K-finite vectors detect nonzero closed invariant subspaces

Fix $g\in G$ and write $k_{\theta }g=a_{t(\theta )}{n}_{x(\theta )}{k}_{\psi (\theta )}$ in the unique smooth $ANK$ coordinates. In a local lift of the angle $\psi$, the bottom row is $v(\theta )=e^{-t(\theta )\mathrm{/}2}(-\sin \psi (\theta ),\cos \psi (\theta ))$. It is also $(-\sin \theta ,\cos \theta )g$, so $\det (v,v^{\mathrm{\prime }})=\det \text{\hspace{0.17em}⁣}\left(\begin{array}{cc}-\sin \theta & \cos \theta \\ -\cos \theta & -\sin \theta \end{array}\right)\det g=1$. The first expression gives $\det (v,v^{\mathrm{\prime }})=e^{-t}{\psi }^{\mathrm{\prime }}$, hence ${\psi }^{\mathrm{\prime }}=e^t=\mathrm{\mid }\alpha (p(k_{\theta },g)){\mathrm{\mid }}^2>0$. Thus $\kappa (\cdot ,g)$ is an orientation-preserving circle diffeomorphism; uniqueness of the $ANK$ factorization gives inverse $\kappa (\cdot ,g^{-1})$. By [F6], $dk=d\theta \mathrm{/}(2\pi )$, and the compactly supported top-form change of variables [F7] applies because $K$ is compact; it gives Haar Jacobian $d{k}^{\mathrm{\prime }}=\mathrm{\mid }\alpha (p(k,g)){\mathrm{\mid }}^{2}dk$.

For smooth $f$, [F1] and step 1.1 give $\mathrm{\parallel }{\mathrm{\Pi }}_{\nu }(g)f{\mathrm{\parallel }}_2^2={\int }_K\mathrm{\mid }\alpha (p(k,g)){\mathrm{\mid }}^{2+2\mathrm{Re}\nu }\mathrm{\mid }f(\kappa (k,g)){\mathrm{\mid }}^{2}dk={\int }_K\mathrm{\mid }\alpha (p({\kappa }^{-1}(k^{\mathrm{\prime }}),g)){\mathrm{\mid }}^{2\mathrm{Re}\nu }\mathrm{\mid }f(k^{\mathrm{\prime }}){\mathrm{\mid }}^{2}d{k}^{\mathrm{\prime }}\le C_g\mathrm{\parallel }f{\mathrm{\parallel }}_2^2$, where $C_g:={\max }_{k\in K}\mathrm{\mid }\alpha (p(k,g)){\mathrm{\mid }}^{2\mathrm{Re}\nu }<\mathrm{\infty }$ by [F8]; this is the same supremum because $\kappa (\cdot ,g)$ is a diffeomorphism. Thus each smooth operator extends uniquely to a bounded operator on $H$, since smooth Fourier sums are dense by [F3].

The bounded extensions satisfy the group law because the original smooth action does and smooth Fourier sums are dense by [F3]. The coefficient $\mathrm{\mid }\alpha (p(k,g)){\mathrm{\mid }}^{2\mathrm{Re}\nu }$ is jointly continuous in $(k,g)$; compactness of $K$ and a finite subcover near any fixed $g_0$ give a local uniform bound $M$ for the operator norms. For smooth $f$, [F1] gives a jointly smooth function ${\mathrm{\Pi }}_{\nu }(g)f(k)$; compactness of $K$ makes every parameter derivative continuous uniformly in $k$, so its orbit map is smooth into $H$. For arbitrary $h\in H$, approximate by a smooth Fourier sum $f$ and use $\mathrm{\parallel }{\mathrm{\Pi }}_{\nu }(g)h-{\mathrm{\Pi }}_{\nu }(g_0)h{\mathrm{\parallel }}_2\le M\mathrm{\parallel }h-f{\mathrm{\parallel }}_2+\mathrm{\parallel }{\mathrm{\Pi }}_{\nu }(g)f-{\mathrm{\Pi }}_{\nu }(g_0)f{\mathrm{\parallel }}_2+\mathrm{\parallel }{\mathrm{\Pi }}_{\nu }(g_0)(f-h){\mathrm{\parallel }}_2$ near $g_0$; this proves strong continuity. For $\nu \mathrm{\notin }i\mathbb{R}$ no unitarity of the $G$-action is asserted.

By [F1] and [F3], ${\mathrm{\Pi }}_{\nu }(k_{\varphi })f_n=e^{in\varphi }{f}_n$. Since these vectors are an orthonormal basis, the $K$-action extends to a unitary action on $H$; it is strongly continuous by step 3.1. Every $K$-finite vector is a finite Fourier sum by [F3]. Joint smoothness in [F1] and compactness of $K$ show that its orbit map is $C^{\mathrm{\infty }}$ as an $H$-valued map. The formulas in [F4] preserve finite Fourier sums, and the $K$-action does too; hence $I_{\epsilon ,\nu }^K$ is a $(\mathfrak{g},K)$-submodule of the smooth vectors.

For $n\equiv \epsilon (\mathrm{m}\mathrm{o}\mathrm{d}2)$, let ${\sigma }_n(k_{\varphi })=e^{in\varphi }$. By step 4.1 the representation of $K$ on $H$ is strongly continuous and unitary, so [F5] defines $P_n$. For each basis vector $f_m$, [F3] gives $P_n{f}_m=\left({\int }_K{e}^{i(m-n)\varphi }dk\right)f_m=⟨f_m,f_n⟩f_m={\delta }_{nm}{f}_m$. Boundedness of $P_n$ and the orthonormal-basis expansion in [F3] therefore give $P_{n}v=⟨v,f_n⟩f_n$ and $v={\sum }_{n\equiv \epsilon (2)}{P}_{n}v$ in Hilbert norm.

If $W$ is $K$-invariant and $v\in W$, every value $\overline{{\sigma }_n(k)}\mathrm{\Pi }(k)v$ in the integrand of [F5] belongs to $W$. The simple-function approximants to its Bochner integral are finite linear combinations of such values, and their norm limit lies in $W$ because $W$ is closed. Thus $P_{n}v\in W$ for every allowed $n$. If $W\mathrm{\ne }\{0\}$, take any $v\in W\setminus \{0\}$; the expansion in step 5.1 has a nonzero term, so $P_{n}v\mathrm{\ne }0$ for some $n$, and this vector is $K$-finite. For $W=\{0\}$ every projection is zero. This proves detection for closed $K$-invariant subspaces.

If $W$ is closed and $G$-invariant, then it is $K$-invariant, so step 6.1 proves $W\mathrm{\ne }\{0\}$ iff $W\cap I_{\epsilon ,\nu }^K\mathrm{\ne }\{0\}$, including $W=\{0\}$ where both sides are false. For $w\in W\cap I_{\epsilon ,\nu }^K$ and real $X\in \mathfrak{g}$, the difference quotients $({\mathrm{\Pi }}_{\nu }(\exp (tX))w-w)\mathrm{/}t$ lie in $W$; their limit $L_{X}w$ also lies in $W$ because $W$ is closed, and is $K$-finite by [F4]. The $K$-action preserves the intersection, so it is a $(\mathfrak{g},K)$-submodule.