Representations of Lie groups in \(L^ 2\)-kernels of elliptic homogeneous operators.

*(English)*Zbl 0531.22011There are already known many cases when irreducible representations of a Lie group G may concretely be realized as kernels of invariant differential operators D acting between spaces of sections of appropriately chosen vector G-bundles [e.g. R. Parthasarathy, Ann. Math., II. Ser. 96, 1-30 (1972; Zbl 0249.22003), M. F. Atiyah and W. Schmid, Invent. Math. 42, 1-62 (1977; Zbl 0373.22001)]. In the paper under review the authors give some general conitions on the operator D which ensure that representations so constructed will be irreducible and square integrable.

The setting is as follows: G is a Lie group, \(K\subset G\) a compact subgroup, \({\mathbb{E}}\), \({\mathbb{E}}'\) finite-dimensional homogeneous bundles over G/K associated to irreducible K-modules E, E’, D: \({\mathcal C}^{\infty}({\mathbb{E}})\to {\mathcal C}^{\infty}({\mathbb{E}}')\) a first order G- invariant differential operator. Denote \({\mathcal H}^ 2({\mathbb{E}})=L^ 2({\mathbb{E}})\cap Ker D\). Let \({\mathfrak g}={\mathfrak k}+{\mathfrak p}\) be a decomposition of the Lie algebra \({\mathfrak g}\) of G, orthogonal with respect to a K-invariant inner product on \({\mathfrak g}\), \({\mathfrak t}\subset {\mathfrak k}\) a maximal Abelian algebra and B - the Borel subgroup of the complexification \(K_{{\mathbb{C}}}\) of K which contains the maximal torus T with Lie alebra \({\mathfrak t}\). Using the Borel-Weil-Bott realization of irreducible K-modules in cohomology spaces of vector bundles induced by B-modules, \(E={\mathcal H}^ s(1_{\lambda +2\rho})\), \(E'={\mathcal H}^ s(1'_{\lambda})\) for suitable s, the authors define the primary symbol \(\sigma_ 0\) of D as a B-module homomorphism \(\sigma_ 0: {\mathfrak p}_{{\mathbb{C}}}\otimes 1_{\lambda +2\rho}\to 1'_{\lambda}\) uniquely determining the usual symbol \(\sigma\) : \({\mathfrak p}\otimes E\to E'\) of D. The main result of the paper is the theorem 3.1, where it is shown that if the primary symbol \(\sigma_ 0\) satisfies certain condition (P) (for positivity) involving roots of \(k_{{\mathbb{C}}}\) and weights occuring in the kernel of \(\sigma_ 0\), then the representation of G in \({\mathcal H}^ 2({\mathbb{E}})\) is irreducible and that the multiplicities of K-modules occuring in \({\mathcal H}({\mathbb{E}})={\mathcal C}^{\infty}({\mathbb{E}})\cap Ker D\) satisfy certain estimates given in terms of the Blattner numbers. The proof of that result uses fairly powerful machinery from the theory of holomorphic vector bundles (Serre’s duality, Griffiths’ vanishing theorem) as well as a certain resolution of the symbol \(\sigma\) (the polynomialization), due to R. Hotta and R. Parthasarathy [Invent. Math. 26, 133-178 (1974; Zbl 0298.22013)]. The scope of the theorem is illustrated by an application to the irreducibility and square integrability of representations obtained this way from the partial Dirac operator, considered by Hotta and Parthasarathy. Let us point out that in the recent work of A. Connes and H. Moscovici [Ann. Math., II. Ser. 115, 291-330 (1982; Zbl 0515.58031)] it is shown that for G- invariant elliptic pseudo-differential operators of arbitrary order the corresponding representation in \({\mathcal H}^ 2({\mathbb{E}})\) finitely decomposes into square integrable representations, provided certain conditions on group G are satisfied.

The setting is as follows: G is a Lie group, \(K\subset G\) a compact subgroup, \({\mathbb{E}}\), \({\mathbb{E}}'\) finite-dimensional homogeneous bundles over G/K associated to irreducible K-modules E, E’, D: \({\mathcal C}^{\infty}({\mathbb{E}})\to {\mathcal C}^{\infty}({\mathbb{E}}')\) a first order G- invariant differential operator. Denote \({\mathcal H}^ 2({\mathbb{E}})=L^ 2({\mathbb{E}})\cap Ker D\). Let \({\mathfrak g}={\mathfrak k}+{\mathfrak p}\) be a decomposition of the Lie algebra \({\mathfrak g}\) of G, orthogonal with respect to a K-invariant inner product on \({\mathfrak g}\), \({\mathfrak t}\subset {\mathfrak k}\) a maximal Abelian algebra and B - the Borel subgroup of the complexification \(K_{{\mathbb{C}}}\) of K which contains the maximal torus T with Lie alebra \({\mathfrak t}\). Using the Borel-Weil-Bott realization of irreducible K-modules in cohomology spaces of vector bundles induced by B-modules, \(E={\mathcal H}^ s(1_{\lambda +2\rho})\), \(E'={\mathcal H}^ s(1'_{\lambda})\) for suitable s, the authors define the primary symbol \(\sigma_ 0\) of D as a B-module homomorphism \(\sigma_ 0: {\mathfrak p}_{{\mathbb{C}}}\otimes 1_{\lambda +2\rho}\to 1'_{\lambda}\) uniquely determining the usual symbol \(\sigma\) : \({\mathfrak p}\otimes E\to E'\) of D. The main result of the paper is the theorem 3.1, where it is shown that if the primary symbol \(\sigma_ 0\) satisfies certain condition (P) (for positivity) involving roots of \(k_{{\mathbb{C}}}\) and weights occuring in the kernel of \(\sigma_ 0\), then the representation of G in \({\mathcal H}^ 2({\mathbb{E}})\) is irreducible and that the multiplicities of K-modules occuring in \({\mathcal H}({\mathbb{E}})={\mathcal C}^{\infty}({\mathbb{E}})\cap Ker D\) satisfy certain estimates given in terms of the Blattner numbers. The proof of that result uses fairly powerful machinery from the theory of holomorphic vector bundles (Serre’s duality, Griffiths’ vanishing theorem) as well as a certain resolution of the symbol \(\sigma\) (the polynomialization), due to R. Hotta and R. Parthasarathy [Invent. Math. 26, 133-178 (1974; Zbl 0298.22013)]. The scope of the theorem is illustrated by an application to the irreducibility and square integrability of representations obtained this way from the partial Dirac operator, considered by Hotta and Parthasarathy. Let us point out that in the recent work of A. Connes and H. Moscovici [Ann. Math., II. Ser. 115, 291-330 (1982; Zbl 0515.58031)] it is shown that for G- invariant elliptic pseudo-differential operators of arbitrary order the corresponding representation in \({\mathcal H}^ 2({\mathbb{E}})\) finitely decomposes into square integrable representations, provided certain conditions on group G are satisfied.

Reviewer: A.Strasburger

##### MSC:

22E30 | Analysis on real and complex Lie groups |

32M10 | Homogeneous complex manifolds |

58J70 | Invariance and symmetry properties for PDEs on manifolds |

22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |