Subtopic Deep Dive
Local Langlands Correspondence for GL(n)
Research Guide
What is Local Langlands Correspondence for GL(n)?
The Local Langlands Correspondence for GL(n) establishes a bijection between irreducible smooth representations of GL(n) over a non-archimedean local field and n-dimensional representations of the Weil-Deligne group.
This correspondence classifies representations of p-adic groups via Langlands parameters. Zelevinsky (1980) developed the structure theory for irreducible representations of GL(n) using Lusztig symbols and Zelevinsky segments (645 citations). Key advances include functoriality results by Kim (2002) for the exterior square of GL_4 (663 citations).
Why It Matters
The local Langlands for GL(n) provides the model case for the full Langlands program, enabling explicit computation of representation characters and Hecke eigenvalues. Kim (2002) proved functoriality for exterior square lifts, impacting global functoriality conjectures. Zelevinsky (1980) classification supports explicit formulas in automorphic forms. Connections to geometric Langlands appear in Kapustin-Witten (2007) via electric-magnetic duality (920 citations), influencing mirror symmetry applications.
Key Research Challenges
Generalizing beyond GL(n)
Extending the bijection to other reductive groups requires new parameterizations. Zelevinsky (1980) solved GL(n) via segments, but Arthur's endoscopic classification for general groups remains partial. Stability of transfer factors poses verification issues (Kim 2002).
Proving full functoriality
Functoriality for symmetric powers and exterior powers needs completion. Kim (2002) established exterior square for GL_4 and symmetric fourth for GL_2 (663 citations). Higher rank cases lack unconditional proofs despite potential automorphy (Barnet-Lamb et al. 2011).
Explicit parameter computation
Computing Weil-Deligne parameters for discrete series requires new algorithms. Lusztig (1983) constructed square-integrable representations via affine Hecke algebras (228 citations). Verification against supercuspidal induction remains computationally intensive.
Essential Papers
Electric-magnetic duality and the geometric Langlands program
Anton Kapustin, Edward Witten · 2007 · Communications in Number Theory and Physics · 920 citations
The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N = 4 super Yang-Mills theory in four dimensions.The key ingredients a...
Functoriality for the exterior square of 𝐺𝐿₄ and the symmetric fourth of 𝐺𝐿₂
Henry Kim · 2002 · Journal of the American Mathematical Society · 663 citations
In this paper we prove the functoriality of the exterior square of cusp forms on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper ...
Induced representations of reductive ${\germ p}$-adic groups. II. On irreducible representations of ${\rm GL}(n)$
Andrei Zelevinsky · 1980 · Annales Scientifiques de l École Normale Supérieure · 645 citations
A Family of Calabi–Yau Varieties and Potential Automorphy II
Tom Barnet-Lamb, David Geraghty, Michael Harris et al. · 2011 · Publications of the Research Institute for Mathematical Sciences · 387 citations
We prove new potential modularity theorems for n-dimensional essentially self-dual l -adic representations of the absolute Galois group of a totally real field. Most notably, in the ordinary case w...
Points on some Shimura varieties over finite fields
Robert Kottwitz · 1992 · Journal of the American Mathematical Society · 302 citations
Potential automorphy and change of weight
Thomas Barnet-Lamb, Toby Gee, David Geraghty et al. · 2014 · Annals of Mathematics · 237 citations
We prove an automorphy lifting theorem for l-adic representations where we impose a new condition at l, which we call "potentential diagonalizability."This result allows for "change of weight" and ...
Moduli of finite flat group schemes, and modularity
Mark Kisin · 2009 · Annals of Mathematics · 237 citations
We prove that, under some mild conditions, a two dimensional p-adic Galois representation which is residually modular and potentially Barsotti-Tate at p is modular.This provides a more conceptual w...
Reading Guide
Foundational Papers
Start with Zelevinsky (1980) for GL(n) representation classification via segments (645 citations), then Kim (2002) for functoriality examples (663 citations), followed by Lusztig (1983) on discrete series (228 citations).
Recent Advances
Barnet-Lamb et al. (2011) on potential automorphy for n-dimensional Galois reps (387 citations); Kisin (2009) on modularity via flat group schemes (237 citations).
Core Methods
Weil-Deligne parameters via monodromy operators; Zelevinsky segments for supercuspidals; Rankin-Selberg integrals for L-functions; affine Hecke modules.
How PapersFlow Helps You Research Local Langlands Correspondence for GL(n)
Discover & Search
Research Agent uses citationGraph on Zelevinsky (1980) to map 645+ citing papers on GL(n) representations, then findSimilarPapers for functoriality extensions like Kim (2002). exaSearch queries 'Local Langlands GL(n) Weil-Deligne parameters p-adic' to surface 250M+ OpenAlex papers with filtered relevance.
Analyze & Verify
Analysis Agent applies readPaperContent to extract Langlands parameters from Kim (2002), then verifyResponse with CoVe to check functoriality claims against Zelevinsky (1980). runPythonAnalysis computes representation dimensions via NumPy on segment data; GRADE scores evidence strength for stability conjectures.
Synthesize & Write
Synthesis Agent detects gaps in n>4 functoriality via contradiction flagging across Kim (2002) and Barnet-Lamb et al. (2011). Writing Agent uses latexEditText for parameter tables, latexSyncCitations to link 920 Kapustin-Witten refs, and latexCompile for proof sketches; exportMermaid diagrams Zelevinsky segment induction.
Use Cases
"Compute character table for principal series GL(3,Q_p) using Zelevinsky classification"
Research Agent → searchPapers 'Zelevinsky GL(n)' → Analysis Agent → readPaperContent + runPythonAnalysis (NumPy matrix for segments) → character values table with GRADE verification.
"Write LaTeX section on Kim's exterior square functoriality proof"
Research Agent → citationGraph Kim(2002) → Synthesis → gap detection → Writing Agent → latexEditText (proof outline) → latexSyncCitations (663 refs) → latexCompile → PDF with diagrams.
"Find code for simulating Weil-Deligne representations in Local Langlands"
Research Agent → exaSearch 'GL(n) Local Langlands code' → Code Discovery → paperExtractUrls → paperFindGithubRepo → githubRepoInspect → Python snippets for parameter simulation.
Automated Workflows
Deep Research workflow scans 50+ papers from Zelevinsky (1980) citationGraph, chains searchPapers → readPaperContent → GRADE, outputs structured report on GL(n) parameter bijection. DeepScan applies 7-step CoVe to verify Kim (2002) functoriality against counterexamples. Theorizer generates conjectures for GL(5) stability from Barnet-Lamb et al. (2011) potential automorphy.
Frequently Asked Questions
What is the definition of Local Langlands for GL(n)?
It bijiects irreducible smooth representations of GL(n,F) for local field F with n-dimensional Weil-Deligne representations over C. Zelevinsky (1980) gives explicit parameterization via segments.
What are the main methods used?
Zelevinsky classification uses Lusztig symbols and parabolic induction. Kim (2002) applies Rankin-Selberg integrals for functoriality. Lusztig (1983) uses affine Hecke algebra representations.
What are the key papers?
Zelevinsky (1980, 645 citations) on GL(n) irreducibles; Kim (2002, 663 citations) on exterior square functoriality; Kapustin-Witten (2007, 920 citations) on geometric aspects.
What open problems remain?
Unconditional functoriality for higher symmetric powers of GL(n). Explicit bijection for non-tempered representations. Generalization to inner forms of GL(n).
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Part of the Advanced Algebra and Geometry Research Guide