A uniqueness theorem for spherical distributions on the tangent space of a symmetric space (Q1357048)

For a semisimple Lie group with finite centre and with an involutive automorphism \(\sigma\), consider the symmetric space \(G/H\), where \(H\) is a connected component of the fixed point set \(G^\sigma\) and \({\mathfrak g}={\mathfrak h}\oplus{\mathfrak q}\) the decomposition of \({\mathfrak g}\) into the sum of eigenspaces of the differential \(D\sigma=\sigma_*\). Following Chevalley's theorem, the symmetric algebra \(S[{\mathfrak q}_{\mathcal C}]^H\) of \(H\)-invariants is isomorphic to the symmetric algebra \(S[{\mathfrak a}_{\mathcal C}]^W\) of \(W\)-invariants, where \({\mathfrak a}\) is a Cartan subalgebra and \(W\) the corresponding Weyl group. The characters of \(S[{\mathfrak a}_{\mathcal C}]^W\) are given by evaluation at points \(\Lambda\in{\mathfrak a}^*_{\mathcal C}\). In this note, the author remarks that if \(u\) is a distribution satisfying the system \[ \partial (P)u=P(\Lambda)u,\quad\forall P\in S[{\mathfrak q}_{\mathcal C}]^H \] with singular support, then \(u=0\). This result is well-known as a theorem of Harish-Chandra in the case of Lie algebras.