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


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