Developments and Retrospectives in Lie Theory pp 385-397 | Cite as

# The Cubic, the Quartic, and the Exceptional Group G_{2}

## Abstract

We study an example first addressed in a 1949 paper of J. A. Todd, in which the author obtains a complete system of generators for the covariants in the polynomial functions on the eight-dimensional space of the double binary form of degree (3,1), under the action of SL_{2} ĂSL_{2}. We reconsider Toddâs result by examining the complexified Cartan complement corresponding to the maximal compact subgroup of simply connected split *G* _{2}. A result of this analysis involves a connection with the branching rule from the rank two complex symplectic Lie algebra to a principally embedded \(\mathfrak{s}\mathfrak{l}_{2}\)-subalgebra. Special cases of this branching rule are related to covariants for cubic and quartic binary forms.

## Key words

Binary formÂ Branching ruleÂ Double binary formÂ G_{2}Â Harmonic polynomialsÂ Principal \(\mathfrak{s}\mathfrak{l}_{2}\)Â Symmetric spaceÂ

## Mathematics Subject Classification (2010):

20G05Â 22E45Â 17B10.Â## Notes

### Acknowledgements

We thank Allen Bell, Nolan Wallach and Gregg Zuckerman for helpful conversations about the results presented here. The first authorâs Ph.D. thesis [17] contains a more thorough treatment of the graded K-multiplicities associated with the symmetric pair (\(G_{2},\mathfrak{s}\mathfrak{l}_{2} \oplus \mathfrak{s}\mathfrak{l}_{2}\)), which was jointly directed by Allen Bell and the second author. The problem concerning the graded decomposition of \(\mathcal{H}_{\mathfrak{p}}\) as a K-representation was suggested by Nolan Wallach (see [19]), while the problem of studying the restriction of a finite-dimensional representation to a principally embedded \(\mathfrak{s}\mathfrak{l}_{2}\)-subalgebra was suggested by Gregg Zuckerman (see [20]).

Finally, we would like to thank the referee for many helpful suggestions, and corrections.

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