Equivariant first order differential operators on boundaries of symmetric spaces
Let \(G\) be a real semisimple Lie group with finite centre and MAN a minimal parabolic subgroup of \(G\). Representations of the principal series are induced representations \(T_{\delta, \mu}= \text{Ind}^G_{ \text{MAN}}(\delta \otimes\mu \otimes 1)\), where \(\delta\) is an irreducible representation of \(M\) on a vector space \(V\), \(a\mapsto a^\mu\) is a character of \(A\). The representation \(T_{\delta,\mu}\) acts on the space of functions \(f\in C^\infty (G,V)\) satisfying the condition \(f(gma)= \delta(m^{-1}) a^{-\mu} f(g)\). In the paper under review all those intertwining operators between representations of the principal series are determined which can be written as differential operators of the first order. Let us formulate the result: Let \(\theta\) be a Cartan involution of \(G\) such that its fixed point subgroup is a maximal compact subgroup \(K\) with \(G=KAN\) and \(M\) the centralizer of \(A\) in \(K\). Let \(a\) be the Lie algebra of \(A\). Let \(\lambda\) be an indivisible restricted root and \(g_\lambda\) the corresponding root subspace of the Lie algebra \(g\) of \(G\). Take an orthonormal basis \(Y_j\) in \(g_{-\lambda}\) (with respect to the scalar product defined via the Killing form \(B\) of \(g)\), then \(\theta Y_j\) is an orthonormal basis in \(g_\lambda\). Let \(\sigma_\lambda \) be the restriction to \(M\) of the adjoint representation of \(G\) on \(g^C_\lambda\). Then \(\delta\otimes \sigma_\lambda\) decomposes multiplicity free into irreducible representations \(\delta_l\). Let \(\text{pr}_{\delta_l}\) denote the corresponding projection. Define \(H_\lambda\in a\) by \(B(H,H_\lambda)= \lambda (H)\), \(H\in a\). Under an irreducible representation \(\pi\) a Casimir element of Lie \(M\) goes to the multiplication by a number \(c(\pi)\). At last, take a basis \(e_k\) in \(V\), let \(e^*_k\) denote the dual basis. Now, if \(\mu\) satisfies the condition \(2\mu(H_\lambda)= c(\delta)+ c(\sigma_\lambda)-c (\delta_l)\), then an operator \(\nabla\) defined by \(\nabla f=\sum_{j,k} Y_j \langle f,e^*_k \rangle\text{pr}_{\delta_l} (\theta Y_j\otimes e_k)\) is a first order differential operator intertwining \(T_{\delta,\mu}\) and \(T_{\delta_l, \mu+ \lambda}\). This construction gives all operators in question.
scientific article; zbMATH DE number 1443408