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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:
22E50 Representations of Lie and linear algebraic groups over local fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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