The spherical complementary series destroys property (T) for SL2(R)

For $0<\nu <1$, the completion of $I_{0,\nu }$ in the normalized complementary-series form is an irreducible strongly continuous unitary representation, and $f_0=1$ has norm one (Unitarity of the complementary series).

The smooth even Fourier vectors $f_{2j}$ have pairwise distinct right $K$-characters $e^{2ij\varphi }$. For $0<\nu <1$, the normalized complementary form has $B_{\nu }(f_{2j},f_{2j})=a_{2j}(\nu )>0$, so these nonzero $K$-lines survive in its weighted Hilbert completion; no ordinary-$L^2$ norm is transferred (K-type decomposition of the SL2(R) principal series, Unitarity of the complementary series).

The normalized spherical coefficient ${\phi }_{\nu }(g)=⟨{\mathrm{\Pi }}_{\nu }(g)f_0,f_0⟩$ tends to $1$ uniformly on compact sets as $\nu \uparrow 1$ (The spherical complementary series converge to the trivial representation).

The classes $[I_{0,\nu }]$ converge to the trivial class in the Fell topology as $\nu \uparrow 1$ (The spherical complementary series converge to the trivial representation, The Fell topology on the unitary dual).

For the unit vector $f_0$ in a unitary representation, $\mathrm{\parallel }{\mathrm{\Pi }}_{\nu }(g)f_0-f_0{\mathrm{\parallel }}^2=2(1-\mathrm{Re}{\phi }_{\nu }(g))$ by expansion of the squared norm.

The A-page proposition proves directly that $G$ fails property (T) by one direct-sum representation built from a cofinal sequence of complementary-series parameters (SL2(R) does not have property (T), Kazhdan's property (T)).

A unit vector is $(Q,\epsilon )$-invariant when its displacement is strictly less than $\epsilon$ at every point of $Q$ (Almost invariant vectors for a unitary representation).

AC is the principle that every family of nonempty sets has a choice function (The Axiom of Choice); it is assumed by the in-run representation suppliers.