K-type decomposition of the SL2(R) principal series

Since $k_{\theta +\pi }=-k_{\theta }$, $f_n(k_{\theta +\pi })=e^{in\pi }{f}_n(k_{\theta })=(-1{)}^n{f}_n(k_{\theta })$. This equals $(-1{)}^{\epsilon }{f}_n(k_{\theta })$ exactly when $n\equiv \epsilon (\mathrm{m}\mathrm{o}\mathrm{d}2)$, proving both directions of the parity criterion. By [F3], for matching parities the inner product is $⟨f_n,f_m⟩=\frac{1}{2\pi }{\int }_0^{2\pi }{e}^{i(n-m)\theta }d\theta =\{\begin{array}{ll}1,& n=m,\\ 0,& n\mathrm{\ne }m\mathrm{.}\end{array}$

The full Fourier basis in [F2], transported by [F3], is $\{f_n:n\in \mathbb{Z}\}$. If $f\in L_{\epsilon }^2(K)$, translation invariance of the coefficient integral by $1\mathrm{/}2$ in the torus coordinate gives $\widehat{f}(n)=(-1{)}^{\epsilon -n}\widehat{f}(n)$. Thus $\widehat{f}(n)=0$ when $n\equiv ̸\epsilon (\mathrm{m}\mathrm{o}\mathrm{d}2)$. The mean-square Fourier expansion of $f$ therefore uses only matching parity indices, proving the asserted orthonormal basis of $L_{\epsilon }^2(K)$.

For $k_{\varphi }\in K$, right translation gives $(k_{\varphi }\cdot f_n)(k_{\theta })=f_n(k_{\theta +\varphi })=e^{in\varphi }{f}_n(k_{\theta })$. The characters $\varphi \mapsto e^{in\varphi }$ are distinct for distinct integers $n$, so these are pairwise inequivalent one-dimensional $K$-types.

Let $W\subseteq L_{\epsilon }^2(K)$ be closed and $K$-invariant, and let $f\in W$. For $J\ge 0$, set $s_J={\sum }_{\mathrm{\mid }m\mathrm{\mid }\le J}\widehat{f}(m)f_m$, so $s_J\to f$ in $L^2$ by [F2]. For an integer $n$, $J\ge \mathrm{\mid }n\mathrm{\mid }$, and $L>J+\mathrm{\mid }n\mathrm{\mid }$, define $Q_{n,L}h=\frac{1}{L}{\sum }_{j=0}^{L-1}{e}^{-2\pi inj\mathrm{/}L}(k_{2\pi j\mathrm{/}L}\cdot h)$. Each $Q_{n,L}f$ belongs to $W$, and $\mathrm{\parallel }{Q}_{n,L}\mathrm{\parallel }\le 1$ because it is an average of unitary operators with coefficients of modulus one. On $s_J$, the finite geometric sum $\frac{1}{L}{\sum }_{j=0}^{L-1}{e}^{2\pi i(m-n)j\mathrm{/}L}$ is $1$ for $m=n$ and $0$ for all other $\mathrm{\mid }m\mathrm{\mid }\le J$, since then $0<\mathrm{\mid }m-n\mathrm{\mid }<L$. Hence $Q_{n,L}{s}_J=\widehat{f}(n)f_n$. Taking $J\to \mathrm{\infty }$ with $L=J+\mathrm{\mid }n\mathrm{\mid }+1$ gives $Q_{n,L}f\to \widehat{f}(n)f_n$; closedness of $W$ implies $f_n\in W$ whenever $\widehat{f}(n)\mathrm{\ne }0$. By the Fourier expansion from step 2.1, every $f\in W$ is the $L^2$ limit of finite sums of modes it contains. Therefore $W$ is their closed span.

A finite sum of the $f_n$ has a finite-dimensional right $K$-orbit span. Conversely, if $f$ is $K$-finite in the local sense stated above, its orbit span $V$ is $K$-invariant and closed by [F4]. Step 3.1 makes $V$ the closed span of the modes it contains. Since distinct $f_n$ are linearly independent by step 1.1, only finitely many can lie in $V$; hence $f$ is a finite sum of them. Thus the $K$-finite vectors are exactly the algebraic direct sum of the parity-matching lines. By step 2.1 their $K$-types each have multiplicity one, and that direct sum is dense in $L_{\epsilon }^2(K)$.