Discrete-spectrum base-change conjecture for complex symmetric spaces

Let GG be a real reductive group and let π\pi be an irreducible unitary representation of G(C)G(\mathbb{C}). Discrete-spectrum base-change conjecture. There is at most one pure inner form GG' of GG over R\mathbb{R}, necessarily the quasi-split inner form, for which π\pi occurs in the discrete spectrum of L2(G(R)\G(C))L^2(G'(\mathbb{R})\backslash G(\mathbb{C})). If it occurs, it has multiplicity one and does not occur in the continuous spectrum for any inner form. The discrete spectrum for G(R)G(\mathbb{R}) is the base change of the discrete spectrum for GopG^{\rm op}; in particular it is tempered and is nonzero exactly when the discrete spectrum for GopG^{\rm op} is nonzero, with the final comparison stated in the source when GopG^{\rm op} is not an inner form of GG. This conjecture concerns the spectral decomposition of complex symmetric spaces and is reported as largely known in parts, but no complete resolution is supplied here.

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Primary source

Dipendra Prasad, “A `relative' local Langlands correspondence”, arXiv:1512.04347 (2015).

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