Theorem

Multiplicity-One Theorem

Number Theory

The multiplicity-one theorem is a result about automorphic representations, the representations of a group realized within the space of square-integrable cusp forms. For the general linear group GL(n), it establishes that each irreducible representation occurs at most once in that space. Jacquet and Langlands proved the case GL(2) in 1970, and Piatetski-Shapiro and Shalika independently proved the case for n greater than 2 in 1979 and 1974 respectively, using the uniqueness of the Whittaker model; the property holds for SL(2) but fails for SL(n) when n exceeds 2, and its strengthening, the strong multiplicity-one theorem, states that two cuspidal automorphic representations of GL(n) are isomorphic whenever their local components agree at all but finitely many places. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts

Classification

Statement Form

Uniqueness Theorem 1

Statement Form

Inequality 1

Statement

For GL(n), every smooth irreducible admissible representation of G(A) occurs with multiplicity at most one in the space of cusp forms of a given central character. 1

Connections

Has Statement Form

Inequality

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

Uniqueness Theorem

Sources

1. Multiplicity-one theorem (Wikipedia)

Definition

Quote, Definition

any smooth irreducible admissible representation of G(A) occurs with multiplicity at most one in the space of cusp forms of central character

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Multiplicity-One Theorem

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