The spherical complementary series converge to the trivial representation

Put $R_{\nu }=A(\nu )\mathrm{/}{c}_0(\nu )$ on the smooth compact picture. By [F3]–[F4], $B_{\nu }(f,h)=⟨R_{\nu }f,h{⟩}_0$, $R_{\nu }{f}_0=f_0$, and $R_{\nu }{\mathrm{\Pi }}_{\nu }(g)={\mathrm{\Pi }}_{-\nu }(g)R_{\nu }$. Therefore ${\phi }_{\nu }(g)=⟨{\mathrm{\Pi }}_{-\nu }(g)f_0,f_0{⟩}_0$. Substituting the compact-picture action at parameter $-\nu$ and $f_0=1$ gives ${\phi }_{\nu }(g)={\int }_K\mathrm{\mid }\alpha (p(k,g)){\mathrm{\mid }}^{1-\nu }dk$, with the exponent $1-\nu$ coming from the intertwiner rather than the original $1+\nu$ action.

For $0<\nu <1$, the normalized intertwiner has $R_{\nu }{f}_n=a_n(\nu )f_n$, and the displayed product weights satisfy $a_n(-\nu )=a_n(\nu {)}^{-1}$ for every even $n$. Consequently, on finite Fourier sums, $B_{-\nu }(R_{\nu }f,R_{\nu }h)=B_{\nu }(f,h)$. The finite Fourier sums are dense in both Hilbert completions by [F5]; the inverse $R_{-\nu }$ from [F8] gives surjectivity. Thus $R_{\nu }$ extends to a unitary operator between the completions. The smooth intertwining identity [F4] extends to the completions by density, since both representation operators and the extended intertwiner are bounded. Hence the two unitary representations are equivalent, so their classes for parameters $\nu$ and $-\nu$ agree. Thus parameter-sign equivalence reduces the limit as $\mathrm{\mid }\nu \mathrm{\mid }\uparrow 1$ to the positive-parameter limit.

Let $Q\subseteq G$ be compact. If $Q=\mathrm{\varnothing }$ the assertion is vacuous, so assume $Q\mathrm{\ne }\mathrm{\varnothing }$ and set $d(k,g)=\mathrm{\mid }\alpha (p(k,g))\mathrm{\mid }$. By [F1], $d$ is positive and continuous on $K\times Q$. The Iwasawa data make $K$ a circle, hence compact; [F9] makes $K\times Q$ compact. For each $(k,g)$ continuity gives a neighborhood on which $d>d(k,g)\mathrm{/}2>0$ and a neighborhood on which $d<d(k,g)+1$. Finite subcovers of these two covers give constants $0<m\le d\le M<\mathrm{\infty }$ on $K\times Q$. Thus $\mathrm{\mid }\log d\mathrm{\mid }\le C:=\max (\mathrm{\mid }\log m\mathrm{\mid },\mathrm{\mid }\log M\mathrm{\mid })$ there. For $0<\nu <1$ and $\mathrm{\mid }x\mathrm{\mid }\le C$, the function $t\mapsto e^{(1-\nu )t}$ is continuously differentiable on the closed interval with endpoints $0,x$ by [F10]. If $x=0$ its increment is zero; otherwise the mean value theorem in [F10] and the derivative bound $(1-\nu )e^{(1-\nu )t}\le (1-\nu )e^C$ on that interval give $\mathrm{\mid }{e}^{(1-\nu )x}-1\mathrm{\mid }\le (1-\nu )C{e}^C$. This gives ${\sup }_{g\in Q}\mathrm{\mid }{\phi }_{\nu }(g)-1\mathrm{\mid }\le (1-\nu )C{e}^C\to 0$ as $\nu \uparrow 1$, using that $dk$ is normalized Haar probability.

A basic Fell neighborhood of the trivial class tests a finite list of its diagonal coefficients on a compact set $Q$, each to a common tolerance $ϵ>0$. Each coefficient is a constant $c_i\ge 0$; if the list is empty or all $c_i=0$, the zero approximants suffice. Otherwise put $C={\max }_i{c}_i>0$. For each $c_i>0$, the vector $\sqrt{c_i}{f}_0$ in $I_{0,\nu }$ gives the diagonal coefficient $c_i{\phi }_{\nu }$; for $c_i=0$, use the zero vector. By step 2.1, choose $\nu$ sufficiently close to $1$ that ${\sup }_{g\in Q}\mathrm{\mid }{\phi }_{\nu }(g)-1\mathrm{\mid }<ϵ\mathrm{/}C$. Then every tested coefficient is within $ϵ$ of its constant on $Q$. Hence every basic Fell neighborhood of the trivial class contains $[I_{0,\nu }]$ for all sufficiently large $\nu <1$, proving $[I_{0,\nu }]\to [1_G]$.

Let $0<{\nu }_j<1$ increase to $1$, and form the strongly continuous Hilbert direct sum $\mathrm{\Pi }={\widehat{⨁}}_j{\mathrm{\Pi }}_{{\nu }_j}$. If ${\xi }_j$ is the unit vector $f_0$ in its $j$th summand, then ${\sup }_{g\in Q}\mathrm{\parallel }\mathrm{\Pi }(g){\xi }_j-{\xi }_j{\mathrm{\parallel }}^2={\sup }_{g\in Q}2(1-\mathrm{Re}{\phi }_{{\nu }_j}(g))\to 0$ for each compact $Q$ by step 2.1. Hence $\mathrm{\Pi }$ has almost invariant unit vectors and $1_G\prec \mathrm{\Pi }$ by [F7]. Each fixed summand has no nonzero invariant vector: it is irreducible by [F5] and has infinitely many K-types by [F2], so it is not the one-dimensional trivial representation. Since the direct sum acts componentwise, it too has no nonzero invariant vector. This is a sequence-level weak-containment conclusion; no such assertion is made for an individual fixed $I_{0,\nu }$.

AC is already declared and used for normalized Haar measure, the unitary completions, and the direct-sum weak-containment suppliers; the coefficient limits and the two displayed estimates are choice-free.