Smooth and K-finite vectors are dense and stable under the derived action

A local smooth probability kernel exists in every identity neighborhood $U$. In the matrix chart $\mathrm{\Psi }(a,b,c)=\left(\begin{array}{cc}1+a& b\\ c& (1+bc)\mathrm{/}(1+a)\end{array}\right)$ around $I$, choose $r>0$ with $\mathrm{\Psi }({\bar{B}}_r)\subset U$ and ${\bar{B}}_r\subset (-1,1{)}^3$. Pull back the function $b_r(x)=\exp (-1\mathrm{/}(r^2-\mathrm{\parallel }x{\mathrm{\parallel }}^2))$ for $\mathrm{\parallel }x\mathrm{\parallel }<r$, extended by zero for $\mathrm{\parallel }x\mathrm{\parallel }\ge r$, and extend it by zero off the chart. This gives a nonnegative smooth compactly supported function $q_U$ in $U$, positive on a nonempty open set. By [F2], $0<{\int }_G{q}_{U}dg<\mathrm{\infty }$; hence $f_U=q_U\mathrm{/}{\int }_G{q}_{U}dg$ has $\int f_U=1$.

For smooth compactly supported $f$, put $f_h(x)=f(h^{-1}x)$. Left invariance and the pairing formula [F1] give $\pi (h)\pi (f)=\pi (f_h)$. Near any fixed $h_0$, choose a relatively compact coordinate neighborhood; the supports of $f_h$ there lie in one compact set $C$, and all parameter derivatives of $f(h^{-1}x)$ are jointly continuous on the product of a compact neighborhood with $C$. Uniform continuity of each derivative shows, by the difference-quotient definition and induction on its order, that $h\mapsto f_h$ is $C^{\mathrm{\infty }}$ in $L^1$ norm (the $L^1$ error is bounded by the uniform error on $C$ times its finite Haar measure). The bound in [F1] makes $h\mapsto \pi (f_h)v$ norm-$C^{\mathrm{\infty }}$. Thus $\pi (f)v\in H^{\mathrm{\infty }}$.

Every irreducible unitary representation $\sigma$ of $K=\mathrm{S}\mathrm{O}(2)$ is one dimensional. By [F4] it is finite dimensional; since $K$ is abelian, each $\sigma (k)$ is a self-intertwiner and hence scalar by Schur's lemma. Irreducibility then forces dimension one. Writing $\chi (k_{\theta })=q(\theta )$ for this character, [F5] makes $q$ differentiable. Its homomorphism law gives $q^{\mathrm{\prime }}(\theta )=q(\theta )q^{\mathrm{\prime }}(0)$; since $\mathrm{\mid }q(\theta )\mathrm{\mid }=1$, write $q^{\mathrm{\prime }}(0)=ia$ with $a\in \mathbb{R}$, and solve to get $q(\theta )=e^{ia\theta }$. The period $k_{\theta +2\pi }=k_{\theta }$ forces $e^{2\pi ia}=1$, so $a\in \mathbb{Z}$. Conversely each ${\sigma }_n(k_{\theta })=e^{in\theta }$ is an irreducible unitary character. Thus these are exactly the irreducibles of $K$.

For each $n$, a nonzero vector in $H_n$ spans a copy of ${\sigma }_n$, while every ${\sigma }_n$-copy lies in $H_n$; since $H_n$ is closed, it is exactly the ${\sigma }_n$-isotypic subspace. By [F3], $H={\widehat{⨁}}_{n\in \mathbb{Z}}{H}_n$, with orthogonal projections $P_n$ onto $H_n$.

Given $v\in H$ and $ϵ>0$, strong continuity supplies an identity neighborhood $U$ with $\mathrm{\parallel }\pi (g)v-v\mathrm{\parallel }<ϵ$ for all $g\in U$. Choose $f_U$ from step 1.1. The pairing formula [F1] and $f_U\ge 0$, $\int f_U=1$ yield $\mathrm{\mid }⟨\pi (f_U)v-v,w⟩\mathrm{\mid }\le ϵ\mathrm{\parallel }w\mathrm{\parallel }$ for every $w\in H$, hence $\mathrm{\parallel }\pi (f_U)v-v\mathrm{\parallel }\le ϵ$. By step 1.2, $\pi (f_U)v\in H^{\mathrm{\infty }}$, so $H^{\mathrm{\infty }}$ is dense in $H$.

If $w\in H^{\mathrm{\infty }}$, then $\pi (g)P_{n}w={\int }_K\overline{{\chi }_n(k)}{F}_w(gk)dk$, where ${\chi }_n(k_{\theta })=e^{in\theta }$ and $dk$ is normalized Haar probability. On each compact coordinate neighborhood in $G$, every derivative in $g$ of the integrand is continuous and uniformly bounded over compact $K$. The Bochner norm inequality [F7] bounds the integral of a uniform difference-quotient error by that same error, so induction on derivative order lets every derivative pass through the integral. Hence $P_{n}w\in H^{\mathrm{\infty }}$, and step 2.1 gives $P_{n}w\in H_n$, so $P_{n}w\in V$. If also $w\in M$, let $Q$ be the orthogonal projection onto $M^{\perp }$. It is bounded, and $Q\pi (k)w=0$ for every $k\in K$; bounded maps commute with the Bochner integral [F8], so $Q{P}_{n}w=0$ and $P_{n}w\in \ker Q=M$.

By step 2.2, $H^{\mathrm{\infty }}$ is dense in $H$; for $w\in H^{\mathrm{\infty }}$, the finite partial sums of ${\sum }_n{P}_{n}w$ converge to $w$ by step 2.1, and each term is in $V$ by step 3.1. Thus $V$ is dense in $H$. For $v\in V$, its finite-dimensional $K$-orbit span is a unitary representation of $K$; applying the compact-group decomposition and step 1.3 shows that it has only finitely many weights $n$. Therefore $V={⨁}_{n\in \mathbb{Z}}{V}_n$.

Let $M\mathrm{\ne }\{0\}$ be closed and $\pi (G)$-invariant. Choose $v\in M$ nonzero and an identity neighborhood $U$ with ${\sup }_{g\in U}\mathrm{\parallel }\pi (g)v-v\mathrm{\parallel }<\mathrm{\parallel }v\mathrm{\parallel }\mathrm{/}2$; use step 1.1 to choose $f_U$. Then $w=\pi (f_U)v$ lies in $H^{\mathrm{\infty }}$ by step 1.2 and satisfies $\mathrm{\parallel }w-v\mathrm{\parallel }<\mathrm{\parallel }v\mathrm{\parallel }\mathrm{/}2$ by the pairing estimate of step 2.2, so $w\mathrm{\ne }0$. For every $z\in M^{\perp }$, $⟨w,z⟩={\int }_G{f}_U(g)⟨\pi (g)v,z⟩dg=0$, since $\pi (g)v\in M$; hence $w\in (M^{\perp }{)}^{\perp }=M$ by [F6]. Since $H={\widehat{⨁}}_n{H}_n$, some $P_{n}w\mathrm{\ne }0$, and step 3.1 gives $P_{n}w\in M\cap V$.

The definition Smooth and K-finite vectors for SL2(R), and the (g,K)-module already proves that $H^{\mathrm{\infty }}$ is stable under every $L_X$ and is a $U({\mathfrak{g}}_{\mathbb{C}})$-module. Within $V$, stability under $K$ follows because right translation of a smooth orbit map is smooth and the $K$-orbit span of $\pi (k)v$ equals that of $v$. It is stable under $L_X$ as well. For $k\in K$, conjugation gives $\pi (k)L_{X}v=L_{\mathrm{Ad}(k)X}\pi (k)v$. If $v\in V$, choose finite bases of ${\mathfrak{g}}_{\mathbb{C}}$ and of $\mathrm{span}\pi (K)v$; the right side lies in the span of the finitely many vectors $L_{X_i}{w}_j$, so $L_{X}v$ is $K$-finite, and it is smooth by the same definition supplier. Thus $V$ is a ${\mathfrak{g}}_{\mathbb{C}}$- and $K$-stable algebraic direct sum of the $V_n$, and the enveloping-algebra action restricts to it. In particular the algebra element $XY$ acts as $L_X{L}_Y$.

Steps 1.2, 2.1, 2.2, 3.1, 4.1, 4.2, and 5.1 establish claims 1–4, including the nonzero-subspace qualification and the zero-Hilbert-space case stated above. No endpoint or parameter case occurs in this lemma.