The Plancherel formula for the symmetric space \(G_ C/G_ R\)
In this short paper the author announces a Plancherel formula for reductive symmetric spaces \(X=G_ C/G_ R\) where \(G_ C\) is a complex linear connected reductive group and \(G_ R\) is its real form. The latter means that \(G_ R\) is an open subgroup of the fixed point subgroup of some involutive automorphism \(\tau\) of \(G_ C\) (as a real group). The symmetric space X is dual in some sense to the group \(G_ R\) viewed as the symmetric space \(G_ R\times G_ R/diagonal.\) Denote \({\mathfrak g}_ C=Lie G_ C\), \({\mathfrak g}_ R=Lie G_ R\). Then \({\mathfrak g}_ C={\mathfrak g}_ R+{\mathfrak q}\) where \({\mathfrak q}=i{\mathfrak g}_ R\). For any Cartan subspace \({\mathfrak j}\) of \({\mathfrak q}\) there is defined some family of \(G_ R\)-invariant distributions on X called spherical distributions. They have a simple explicit expression. Let \({\mathfrak j}_ 1,...,{\mathfrak j}_ m\) be a maximal set of Cartan subspaces such that no two of them are conjugate under \(K\cap G_ R\) (K being a maximal compact subgroup of \(G_ C\) invariant under \(\tau\)). Let f be a function in \(C_ c^{\infty}(X)\) and \(f_ 0\) a function in \(C_ c^{\infty}(G_ C)\) which gives f through integrating over cosets. Let \(f^*_ 0(g)=\overline{f_ 0(g^{-1})}\). The Plancherel formula decomposes the scalar square of f in \(L^ 2(X)\) into the values of spherical distributions corresponding to \({\mathfrak j}_ 1,...,{\mathfrak j}_ m\) at \(f_ 0*f^*_ 0\). The Plancherel measure is shown explicitly. In the paper some symbols remain unexplained.
complex linear connected reductive group
scientific article; zbMATH DE number 4214681