  Let $F$ be a local non-archimedian field and let $G$ be a group of points of
a split reductive group over $F$. For a parabolic subgroup $P$ of $G$ we set
$X_P=G/[P,P]$. For any two parabolics $P$ and $Q$ with the same Levi component
$M$ we construct an explicit unitary isomorphism $L^2(X_P)\to L^2(X_Q)$ (which
depends on a choice of an additive character of $F$). The formula for the above
isomorphism involves the action of the principal nilpotent element in the
Langlands dual group of $M$ on the unipotent radicals of the corresponding dual
parabolics. We use the above isomorphisms to define a new space $\calS(G,M)$ of
functions on $X_P$ (which depends only on $P$ and not on $M$). We explain how
this space may be applied in order to reformulate in a slightly more elegant
way the construction of $L$-functions associated with the standard
representation of a classical group due to Gelbart, Piatetski-Shapiro and
Rallis.
