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Normalized intertwining operators and nilpotent elements in the Langlands dual group. (English) Zbl 1022.22015
Let $$\mathbb G$$ be a split reductive group over a nonarchimedean local field $$F$$ whose derived group is simply connected, and let $$\psi: F \to \mathbb C^\times$$ be a nontrivial additive character. Given parabolic subgroups $$P$$ and $$Q$$ of $$G= \mathbb G(\mathbb R)$$ with the same Levi component $$M$$, the authors construct an explicit unitary isomorphism $$\mathcal F_{P,Q,\psi}: L^2 (G/[P,P]) \to L^2 (G/[Q,Q])$$ commuting with the natural actions of the group $$G\times M/[P,P]$$, which reduces to the Fourier transform in some special cases. The operators $$\mathcal F_{P,Q,\psi}$$ are used to define the Schwartz space $$\mathcal S (G,M)$$, which contains the space $$\mathcal C_2 (G/[Q,Q])$$ of locally constant compactly supported functions on $$G/[P,P]$$ for every $$P$$ with Levi component $$M$$. The authors calculate the space of spherical vectors in $$\mathcal S (G,M)$$ and discuss its global analogue. They also apply these results to give an alternative treatment of automorphic $$L$$-functions associated to standard representations of classical groups.

##### MSC:
 2.2e+51 Representations of Lie and linear algebraic groups over local fields 2.2e+56 Representations of Lie and linear algebraic groups over global fields and adèle rings
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