Quotient branching laws and Gan-Gross-Prasad relevance for general linear groups

Abstract: This paper proves the branching laws for the full class of unitarizable representations of general linear groups in non-Archimedean local fields, extending the original notion of Gan-Gross-Prasad relevant pair for Arthur-type representations \cite{GGP2, Gur, Cha_crelle}. Further, we provide an explicit computable algorithm to determine the generalized GGP relevant pair, as developed in \cite{Cha_qbl}. In particular, we show that if $π$ and $π'$ are any irreducible smooth representations of $\mathrm{GL_{n+1}(F)}$ and $\mathrm{GL_{n}(F)}$ respectively, and their Langlands data or Zelevinsky data are given in terms of multisegments, then through an algorithmic process we can determine whether the space $\mathrm{Hom}_{\mathrm{GL_n(F)}}(π, π')$ is non-zero. Finally, when one of the represntations $π$ and $ π'$ is a generalized Speh representation, we give a complete classification for the other one for which the Hom space is non-zero.

• The paper introduces an explicit combinatorial algorithm to determine non-vanishing Hom-spaces in branching laws for p-adic GL groups.
• It extends the Gan-Gross-Prasad relevance criterion to all unitarizable representations using multisegment and RdLi-commutativity techniques.
• The work resolves quotient branching problems by completely characterizing representation restrictions with computationally tractable procedures.

Quotient Branching Laws and Gan-Gross-Prasad Relevance for General Linear Groups

Overview and Context

The paper "Quotient branching laws and Gan-Gross-Prasad relevance for general linear groups" (2512.11696) addresses significant open problems in the representation theory of $p$-adic general linear groups, specifically those relating to the quotients of restricted representations and the precise combinatorial conditions under which irreducible smooth representations appear in such quotients. It extends the scope of branching laws previously established for certain special classes (tempered or Arthur-type) to fully encompass the class of unitarizable representations. The paper generalizes the Gan-Gross-Prasad (GGP) relevance criterion and establishes a robust equivalence of combinatorial branching data with the original GGP conditions. It also provides an explicit, computational algorithm for determining the non-vanishing of Hom-spaces in restriction problems, framed in the language of multisegments and derivatives/integrals in the Langlands classification.

Technical Contributions

Generalized Quotient Branching Laws

The author proves a branching law for all unitarizable representations of $\mathrm{GL}_{n+1}(F)$ restricted to $\mathrm{GL}_n(F)$ over non-archimedean local fields, characterizing when $\mathrm{Hom}_{\mathrm{GL}_n(F)}(\pi, \pi')$ is nonzero purely in terms of explicit combinatorial data (Langlands or Zelevinsky multisegments). The fundamental result is that the Hom-space is nonzero if and only if the pair $(\pi, \pi')$ is "generalized GGP relevant", as defined via intricate conditions on derivatives, integrals, and the newly formalized notion of RdLi-commutativity.

Generalized GGP Relevance and RdLi-Commutativity

A core technical development is the combinatorial definition of the generalized GGP relevance for pairs of representations, extending the original criteria to all irreducible smooth pairs. This is construced using

• Right and left derivatives/integrals: The paper develops algorithmic procedures (based on the Langlands classification) for iteratively producing Jacquet modules and induced representations that mirror the branching process.
• $\eta$-invariants: These serve as combinatorial fingerprints of representation behavior under segmentation.
• RdLi-commutativity: The crucial new property governing when multisegments associated to representations interact in a manner compatible with nonzero Hom-spaces.

The author gives a precise equivalence between the generalized combinatorial condition (involving minimal derivatives and strong RdLi-commutativity of multisegments) and the classical Gan-Gross-Prasad relevance criterion for the appearance of $\pi'$ as a quotient of the restriction of $\pi$.

Equivalence of Generalized and Classical GGP Relevance

A central theorem demonstrates that for unitary representations, the generalized GGP relevant pair criterion exactly coincides with the original Gan-Gross-Prasad relevance (as defined through explicit correspondence conditions on parameters in Speh and Arthur-type decompositions). This result directly implies the local non-archimedean Gan-Gross-Prasad conjecture for general linear groups and provides an alternative proof using entirely local representation-theoretic, algorithmic methods.

Branching for Generalized Speh Representations

The paper also delivers complete classification results for the branching laws where either member of the pair is a generalized Speh (ladder) representation. It generalizes key previous results, such as those of Prasad and Venketasubramanian, and formulates sharp structural theorems for $$\mathrm{Hom}_{\mathrm{GL}_n(F)}(\pi, \mathrm{Speh})$$ and its dual problems.

Explicit Combinatorial Algorithms

A substantial practical outcome of this work is the provision of a concrete, step-by-step algorithm for deciding the non-vanishing of Hom-spaces for arbitrary irreducible pairs, based only on their multisegment data. This algorithm reduces the quotient branching law check to a finite sequence of combinatorial operations (reduction, interchange, admissibility), thus making the branching law for general linear groups algorithmically tractable.

Significant Results and Claims

• The dimension of $\mathrm{Hom}_{\mathrm{GL}_n(F)}(\pi, \pi')$ is at most one (multiplicity one), aligning with previous work, but the criterion for non-vanishing is completely characterized via the generalized GGP relevance [(2512.11696), Theorem 4.1].
• For unitary pairs, Hom is nonzero if and only if the parameters of the Speh decomposition or complementary series match per the extended GGP relations (bijections on parameter sets satisfying shift and duality rules).
• For generalized Speh representations, the classification of branching is resolved: nonzero Hom spaces correspond precisely to explicit forms of Langlands multisegments (either generic, tower, or union-intersection structures).
• The new combinatorial algorithm yields results compatible with previous conjectures and theorems, reproducing known results as corollaries.

Implications

Theoretical Implications

• This work establishes an effective bridge between intricate combinatorial representation theory (multisegments and Jacquet modules) and geometric conjectures (GGP).
• It resolves the quotient branching law problem for all irreducible smooth representations of $p$-adic general linear groups in the local setting.
• The equivalence theorem strengthens the conceptual foundations of the local Gan-Gross-Prasad conjecture and unifies disparate cases (tempered, generic, Arthur-type, unitary).

Practical and Computational Implications

• The step-by-step algorithm enables explicit computation of branching laws for arbitrary irreducible smooth representations when segmented data is available. This directly benefits computational approaches in automorphic forms and explicit local representation analysis.
• The method is highly constructive; every step can be implemented, reducing reliance on abstract or non-computable criteria.

Future Directions

• The extension to archimedean or global fields, synergizing with recent results in those contexts, appears promising.
• The combinatorial machinery developed here could influence approaches to Ext-branching laws, character computations, and the study of distinction properties in broader reductive contexts.
• Potential applications in the analysis of automorphic periods and explicit harmonic analysis over $p$-adic groups.

Conclusion

This paper systematically generalizes and algorithmizes the quotient branching laws and the Gan-Gross-Prasad relevance criterion for $p$-adic general linear groups, providing both rigorous theoretical structure and explicit computational tools. It closes substantial gaps in the understanding of when irreducible representations appear in restricted quotients, connecting local representation theory, combinatorics, and broader branching conjectures. The equivalence and classification results for unitary and Speh representations, along with the explicit algorithm, will serve as essential technical infrastructure for further research in the field.