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In mathematics, the '''Langlands fundamental lemma''', named after ] and Diana Shelstad, is a family of conjectural identities between orbital integrals for ''G''(''F'') and ]s for ]s of ''G'', a conjectural identity between ] ]s. They are part of the ]. The '''Langlands Fundamental Lemma''' is a family of conjectural identities between orbital integrals for G(F) and orbital integrals for endoscopic groups of G, a conjectural identity between p-adic integrals. They are part of the ].


It was proved by Ngo Bao Chau<ref></ref><ref>, Time</ref><ref>, Bill Casselman</ref>. It was proved by Ngo Bao Chau.<ref></ref><ref>, Time</ref><ref>, Bill Casselman</ref>


==The fundamental lemma== ==The fundamental lemma==
For every <math>\gamma \in H(F)</math> that is strongly <math>G</math>-regular semisimple,

For every <math>\gamma \in H(F)</math> that is strongly ''G''-regular semisimple, :<math>\Lambda_{G,H}(\gamma ) = \Lambda_{H}^{st}(\gamma ).</math>

:<math>\Lambda_{G,H}(\gamma ) = \Lambda_H^{st}(\gamma ).</math>


The superscript <math>st</math> stands for ''stable''.<ref>, Thomas C. Hales</ref> The superscript <math>st</math> stands for ''stable''.<ref>, Thomas C. Hales</ref>

Revision as of 07:59, 15 December 2009

The Langlands Fundamental Lemma is a family of conjectural identities between orbital integrals for G(F) and orbital integrals for endoscopic groups of G, a conjectural identity between p-adic integrals. They are part of the Langlands program.

It was proved by Ngo Bao Chau.

The fundamental lemma

For every γ ∈ H ( F ) {\displaystyle \gamma \in H(F)} that is strongly G {\displaystyle G} -regular semisimple,

Λ G , H ( γ ) = Λ H s t ( γ ) . {\displaystyle \Lambda _{G,H}(\gamma )=\Lambda _{H}^{st}(\gamma ).}

The superscript s t {\displaystyle st} stands for stable.

References

  1. Not even wrong
  2. Top 10 Scientific Discoveries of 2009, Time
  3. The fundamental lemma, Bill Casselman
  4. A Statement of the Fundamental Lemma, Thomas C. Hales
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