Does anyone know any book or article proving that the unitary representation $\pi$ of $SL(2,\mathbb{R})$ into $L^2(SL(2,\mathbb{R}))$ has spectral gap? And what happens if we replace $SL(2,\mathbb{R})$ by $\Gamma\backslash SL(2,\mathbb{R})$ where $\Gamma$ is a lattice in $SL(2,\mathbb{R})$. Thank you.



Spectral gap of unitary representation
For your first question, $SL\_2(\mathbb{R})$ is non-amenable, so its regular representation has a spectral gap. (Why is it non-amenable? E.g because it contains the free group on 2 generators as a discrete subgroup. Or because its action on the projective line $P^1(\mathbb{R})$ has no invariant probability measure.)


For your second question, here is a paper by Bekka and Lubotzky that puts an end to a long story:


<http://perso.univ-rennes1.fr/bachir.bekka/Alex-Bachir-Final-v2-sep2009.pdf>


It proves that, for any lattice $\Gamma$ in the group $G(K)$ of $K$-rational points of a simple algebraic group $G
$ over a local field $K$, the representation of $G(K)$ on $L^2(G(K)/\Gamma)$ has a spectral gap.


