## The Dirac operator on homogeneous spaces and representations of reductive Lie groups. I.(English)Zbl 0649.58031

From the introduction: “Let G be a real connected, reductive Lie group and K a maximal compact subgroup containing a given compact, connected subgroup L of G. The reductive homogeneous space G/L becomes Riemannian via an L-invariant metric $$(.,.)$$ and take a left G-invariant, metric connection $$\gamma$$ on the tangent bundle. If $$G/L$$ is G-spin, then associated to $$((.,.),\gamma)$$ there is a twisted Dirac operator $$D_ V$$, a first order left G-invariant, elliptic, essentially self-adjoint differential operator, on spinors with values in an induced vector bundle $$\underline V$$ via a finite-dimensional L-module V. The left regular representation of G gives a unitary representation $$\ell$$ of G on the $$L^ 2$$-kernel of $$D_ V$$, Ker $$D_ V$$ (the space of harmonic spinors). We are interested in how (Ker $$D_ V,\ell)$$ decomposes into irreducible unitary representations of G when L is any compact subgroup.
When G is noncompact semisimple with rank K= rank G, it is known how to construct the discrete series representations of G using Dirac operators with $$\gamma$$ the reductive (or Levi-Civita) connection on the symmetric space G/K; and this was extended to the relative discrete series of a reductive Lie group in Harish-Chandra’s class.
In discussing the general problem two main techniques will be used; that of twisting a twisted Dirac operator by a G-module (Section 2 of this paper), and inducing in stages with Dirac operators in paper II. Having reduced the case of G/L to that of G/K and K/L (which we consider later) our immediate concern is to determine the ‘space of harmonic spinors’ as a unitary G-module, when $$G=K$$ is compact. In fact harmonic spinors are rather more readily obtained for a Dirac operator on symmetric space, therefore the more substantial part of our work will be $$D_ V$$ on compact $$K/L$$. The first step is to take $$D_ V$$ over K/H where H is a maximal torus of K; then with $$V=E_{\lambda}$$, $$\lambda\in \Lambda$$ (one dimensional) Theorem 1 in (3.5) says that Ker $$D_{\lambda}is$$ zero if $$\lambda$$ is singular and a simple K-module of highest weight $$w\lambda-\rho$$, if $$\lambda$$ is nonsingular. Therefore we get a(n) (alternative to Borel-Weil) geometric construction of the irreducible representations of a compact Lie group.”
Reviewer: G.Warnecke

### MSC:

 58J10 Differential complexes 58J40 Pseudodifferential and Fourier integral operators on manifolds 22E15 General properties and structure of real Lie groups
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