Harmonic Analysis on Real Reductive Groups

adjoint group analytic diffeomorphism analytic function assertion assume automorphism bijection Cartan decomposition Cartan involution centralizer compact CSG completely invariant open conjugate connected component constant c>0 containing Corollary corresponding CSA's cusp forms defined denote differential operator direct sum discrete series distribution on g e-stable eigenvalues element elliptic estimate exp H following property G-invariant H₂ Haar measure Harish-Chandra Hence homomorphism I-finite implies invariant distribution invariant open subset isomorphism Lemma Let G Let notation Lie algebra Lie group linear locally integrable M₁ maximal compact subgroup Moreover nilpotent nonzero open neighborhood open subset orthogonal polynomial positive system proof Proposition prove regular representation resp Schwartz space Section seminorms semiregular semisimple subalgebra subgroup of G subspace sufficiently small supp Suppose Theorem unique V₁ vector space w₁ Weyl group write xe G αερ