is extended to a larger space of unipotent compact characters, isomorphic to the rigid quotient. The authors define a basis for this space using triples $p$2 in $p$3 involving compact and essentially elliptic pairs across Levi subgroups and classify the induced involution by flipping $p$4.

Compatibility with various operations—parabolic induction, restriction, and passages between disconnected and connected groups—is systematically established, and explicit isomorphisms are constructed (Theorem \ref{t:cpt-intro}), including geometric interpretations related to the cocenter of the affine Hecke algebra (cf. [BKK], [LNY], [BNP]).

Main Results

Isomorphism of Representation Spaces

A central result (Theorem \ref{t:cpt-intro}) establishes that the compact representation space $p$5 is explicitly isomorphic to the rigid quotient and also to the space of $p$6-orbits of compact pairs modulo suitable equivalence. This extends conjectures confirmed for split, semisimple groups to arbitrary reductive groups and proves Conjecture 9.1 from [ACR]:

$p$7

The proof leverages bijections between different parametrizations of unipotent representations (compact/essentially elliptic pairs, Levi subgroup decompositions) and employs Hopf system formalism as in [Da], mirroring decomposition results for affine Hecke algebra cocenters.

Fourier Transform Compatibility and Parahoric Restriction

The nonabelian Fourier transform on $p$8, defined as flipping $p$9 in each triple $L$0, is shown to be compatible with parahoric restriction to maximal compact subgroups. For the classical groups $L$1, $L$2, and $L$3, commutativity (up to root of unity factors) is rigorously established for the diagrams relating the compact Fourier transform, parahoric restriction, and induction, reproducing conjectural compatibility diagrams (Conjecture \ref{c:compact}, Theorem \ref{t:typeA-intro}, Corollary \ref{c:GL-G2}). Compatibility extends to all Levi subgroups, reducing the conjecture for a group $L$4 to its Levi factors (Corollary \ref{c:levis-imply-compact}).

Explicit Classification and Numerical Results

The paper provides explicit enumerations of compact pairs for $L$5 and $L$6 (see Lemma \ref{lem-pairs-pgln}, Proposition 5.5), demonstrating that for $L$7 prime, the number of compact pairs in $L$8 is $L$9 for $G^\vee$0 and $G^\vee$1 for $G^\vee$2. The compatibility claims are checked concretely for these cases, and root of unity factors relating the parahoric restriction and dual Fourier transforms are computed explicitly.

Implications and Future Directions

The geometric nature of the isomorphisms between representation spaces points to deeper categorical correspondences, as seen in the cocenter identification with Springer-type functions on commuting varieties, and current/forthcoming works on geometric realizations of nonabelian Fourier transforms ([BKK], [LNY], [BNP]). The commutativity with induction and restriction operations provides a robust mechanism for reducing global conjectures to local or Levi subgroup cases, furthering the understanding of $G^\vee$3-packet stability and local proof strategies in the local Langlands program.

Practically, the compatibility results facilitate computations of character values and enhance modularity in algorithmic approaches to branching problems for $G^\vee$4-adic groups. The explicit root of unity factors indicate subtleties in the passage between representation spaces tied to inner twists/Levi subgroup decompositions and hint at links with harmonic analysis and endoscopic transfer.

Future developments likely include:

• Extension to more general classes of representations (non-unipotent, non-tempered),
• Further categorification and geometric interpretations,
• Analysis for larger classes of disconnected or non-split groups,
• Application of the formalism to automorphic induction, trace formula stabilization, and explicit $G^\vee$5-packet analysis in non-classical types.

Conclusion

The paper achieves a systematic generalization of the nonabelian Fourier transform for tempered unipotent representations to the compact (rigid) setting, substantiates compatibility with parahoric restriction and induction, and establishes explicit isomorphisms across representation spaces for several classical types. The explicit computation and confirmation of root of unity commutativity in type $G^\vee$6 groups and $G^\vee$7 reinforce the conjectural framework and open avenues toward further geometric and categorical advancements in $G^\vee$8-adic representation theory and the Langlands program.