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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.

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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