The compact picture of the SL2(R) principal series

By [F2], write each $g\in G$ uniquely as $g=n_x{a}_{t}k$. Since $n_x{a}_t=a_t{n}_{e^{-t}x}$, this is also the unique smooth factorization $g=a_t{n}_{e^{-t}x}k$ with $a_t{n}_{e^{-t}x}\in AN$. Thus restriction to $K$ is injective on the induced function model, because covariance determines $\phi$ on every $g=pk$.

By [F2], write uniquely $g=k{a}_t{n}_x$ and set $\rho (g)=e^{-t}$. For $p_0=m{a}_s{n}_y$, direct multiplication using centrality of $M$ gives $g{p}_0=km{a}_{t+s}{n}_{e^{-s}x+y}$; hence $\rho (g{p}_0)=e^{-s}\rho (g)={\mathrm{\Delta }}_P(p_0){\mathrm{\Delta }}_G(p_0{)}^{-1}\rho (g)$ by [F1], [F2], and [F6]. Thus $\rho$ is a positive continuous rho-function by [F5], and $\rho (k)=1$ for $k\in K$. The continuous map $q:K\to G\mathrm{/}P$, $q(k)=kP$, is onto because $G=KAN$; therefore $G\mathrm{/}P$ is compact by [F7] and [F8]. Define $\mathrm{\Lambda }(\psi )={\int }_K\psi (kP)dk$ on $C_c(G\mathrm{/}P;\mathbb{R})$. This is positive and $\mathrm{\Lambda }(1)=1$, so [F9] represents it by a Radon probability $\overline{dk}$, the pushforward of $dk$. Give $P=M\times AN$ left Haar measure $dp$ by normalized counting on $M$ and $dtdx$ on $AN$; left multiplication by $a_s{n}_y$ sends $(t,x)$ to $(s+t,x+e^{-t}y)$ with Jacobian $1$, and $M$ is central. The Radon property follows from [F17] and finite disjoint union over $M$. For $h\in C_c(G)$, subgroup averaging [F7] makes $kP\mapsto {\int }_{P}h(kp)\rho (kp{)}^{-1}dp$ a member of $C_c(G\mathrm{/}P)$. The Weil formula [F10], applied to $h\mathrm{/}\rho$, gives ${\int }_{G}h(g)dg={\int }_{G\mathrm{/}P}{\int }_{P}h(kp)\rho (kp{)}^{-1}dpd{\mu }_{\rho }(kP)$. With $\overline{dk}$ instead, the right side is ${\int }_K\frac{1}{2}{\sum }_{m\in M}{\int }_{{\mathbb{R}}^2}h(km{a}_s{n}_y)e^{s}dydsdk={\int }_K{\int }_{{\mathbb{R}}^2}h(k{a}_s{n}_y)e^{s}dydsdk$, since $dk$ is right-$M$-invariant by [F16]. This is the KAN Haar integral [F2], so uniqueness in [F10] gives $\overline{dk}={\mu }_{\rho }$.

Identify $K$ with the circle by $k_{\theta }\leftrightarrow [\theta \mathrm{/}(2\pi )]$. On continuous functions the projection $P_{\epsilon }f(k)=\frac{1}{2}(f(k)+(-1{)}^{\epsilon }f(-k))$ has norm at most $1$ and range $C_{\epsilon }(K)$. By [F14], trigonometric polynomials approximate each continuous function uniformly; applying $P_{\epsilon }$ gives parity trigonometric polynomials approximating each continuous parity function uniformly. By [F15], continuous functions are dense in $L^2(K)$; applying the same bounded projection, which is an $L^2$ contraction by [F16], shows continuous parity functions are dense in $L_{\epsilon }^2(K)$. Therefore parity trigonometric polynomials, which are $K$-finite by [F13], are dense in $L_{\epsilon }^2(K)$.

If $\phi \in I_{\epsilon ,\nu }$, its $M$-covariance gives $f(-k)=(-1{)}^{\epsilon }f(k)$ for $f=\phi {\mathrm{\mid }}_K$. Conversely, for $f\in C_{\epsilon }^{\mathrm{\infty }}(K)$ define $\phi (a_t{n}_{x}k)=e^{(1+\nu )t\mathrm{/}2}f(k)$ in the unique $ANK$ coordinates. Left multiplication by $a_s{n}_y$ changes $t$ to $s+t$ and the $K$ factor remains fixed; left multiplication by $m\in M$ replaces $k$ by $mk$ and multiplies $f$ by ${\sigma }_{\epsilon }(m)$. Thus $\phi (pg)={\chi }_{\epsilon ,\nu }(p)\phi (g)$ and $\phi$ is smooth. For any factorization $g=pk$ with $p=m{a}_t{n}_x$, centrality of $m$ gives the same canonical factorization $g=a_t{n}_x(mk)$, so $e^{(1+\nu )t\mathrm{/}2}{\sigma }_{\epsilon }(m)f(k)=e^{(1+\nu )t\mathrm{/}2}f(mk)$; this proves the inverse formula and shows its independence of the $M$-ambiguity.

For $\nu \in i\mathbb{R}$, define $U\phi (x)=\rho (x{)}^{-1\mathrm{/}2}\phi (x^{-1})$. By [F5] and [F6], $\rho (xp{)}^{-1\mathrm{/}2}={\mathrm{\Delta }}_P(p{)}^{-1\mathrm{/}2}\rho (x{)}^{-1\mathrm{/}2}$, since ${\mathrm{\Delta }}_G=1$. Also ${\chi }_{\epsilon ,\nu }={\delta }_P^{1\mathrm{/}2}{\tau }_{\nu }={\mathrm{\Delta }}_P^{-1\mathrm{/}2}{\tau }_{\nu }$, so $\phi (p^{-1}{x}^{-1})={\chi }_{\epsilon ,\nu }(p{)}^{-1}\phi (x^{-1})$ cancels the modular factor and gives $U\phi (xp)={\tau }_{\nu }(p{)}^{-1}U\phi (x)$. The resulting continuous section has compact quotient support because $G\mathrm{/}P$ is compact by step 1.2. For $g_0\in G$, direct substitution yields $U({\mathrm{\Pi }}_{\nu }(g_0)\phi )(x)={\left(\frac{\rho (g_0^{-1}x)}{\rho (x)}\right)}^{1\mathrm{/}2}U\phi (g_0^{-1}x)=D_{g_0}(xP{)}^{1\mathrm{/}2}U\phi (g_0^{-1}x)$ by [F12]. Hence $U$ intertwines right translation on the left-covariant model with the cocycle-corrected left action on the right-$P$-covariant model of ${\mathrm{Ind}}_P^G({\tau }_{\nu })$; the parameter remains $\nu$.

The quotient norm of $U\phi$ is, by [F10] and step 1.2, $\mathrm{\parallel }U\phi {\mathrm{\parallel }}^2={\int }_{G\mathrm{/}P}\mathrm{\mid }U\phi (x){\mathrm{\mid }}^{2}d{\mu }_{\rho }(xP)={\int }_K\mathrm{\mid }U\phi (k){\mathrm{\mid }}^{2}dk={\int }_K\mathrm{\mid }\phi (k^{-1}){\mathrm{\mid }}^{2}dk$. Inversion invariance of normalized Haar measure [F16] makes this ${\int }_K\mathrm{\mid }f(k){\mathrm{\mid }}^{2}dk$, so restriction is isometric for the normalized induced inner product.

For $g_0\in G$, factor $k{g}_0=a_{t(k,g_0)}{n}_{x(k,g_0)}\kappa (k,g_0)$ by the canonical $ANK$ coordinates. Then $({\mathrm{\Pi }}_{\nu }(g_0)\phi )(k)=\phi (k{g}_0)=e^{(1+\nu )t(k,g_0)\mathrm{/}2}f(\kappa (k,g_0))$. In a general $PK$ factorization the same value is $\mathrm{\mid }\alpha (p){\mathrm{\mid }}^{1+\nu }{\sigma }_{\epsilon }(m_p)f(\kappa )$; the parity relation makes this independent of the $M$-choice. Since $-I$ is central, replacing $k$ by $-k$ leaves $t,x$ fixed and replaces $\kappa$ by $-\kappa$, so the formula preserves the parity-$\epsilon$ subspace. This is the stated Iwasawa-cocycle action, with $m_p=I$ in the canonical $ANK$ coordinates.

The coordinate construction in step 2.1, together with the inverse of $U$ from step 2.2, identifies continuous right-$P$-covariant sections with continuous parity-$\epsilon$ functions on $K$. Step 2.3 identifies their quotient norm with the $L^2(K)$ norm, and step 1.3 shows the smooth parity functions are dense in $L_{\epsilon }^2(K)$; thus the compact-picture isometry extends onto the completed induced Hilbert space. By [F11], ${\mathrm{Ind}}_P^G({\tau }_{\nu })$ is strongly continuous and unitary because ${\tau }_{\nu }$ is a continuous unitary character for $\nu \in i\mathbb{R}$. The intertwining identity in step 2.2 transfers these properties to the compact-picture action.