Let $G$ be a locally compact group and $\Gamma$ a lattice in $G$. Is it known whether the space $$ \mathrm{Hom}_G\left(\pi,L^2(\Gamma\backslash G)\right) $$ is finite dimensional for $\pi\in\widehat G$? This is true if $\Gamma$ is cocompact, but in general?

Here Hom$_G$ refers to $G$-equivariant, continuous linear maps.

  • $\begingroup$ What is $\pi$? A unitary representation of $G$? How do you define $\mathrm{Hom_G}$? $\endgroup$ – Adrián González-Pérez Nov 8 '18 at 9:44
  • $\begingroup$ @Adrian Gonzalez-Perez: $\pi$ is supposed to be an irreducible unitary representation of $G$ and $Hom$ refers to $G$-equivariant continuous linear maps. $\endgroup$ – Zero Nov 8 '18 at 10:01

For $\Gamma$ and $G$ to which Langlands SLN 544 applies, and to adele groups as in Moeglin-Waldspurger, I think we essentially know this finite multiplicity result, from the complete spectral decomposition of $L^2$.

First, we know it for the subspace of ($L^2$) cuspforms, by the compactness on that space of the integral operators coming from test functions on the adele group. (These are sometimes called convolution operators, but the asymmetry makes this potentially misleading terminology: the same operators act on any repn space, whether or not it's a space of functions on the group.) In principle, this part is well known.

The rest of the discrete spectrum consists of (multi-) residues of Eisenstein series attached to cuspidal data on Levi components of parabolics. For maximal proper parabolics, the only residues which play a role are in the right half-plane from (the general analogue of) the line $\Re(s)=1/2$, and there are finitely-many such residues for each (strong-sense) cuspidal datum, and they are all $L^2$. By thinking in terms of inducing in stages, the local representations can be identified.

For multi-residues of cuspidal-data Eisenstein series attached to non-maximal parabolics, the corresponding discussion becomes murkier, I think, though, in principle, SLN 544 treated it. Explicit determination of poles and residues (at this moment, late 2018) seems known only for $GLn$, by Moeglin-Waldspurger (conjectured by Jacquet), and $GSp4$.

But, vaguely/qualitatively, a similar result should hold, namely that there are at most finitely-many $L^2$ (multi-) residues attached to given cuspidal data. (Taking residues is a $G$-hom, and the local isomorphism classes can be estimated.)

So, maybe, "yes". :)


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