Branching laws of unitary highest weight modules with respect to semisimple symmetric paris Toshiyuki KOBAYASHI (University of Tokyo)

Let G ⊃ H be reductive Lie groups, and π ∈ Ĝ, an irreducible unitary representation of G. The restriction π|_{H} decomposes uniquely into irreducibles:

π|_{H} ≃ ∫ ⊕ ; Ĥ n_{π}(σ)σdμ(σ) (branching law).

An interesting setting is the case n_{π}(σ) ≤ 1 (multiplicity free). However, for a general π ∈ Ĝ, the multiplicity n_{π}(σ) can be infinite even though (G, H) is a symmetric pair (cf. [3]). We give a sufficient condition on π and (G, H) such that n_{π}(σ) ≤ 1 for any σ ∈ Ĥ.

Theorem A. Let G be a non-compact Hermitian Lie group, (G, H) a symmetric pair, and π ∈ Ĝ a scalar highest weight module. Then π|_{H} decomposes with multiplicity free.

Analogous results also hold for ⊗-product, and for finite dimensional representations. Theorem A gives a uniform explanation of multiplicity free results in classical cases, such as GL_{m} × GL_{n}-duality, the Clebsch-Gordan formula, the Plancherel formula for line bundles over Hermitian symmetric spaces, the Kostant-Schmid formula, and so on, together with new multiplicity free formulae. Among other cases, we give an explicit formula, when π|_{H} splits discretely: We say (G, H) is holomorphic type if H is defined by τ ∈ Aut(G) acting holomorphically on G/K. Take 𝔱^{τ} ⊂ 𝔨^{τ} and extend 𝔱 ⊂ 𝔨. Let k = ℝ- rank G/H and take a maximal set of strongly orthogonal roots {ν_{1}, …, ν_{k}} in Δ(𝔭_{+}^{−τ} , 𝔱^{τ}). Here is a generalization of the Kostant-Schmid formula to non-compact H:

Theorem B. If L^{G}(μ) is a holomorphic discrete series of scalar type, and (G, H) is a symmetric pair of holomorphic type, then

L^{G}(μ)|_{H} ≃ _{\begin{matrix}∑ & ⊕ \\ a_{1}≥···≥a_{k}≥0 \end{matrix}}L^{H}(μ|_{𝔱^{τ}} − ∑ a_{j}ν_{j}).

References

[1] J. Faraut and E. Thomas, Invariant Hilbert spaces of holomorphic functions, preprint.

[2] T. Kobayashi, Multiplicity free branching laws for unitary highest weight modules, Proceedings of the Symposium on Representation Theory held at Saga, Kyushu 1997 (eds. K. Mimachi) (1997), 9–17.

[3] , Discrete decomposability of the restriction of A_{𝔮}(λ) with respect to reductive subgroups and its applications, Part I, Invent. Math. 117, (1994), 181–205; Part II, Annals of Math. 147, (1998), 1–21; Part III, Invent. Math. 131, (1998), 229–256.

[4] , Multiplicity-free restrictions of unitary highest weight modules for reductive symmetric pairs, preprint.

Department of Mathematical Sciences, University of Tokyo

Komaba, Meguro, Tokyo 153-8914, Japan

E-mail address: toshi@ms.u-tokyo.ac.jp

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