Subtopic Deep Dive
L-functions for GL(n)
Research Guide
What is L-functions for GL(n)?
L-functions for GL(n) are Dirichlet series attached to automorphic representations of the general linear group GL(n) over number fields, studied for their meromorphic continuation, functional equations, and special values.
These L-functions generalize the Riemann zeta function and extend to higher rank via Langlands functoriality. Key results include exterior square functoriality for GL(4) (Kim, 2002, 663 citations) and classification of irreducible representations of GL(n) (Zelevinsky, 1980, 645 citations). Over 20 papers in the list address representations and automorphic forms for GL(n).
Why It Matters
L-functions for GL(n) encode arithmetic data central to Langlands reciprocity conjectures, with applications in modular forms and geometric Langlands (Kapustin and Witten, 2007, 920 citations). They enable proofs of functoriality transfers like exterior square from GL(4) to GL(6) (Kim, 2002). In representation theory, Zelevinsky's classification (1980) underpins decomposition of induced representations, impacting p-adic groups and automorphic L-functions.
Key Research Challenges
Functoriality Conjectures
Proving Langlands functoriality for general automorphic representations on GL(n) remains open beyond specific cases like exterior square (Kim, 2002). General transfers require new analytic methods for L-function continuation. Over 10 papers cite Kim's result as partial progress.
Meromorphic Continuation
Establishing analytic continuation and functional equations for higher rank GL(n) L-functions demands advanced estimates. Zelevinsky's representation theory (1980) aids cusp form analysis but full meromorphy is unresolved. Challenges persist in non-tempered representations.
Special Values Computation
Computing special values at critical points links to arithmetic invariants, building on geometric Langlands duality (Kapustin and Witten, 2007). Explicit formulas are limited to low n. Kač and Peterson's infinite-dimensional algebras (1984) suggest theta function approaches.
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...
Infinite-dimensional Lie algebras, theta functions and modular forms
Victor G. Kač, Dale H. Peterson · 1984 · Advances in Mathematics · 665 citations
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
Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism
Pavel Etingof, Victor Ginzburg · 2002 · Inventiones mathematicae · 631 citations
Pseudo-Hermitian structures on a real hypersurface
S. M. Webster · 1978 · Journal of Differential Geometry · 548 citations
A classification of irreducible prehomogeneous vector spaces and their relative invariants
Masahide Sato, T. Kimura · 1977 · Nagoya Mathematical Journal · 536 citations
Let G be a connected linear algebraic group, and p a rational representation of G on a finite-dimensional vector space V , all defined over the complex number field C . We call such a triplet ( G, ...
Reading Guide
Foundational Papers
Start with Zelevinsky (1980, 645 citations) for GL(n) irreducible representations, then Kim (2002, 663 citations) for exterior square functoriality; Kapustin-Witten (2007, 920 citations) provides geometric context.
Recent Advances
Key advances include Kim (2002) functoriality and Etingof-Ginzburg (2002, 631 citations) on symplectic algebras relevant to representations.
Core Methods
Langlands functoriality, Zelevinsky classification of induced representations, electric-magnetic duality in geometric settings.
How PapersFlow Helps You Research L-functions for GL(n)
Discover & Search
Research Agent uses searchPapers and citationGraph on 'L-functions GL(n)' to map 250M+ papers, surfacing Henry Kim's functoriality paper (2002, 663 citations) with 50+ citing works. exaSearch refines for 'GL(4) exterior square', while findSimilarPapers links Zelevinsky (1980) to representation theory clusters.
Analyze & Verify
Analysis Agent applies readPaperContent to extract L-function proofs from Kim (2002), then verifyResponse with CoVe checks functoriality claims against citations. runPythonAnalysis computes pole locations via NumPy for sample Dirichlet series; GRADE scores evidence strength on meromorphic properties.
Synthesize & Write
Synthesis Agent detects gaps in GL(n) functoriality post-Kim (2002), flagging open transfers; Writing Agent uses latexEditText and latexSyncCitations to draft proofs citing Kapustin-Witten (2007), with latexCompile generating formatted sections and exportMermaid for functional equation diagrams.
Use Cases
"Compute sample L-function poles for GL(2) cusp form using Python."
Research Agent → searchPapers('GL(2) L-function examples') → Analysis Agent → runPythonAnalysis(NumPy Dirichlet series code) → matplotlib pole plot and GRADE-verified output.
"Draft LaTeX section on Zelevinsky's GL(n) representations."
Research Agent → citationGraph(Zelevinsky 1980) → Synthesis Agent → gap detection → Writing Agent → latexEditText(content) → latexSyncCitations(Zelevinsky) → latexCompile(PDF section).
"Find GitHub code for GL(n) representation computations."
Research Agent → paperExtractUrls(Kač Peterson 1984) → Code Discovery → paperFindGithubRepo → githubRepoInspect(Lie algebra code) → verified tensor product examples.
Automated Workflows
Deep Research workflow scans 50+ GL(n) papers via searchPapers → citationGraph, producing structured reports on functoriality progress from Kim (2002). DeepScan applies 7-step CoVe analysis to Kapustin-Witten (2007), verifying geometric Langlands L-function links. Theorizer generates hypotheses on GL(n) special values from Zelevinsky representations.
Frequently Asked Questions
What defines L-functions for GL(n)?
Dirichlet series from automorphic representations on GL(n), with meromorphic continuation and functional equations central to Langlands program.
What are key methods in this area?
Functoriality transfers (Kim, 2002), induced representation classification (Zelevinsky, 1980), and geometric duality (Kapustin and Witten, 2007).
What are the most cited papers?
Kapustin-Witten (2007, 920 citations) on geometric Langlands; Kim (2002, 663 citations) on GL(4) exterior square; Zelevinsky (1980, 645 citations) on GL(n) irreducibles.
What open problems exist?
General functoriality for GL(n), full meromorphic continuation beyond low ranks, and explicit special value formulas.
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Part of the Advanced Algebra and Geometry Research Guide