Normalized intertwining operators and nilpotent elements in the Langlands dual group

Let $F$ be a local non-archimedean field and $\mathbf G$ be a split reductive group over $F$ whose derived group is simply connected. Set $G=\mathbf G(F)$. Let also $\psi\colon F\to\mathbb C^\times$ be a nontrivial additive character of $F$. For two parabolic subgroups $P$, $Q$ in $G$ with the same Levi component $M$, we construct an explicit unitary isomorphism $\mathcal F_{P,Q,\psi}\colon L^2(G/[P,P])\overset\sim\to L^2(G/[Q,Q])$ commuting with the natural actions of the group $G\times M/[M,M]$ on both sides. In some special cases, $\mathcal F_{P,Q,\psi}$ is the standard Fourier transform. The crucial ingredient in the definition is the action of the principal $\mathfrak{sl}_2$-subalgebra in the Langlands dual Lie algebra $\mathfrak m^\vee$ on the nilpotent radical a $\mathfrak u_\mathfrak p^\vee$ of the Langlands dual parabolic.
For $M$ as above, we use the operators $\mathcal F_{P,Q,\psi}$ to define a Schwartz space $S(G,M)$. This space contains the space $C_c{(G/[P,P])}$ of locally constant compactly supported functions on $G/[P,P]$ for every $P$ for which $M$ is a Levi component (but does not 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.