There is a nice tool in representation theory, the Howe duality, which as I know works for certain pairs of classical Lie algebras (the reference to the complete list of Howe dual pairs is appreciated very much). Is there any extension of the Howe duality for exceptional algebras?


There is a theory of dual reductive pairs and examples for exceptional Lie algebras. For example, in $E_8$ we have dual reductive pairs $(A_1,E_7)$, $(A_2,E_6)$, $(G_2,F_4)$, $(D_4,D_4)$. These are used implicitly in constructions of $E_8$; for example $(G_2,F_4)$ corresponds to the Freudenthal-Tits construction and $(D_4,D_4)$ corresponds to the triality construction.

A reference is: MR1264015 (95b:17010) Rubenthaler, Hubert
Les paires duales dans les algèbres de Lie réductives. (French) [Dual pairs in reductive Lie algebras]
Astérisque No. 219 (1994), 121 pp.

These seem to be of interest from the point of view of automorphic forms which I know nothing about.

  • $\begingroup$ Could you please give some references on the review (if any) of this exceptional Howe duality? $\endgroup$ – Eugene Starling Jun 16 '10 at 8:49

Yes, and it was studied both from the point of view of the local theory (correspondence of the infinitesimal characters) and automorphic forms. The following review of Jian-Shu Li's paper contains a nice selection of references:

MR1682229 (2000b:22014) Li, Jian-Shu, The correspondences of infinitesimal characters for reductive dual pairs in simple Lie groups. Duke Math. J. 97 (1999), no. 2, 347--377.

More recent work was done by Gordan Savin and Wee Teck Gan.

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    $\begingroup$ For historical purposes, Rubenthaler's paper (Asterisque 1994) (see B. Westbury's answer) is the first (I think) in which there's a classification of dual reductive pairs for semisimple complex Lie algebras. Gordan Savin's work on the subject (Invent. Math. 1994) predates that of Jian-Shu Li (Duke 1999). Also among (if not) the first work on the subject is Rallis-Schiffmann (J. Math. Kyoto, 1995). Another interesting perspective on exceptional dual pairs of complex Lie algebras is in Deligne-Gross (Comptes Rendu 2002). $\endgroup$ – Marty Jun 16 '10 at 15:08
  • $\begingroup$ Oh, I know, I wasn't trying to assign priorities, merely pointing out a convenient reference (all papers mentioned in the review of J.-S. Li's paper predate it, naturally!) The first Rallis-Schiffman paper on $G_2$ (Amer J Math) is from 1989. I didn't know Deligne-Gross paper, though. One thing worth keeping in mind is that the phrase "Howe duality" has too many meanings and the sets of people who worked on the archimedean, p-adic and automorphic theory have very small overlap. $\endgroup$ – Victor Protsak Jun 16 '10 at 22:08

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