Embeddings of Discrete Series into Principal Series

Abstract

Let G be a connected real semisimple Lie group, σ an involution of G, and H an open subgroup of the group G ^σ of fixed points for σ. For simplicity we assume that G has a complexification G _c . We fix a Cartan involution θ of G with σθ = θσ. The involutions of the Lie algebra g of G induced by σ and θ are denoted by the same letters, respectively. Let $g\;{\rm{ = }}\;h\;{\rm{ + }}\;q\,\;{\rm{and }}\,t\;{\rm{ + }}\;p\,\,$ be the decompositions of g into +1 and −1 eigenspaces for σ and θ, respectively. Let ${g^d}$, t^d and ${h^d}$ be the subalgebras of the complexification ${g_c}$ of g defined by

$$\begin{array}{l} {g^d}\;{\rm{ = }}\;{\rm{t}}\; \cap \;h\;{\rm{ + }}\;\sqrt { - 1} (t\; \cap \;{\rm{q}})\;{\rm{ + }}\;\sqrt { - 1} (p\; \cap \;h)\;{\rm{ + }}\;{\rm{(p}}\; \cap \;{\rm{q),}}\\ {{\rm{t}}^d}\;{\rm{ = }}\;{\rm{t}}\; \cap \;h\;{\rm{ + }}\;\sqrt { - 1} (p\; \cap \;h),\;{h^d}\;{\rm{ = }}\;{\rm{t}}\; \cap \;h\;{\rm{ + }}\;\sqrt { - 1} (t\; \cap \;{\rm{q}}), \end{array}$$

and let K, G ^d, K ^d and H ^d be the analytic subgroups of G _c with the Lie algebras t, ${g^d}$ , t^d and ${h^d}$, respectively. Then the homogeneous space X ^d = G ^d/K ^d is a Riemannian symmetric space of the non-compact type and called the non-compact Riemannian form of the semisimple symmetric space X = G/H. The ring D(X) of the invariant differential operators on X is naturally isomorphic to the ring D(X ^d) of invariant differential operators on X ^d.