Embeddings of unitary highest weight representations and generalized Dirac operators (Q1359506)

Let \(G\) be a simple connected finite-dimensional real Lie group with center \(Z\), let \(G_0= G/Z\) and \(K=p^{-1}(K_0)\), where \(p\) is the canonical projection of \(G\) onto \(G_0\) and \(K_0\) is a maximal compact subgroup of \(G_0\). Let \(\mathfrak k\) be the complexified Lie algebra of \(K\) looked upon as a Lie subalgebra of the complexified Lie algebra \(\mathfrak g\) of \(G\) and assume that \(G/K\) is a Hermitian symmetric space. Then \(K\) has a one-dimensional center and we can choose a decomposition \({\mathfrak k}= CH\oplus[{\mathfrak k},{\mathfrak k}]\) with \(\text{ad }H\) having eigenvalues \(-1\), \(0\), \(1\). If \({\mathfrak u}= \{X|X\in{\mathfrak g}\) and \([H,X]= X\}\) and \(\overline{{\mathfrak u}}= \{X|X\in {\mathfrak g}\) and \([H,X]=-X\}\), then \({\mathfrak q}={\mathfrak k}\oplus{\mathfrak u}\) is a \(\theta\)-stable parabolic subalgebra of \({\mathfrak g}\), where \(\theta\) is the Cartan involution of \(\mathfrak g\) with respect to \(\mathfrak k\). Fix a Cartan subalgebra \(\mathfrak h\) of \(\mathfrak g\) with \({\mathfrak h}\subset{\mathfrak k}\) and let \(\mathfrak b\) be a \(\theta\)-stable Borel subalgebra of \(\mathfrak g\) containing \(\mathfrak h\) and contained in \(\mathfrak q\). There exists a parabolic subalgebra \({\mathfrak q}'\) of \(\mathfrak g\), and a corresponding parabolic subgroup \(Q'\) of \(G\), such that \({\mathfrak q}'\cap{\mathfrak g}_0\) is a real parabolic subalgebra of the (real) Lie algebra \({\mathfrak g}_0\) of \(G\) and \({\mathfrak q}\cap{\mathfrak q}'\) is a Levi factor of both \(\mathfrak q\) and \({\mathfrak q}'\). The Hermitian symmetric pair \((G,K)\) is assumed to be of tube type, i.e., the largest root of the system \(\Phi^+\) of positive roots corresponding to \(\mathfrak b\) of the root system of \(\mathfrak g\) with respect to \(\mathfrak h\) is the biggest element \(\gamma_l\) of the Harish-Chandra's system \(\gamma_1<\gamma_2<\cdots<\gamma_l\) of strongly orthogonal roots for \(\Phi^+_n= \{\alpha|\alpha\in\Phi^+\) and \(\alpha(H)=1\}\). Let \(\sigma\) be the conjugation of \(\mathfrak g\) with respect to \({\mathfrak g}_0\), choose \(X_i\in{\mathfrak u}\), \(Y_i\in\overline{{\mathfrak u}}\) such that \(Y_i=\sigma(X_i)\), \([X_i,Y_i]= H_i\), \([H_i,Y_i]= 2X_i\), \([H_i,Y_i]=- 2Y_i\), \([h,X_i]= \gamma_i(h)X_i\) for all \(h\in{\mathfrak h}\), and let \(\overline N'\) be the nilpotent subgroup of \(G\) generated by \(\{\exp X|X\in\overline{{\mathfrak n}}_0'\}\) with \(\overline{{\mathfrak n}}_0'= \{X|X\in{\mathfrak g}_0\) and \([h_0,X]= -X\}\), where \(h_0={1\over 2}(X_1+ Y_1+\cdots+ X_l+Y_l)\). Suppose \(V\) is an irreducible finite-dimensional \(K\)-module with highest weight \(\lambda\) and, with the nilradical of \(\mathfrak q\) acting by zero on \(V\), let \(L\) be the irreducible \({\mathfrak g}\)-module with highest weight \(\lambda\). There is an automorphism \(\tau\) of \(\mathfrak g\) which carries \({\mathfrak q}\) in \({\mathfrak q}'\) and so induces a mapping of \({\mathfrak q}\)-modules into \({\mathfrak q}'\)-modules. The authors show that \(L\) embeds in the degenerate principal series \(G\)-module \(I_{Q',\tau(V)}\) induced by the \(Q'\)-module \(\tau(V)\). They also show that if, in addition, \(L\) is unitarizable and not isomorphic to the algebraically induced \({\mathfrak g}\)-module \(N= U({\mathfrak g})\otimes_{U({\mathfrak q})}V\), then it can be realized in the kernel of an intertwining operator \(D: I_{Q',\tau(F^*\otimes W)}\to L_{Q',\tau(W)}\) of degenerate principal series \(G\)-modules, with well-defined \({\mathfrak q}\)-modules \(F\) and \(W\). In particular, \(F\) depends only on the level of reduction of \(L\), i.e., on the minimum integer \(d\) with \(J^d\neq 0\), where \(J\) is the graded \({\mathfrak g}\)-submodule of \(N\) such that the short exact sequence \(0\to J\to N\to L\to 0\) is exact. The operator \(D\) is called a generalized Dirac operator since it is given by \(D(\sum e^i\otimes f_i)= \sum\partial(e^i)f_i\), where the \(e^i\) form a basis of \(\tau(F^*)\), the functions \(f_i\) may be regarded as real analytic \(W\)-valued functions on \(\overline N'\), and the \(\partial(e^i)\) are homogeneous differential operators of degree \(d\) in the variables \(e^i\).