To Yuri Ivanovich Manin on the occasion of his 65th birthday NORMALIZED INTERTWINING OPERATORS AND NILPOTENT ELEMENTS IN THE LANGLANDS DUAL GROUP

Let F be a local non-archimedian field and G be a split reductive group over F whose derived group is simply connected. Set G = G(F ). Let also ψ : F → C be a non-trivial additive unitary character of F . For two parabolic subgroups P and Q in G with the same Levi component M we construct an explicit unitary isomorphism FP,Q,ψ : L (G/[P, P ])→̃L(G/[Q,Q]) commuting with the natural actions of the group G × M/[M,M ] on both sides. In some special cases FP,Q,ψ is the standard Fourier transform. The crucial ingredient in the definition is the action of the principal sl2-subalgebra in the Langlands dual Lie algebra m ∨ on the nilpotent radical u∨p of the Langlands dual parabolic. For M as above and using the operators FP,Q,ψ we define a Schwartz space S(G,M). This space contains the space Cc(G/[P, P ]) of locally constant compactly supported functions on G/[P, P ] for every P for which M is a Levi component (but doesn’t depend on P ). We compute the space of spherical vectors in S(G,M) and study its global analogue. Finally we apply the above results in order to give an alternative treatment of automorphic L-functions associated with standard representations of classical groups (thus reproving the results of [10] using the same method as [9]).