Noncommutative Harmonic Analysis pp 291-353 | Cite as

# A branching law for subgroups fixed by an involution and a noncompact analogue of the Borel-Weil theorem

## Abstract

Let g be a complex semisimple Lie algebra and let t be the subalgebra of fixed elements in g under the action of an involutory automorphism of g. Any such involution is the complexification of the Cartan involution of a real form of g. If *V* _{λ} is an irreducible finite-dimensional representation of g, the Iwasawa decomposition implies that *V* _{λ} is a cyclic *U*(t)module where the cyclic vector is a suitable highest weight vector *v* _{λ}. In this paper we explicitly determine generators of the left ideal annihilator *L* _{λ} (t) of *v* _{λ} in *U*(t). One of the applications of this result is a branching law which determines how *V* _{λ} decomposes as a module for t Other applications include (1) a new structure theorem for the subgroup *M* (conventional terminology) and its unitary dual, and (2) a generalization of the Cartan-Helgason theorem where, in the generalization, the trivial representation of *M* (using conventional terminology) is replaced by an arbitrary irreducible representation τ of *M*. For the generalization we establish the existence of a unique minimal representation of g associated to τ.

Another application (3) yields a noncompact analogue of the Borel-Weil theorem. For a suitable semisimple Lie group G and maximal compact subgroup *K*, where g = (Lie *G*)_{ℂ} and t = (Lie *K)* _{ℂ} the representation *V* _{λ} embeds uniquely (see [W], §8.5) as a finite dimensional (g, *K)* submodule *V* _{λ} of the Harish-Chandra module *H*(*δ*, *ξ*) of a principal series representation of *G*. The functions in *H*(*δ*, *ξ*) are uniquely determined by their restriction to *K*. As a noncompact analogue of the Borel-Wei! theorem the functions in *V* _{λ} are given as solutions of differential equations arising from the generators of *L* _{λ}(t) (rather than Cauchy-Riemann equations as in the Borel-Weil theorem).

## Keywords

Disjoint Union Left Ideal Cartan Subalgebra Maximal Compact Subgroup Principal Series## Preview

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## References

- [A]S. Araki, On root systems and an infinitesimal classification of irreducible symmetric spaces,
*J. Math. Osaka City Univ.***13**(1962), 1–34.MathSciNetGoogle Scholar - [BB]A. Beilinson and J. Bernstein, A generalization of Casselman’s submodule theorem,
*Representation Theory of Reductive Groups*, Birkhäuser, Progress in Mathematics, 1982, pp. 35–67.Google Scholar - [C]W. Casselman, Differential equations satisfied by matrix coefficients, preprint.Google Scholar
- [K1]B. Kostant, On the existence and irreducibility of certain series of representations,
*Lie groups and their representations*, I.M. Gelfand, ed., Halsted Press, John Wiley, 1975, pp. 231–329.Google Scholar - [K2]_, On Whittaker vectors and representation
*theory, lnventiones Math.***48**(1978), 101–184.MathSciNetMATHGoogle Scholar - [K3]_, Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, the
*p*-decomposition,*C*(g) = End*Vp*⊗*C*(*P*), and the g-module structure of ∧g,*Adv. in Math.*125 (1997), 275–350.MathSciNetMATHCrossRefGoogle Scholar - [LW]J. Lepowsky and N. Wallach, Finite- and infinite-dimensional representations of linear semisimple groups,
*Trans. Amer. Math. Soc.***184**(1973), 223–246.MathSciNetGoogle Scholar - [PRV]K. R. Parthasarathy, R. Ranga Rao, and V. S. Varadarajan, Representations of complex semisimple Lie groups and Lie algebras,
*Ann. of Math.*(2), 85 (1967), 383–429.MathSciNetMATHCrossRefGoogle Scholar - [V]D. Vogan, Irreducible characters of semisimple Lie groups IV. Charactermultiplicity duality,
*Duke Math. J.***49**(1982), 943–1073.MathSciNetMATHCrossRefGoogle Scholar - [W]