Discrete automorphic spectrum

Let $G$ be a connected reductive group over a global field $F$ . After fixing a unitary central character or removing the split-center direction, the discrete automorphic spectrum is the maximal closed $G(\mathbb A_F)$ -subrepresentation of the automorphic $L^2$ -space that is a Hilbert direct sum of irreducibles:

L^2_{\mathrm{disc}} \cong \widehat{\bigoplus}_{\pi} m_{\mathrm{disc}}(\pi)\,\pi.

The integers $m_{\mathrm{disc}}(\pi)$ are the discrete automorphic multiplicities.

The ambient quotient

One common convention uses $G(F)\backslash G(\mathbb A_F)^1$ , where

G(\mathbb A_F)^1 = \bigcap_{\chi\in X_F^*(G)} \ker |\chi|_{\mathbb A}.

Another fixes a unitary central character and works modulo an appropriate central subgroup. The convention must be stated because $G(F)\backslash G(\mathbb A_F)$ can have infinite volume in split central directions.

Cuspidal and residual parts

There is an orthogonal decomposition

L^2_{\mathrm{disc}} = L^2_{\mathrm{cusp}} \oplus L^2_{\mathrm{res}},

where the first term is generated by cuspidal automorphic representations and the second is the residual spectrum. The residual part is built from poles and residues of Eisenstein series attached to cuspidal data on proper Levi subgroups.

Discrete does not mean local discrete series

“Discrete automorphic” describes occurrence in a global $L^2$ -quotient. It does not require each local component to be a discrete-series representation. Likewise, it is unrelated to discreteness of the adelic group itself.

Classification problem

A global parameter is expected to determine a packet of discrete automorphic representations, while an Arthur multiplicity formula selects the members and multiplicities that actually occur. This makes the discrete spectrum a central testing ground for endoscopy and functoriality.

References

James Arthur, “A trace formula for reductive groups I: Terms associated to classes in $G(\mathbb Q)$ ,” Duke Mathematical Journal 45 (1978), 911–952. DOI.
C. Mœglin and J.-L. Waldspurger, Spectral Decomposition and Eisenstein Series, Cambridge University Press, 1995. DOI.