A limit of discrete series is not square-integrable

Given: AC, $G={\mathrm{S}\mathrm{L}}_2(\mathbb{R})$, its fixed left Haar measure, the odd compact-picture model $I_{1,0}$, and the holomorphic discrete-series models for $n\ge 2$.

The unit vector $f_1(k_{\theta })=e^{i\theta }$ lies in the closed limit summand $D_1^{+}$ of $I_{1,0}$, and $D_1^{+}$ is an irreducible strongly continuous unitary representation (The two limits of discrete series, Unitarity and irreducibility of the limits of discrete series).

Matrix coefficients are $c_{v,w}(g)=⟨\pi (g)v,w⟩$ with pairing linear in the first variable, and they are continuous for strongly continuous unitary representations (Matrix coefficient of a unitary representation).

In the compact picture, $c(g)=⟨{\mathrm{\Pi }}_0(g)f_1,f_1⟩$ satisfies $c(a_{\tau })=\mathrm{sech}(\tau \mathrm{/}2)$ for every $\tau \in \mathbb{R}$ (Matrix-coefficient formulas and decay for the discrete and principal series(b)).

For a continuous function with unitary-character left and right $K$-transformation, its modulus is $K$-bi-invariant, and the fixed Haar measure gives ${\int }_G\mathrm{\mid }c(g){\mathrm{\mid }}^{2}dg=2\pi {\int }_0^{\mathrm{\infty }}\mathrm{\mid }c(a_{\tau }){\mathrm{\mid }}^2\sinh \tau d\tau$ (KAK integration formula for K-bi-invariant functions on SL2(R)).

For each $n\ge 2$, the normalized extremal vector $u_n$ in a genuine discrete-series model has coefficient $⟨{\pi }_n(a_{\tau })u_n,u_n⟩=\cosh (\tau \mathrm{/}2{)}^{-n}$ (Matrix-coefficient formulas and decay for the discrete and principal series(a)).

The square-integrability condition used here requires every matrix coefficient of an irreducible unitary representation to lie in $L^2(G)$ (The limits of discrete series are not square-integrable).

If $0\le g_m\uparrow g$ pointwise, then $\int g_{m}d\mu \uparrow \int gd\mu$ (Monotone convergence for the integral).

AC is assumed and inherited through the normalized principal-series and fixed-Haar model interfaces (The Axiom of Choice).