Decomposition into irreducibles of the representation $L^2(SL_2(\mathbb{C})/\Gamma)$ for $\Gamma$ geometrically finite

Question

I am trying to understand the decomposition $$L^2(SL_2(\mathbb{C})/\Gamma)=\oplus_i C_i \oplus V_{temp}$$ where $C_i$ are complementary series representations corresponding to eigenfunctions of the laplacian on $L^2(\mathbb{H}^3/\Gamma)$ and $V_{temp}$ a tempered representation.
I read that one can decompose it into a spherical and a non-spherical part. Why is the non-spherical part tempered ? How is the decomposition of the spherical part related to the spectral decomposition of $L^2(\mathbb{H}^3/\Gamma)$ with respect to $\Delta$ ?

Answer 1

Because only infinite dimensional, unitary representation of $SL(2,F)$ for a local field $F$ can fail to be tempered, if they are spherical. This follows from the classification. This is true for $GL(2)$ as well modulo twisting by one-dimensional representations, which are trivial for $SL(2)$. I think Wallach "Real reductive groups vol 1+2" and Knapp "Representation theory of semisimple groups" covers this for $\mathbb{R}$ and $\mathbb{C}$.

As Asaf points out, looking at the $K=SU(2)$-invariant vectors will do the job. Be careful, you could think taking representations of $SU(2)$ to get something non-spherical is sufficient, but you need to take care that they are not contained in the Restriction of a spherical one;)

Getting to the finite-volume setting: You will obtain Eisenstein series and cusp forms and constant functions. There you actually have non-tempered representations besides the trivial representation. There are atmost finitely many. For $\Gamma$ a congruence subgroup of an imaginary quadratic field $k$, it is an important conjecture that there are non (the analogue of the Selberg eigenvalue conjecture). Then you actually have a much bigger group acting, i.e., $SL_2(A_k)$ via strong approximation, i.e. there exists an open subgroup$K_\Gamma$ of $SL_2(A_{k,f})$ (finite adeles) $$ \Gamma \backslash SL_2(\mathbb{C}) = SL_2(k) \backslash SL_2(A_k) / K_\Gamma.$$

If you consider an irreducible representation in there, it factors into representations of $SL_2(k_v)$ for each place as a tensor product, all but finitely many are spherical. It is assumed that they should all be tempered (besides the one-dimensional representations). This is know as the Ramanujan Peterson conjecture. Non trivial bounds are known due to Blomer and Brumley (they actually work with $GL(2)$). Here is survey: Blomer + Brumley -The role of the Ramanujan conjecture in analytic number theory, Bulletin AMS 50 (2013), 267-320

Usually the decomposition is $V_{cusp} + V_{const} + V_{cont}$, where $V_{cusp}$ are the cuspidal representations, $V_{const}$ the one-dimensional representation and $V_{cont}$ is the continuous representation. $V_{cont}$ is known to be tempered and can be explicitly given. $V_{cusp}$ is fairly unkown and only very few are conjectured to be related to Galoisrepresentations, which would imply temperedness automatically.

Answer 2

You can look at M.Cowling Almost $L^2$ matrix coefficients and Lax The Asymptotic Distribution of Lattice Points in Euclidean and Non-Euclidean Spaces.

I know the case of $SL_2(\mathbb R)$, but I am not familar with $SL_2(\mathbb C)$. For $SL_2(\mathbb R)$, we have a decomposition $L^2(\Gamma\backslash\mathbb H^2)=\mathbb Cv_0\oplus\cdots\oplus\mathbb Cv_N\oplus H$, where $v_j$ are all the eigenfunctions of $\Delta$ with eigenvalue in $(-1/4,0]$. In the paper of Lax, we have an estimation for a functions $\xi$ in $H$ $$(\pi(x)\xi,\xi)_{L^2(\Gamma\backslash SL_2(\mathbb R))}\in L^{2+\epsilon}(SL_2(\mathbb R)) $$
where $\pi$ is the right regular representation in $L^2(\Gamma\backslash SL_2(\mathbb R)$, $\xi$ regarded as a right $K$-invariant function. Then the theorem 1 in the paper of Cowling tells us the represention space generated by $\xi$ is weakly contained in the regular representaion of $L^2(SL_2(\mathbb R))$, so it is a tempered representation.

Now suppose $L^2(\Gamma\backslash SL_2(\mathbb R))=V_{v_0}\oplus\cdots\oplus V_{v_N}\oplus V_{temp}$. Using the argument above and Zorn's lemma, we can find a subspace of $V_{temp}$ which is tempered, and the rest space has no nonzero right $K$-invariant element. In the case of $SL_2(\mathbb R)$ we have to consider discret series, but in $SL_2(\mathbb C)$ we have done.