# The Vanishing of Scalar Curvature on 6 Manifolds, Einstein’s Equation, and Representation Theory

Chapter

## Abstract

Let *G* be a real semisimple Lie Group, which we may assume to be connected or even simply connected, and let g_{0} = Lie(*G*). The dual vector space of g_{0} will be denoted by g_{0}*. Also, g and g* will denote the complexifications of g_{0} and g_{0}*, respectively. We recall that the group *G* and the Lie algebra g operate on g by the adjoint representation and on g* by the so-called coadjoint representation. Moreover, set *G* _{ℂ} = *Ad*(g). Since the bilinear form (*x*, *y*) = *tr*(*ad x ady*) on g × g is nonsingular, we can identify g and g*, which we shall do whenever it is convenient.

### Keywords

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### References

- [K]Kostant, B., “The Principle of Triality and A Distinguished Unitary Representation of
*SO(4, 4)*,”*Differential Geometrical Methods in Theoretical Physics*, edited by K. Bleuler and M. Werner, Kluwer Academic Publishers, 1988, 65–108Google Scholar - [K1]Kostant, B., “The Vanishing of Scalar Curvature and the Minimal Representation of
*SO(4*,*4)*,*” Operator Algebras*,*Unitary Representations*,*Enveloping Algebras*,*and Invariant Theory*, Birkhäuser Boston: Prog. Math.**92**1990, 85–124MathSciNetGoogle Scholar - [B]Borho, W.,
*A Survey on Eveloping Algebras of Semisimple Lie Algebras*,*I*, CMS Conference Proceedings,**5**1986, 19–50MathSciNetGoogle Scholar - [C]Chevalley, C.,
*The Algebraic Theory of Spinors*, Columbia University Press, 1954MATHGoogle Scholar - [HE]Hawking, S. and Ellis, G.,
*The large scale structure of space-time*, Cambridge University Press, 1973CrossRefGoogle Scholar - [V]Vogan, D.,
*Gelfand-Kirillov dimension for Harish-Chandra modules*, Inventions Math,**48**(1978), 75–98MathSciNetMATHCrossRefGoogle Scholar

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© Birkhäuser Boston 1992