The invariant pairing between opposite principal-series parameters

Given: AC, $\epsilon \in \{0,1\}$, $\nu \in \mathbb{C}$, and smooth compact-picture vectors $f,h$.

The compact-picture action is $({\mathrm{\Pi }}_{\lambda }(g)u)(k)=\mathrm{\mid }\alpha (p(k,g)){\mathrm{\mid }}^{1+\lambda }u(\kappa (k,g))$ for $u\in C_{\epsilon }^{\mathrm{\infty }}(K)$, where $kg=p(k,g)\kappa (k,g)$ is the canonical $AN\times K$ factorization. Here the $AN$ factor has trivial $M$-character (The compact picture of the SL2(R) principal series).

The $NAK$ coordinates are smooth global coordinates; using $a_t{n}_x=n_{e^{t}x}{a}_t$ converts them by a smooth coordinate change into the unique smooth $ANK$ coordinates (Iwasawa decomposition and Haar integration formula for SL2(R), Iwasawa and minimal-parabolic data for SL2(R)).

The model parameter and its normalized inducing character are fixed by The normalized principal series I(epsilon, nu).

The parity basis is $f_n(k_{\theta })=e^{in\theta }$ for $n\equiv \epsilon (\mathrm{m}\mathrm{o}\mathrm{d}2)$, and is orthonormal for normalized Haar measure on $K$ (K-type decomposition of the SL2(R) principal series).

Under $k_{\theta }\mapsto [\theta \mathrm{/}(2\pi )]\in \mathbb{R}\mathrm{/}\mathbb{Z}$, normalized Haar probability on $K$ is $dk=d\theta \mathrm{/}(2\pi )$: the torus definition gives this normalized translation-invariant probability, and uniqueness of normalized Haar probability identifies its pullback with $dk$ (Iwasawa and minimal-parabolic data for SL2(R), The one-dimensional torus and its normalized Haar integral, Normalized Haar measure on a compact Lie group).

An orientation-preserving diffeomorphism of the circle preserves the integral of a smooth top form; every smooth top form on compact $K$ has compact support (Change of variables on oriented manifolds).

The complex exponential obeys $e^{z+w}=e^z{e}^w$, $\mathrm{\mid }{e}^z\mathrm{\mid }=e^{\mathrm{Re}z}$, and $\overline{e^z}=e^{\bar{z}}$ ($\exp (z+w)=\exp z\exp w$, and the complex exponential extends the real exponential, $\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$).

The product of smooth functions is smooth by the coordinatewise product rule, and complex conjugation is the coordinate map $(x,y)\mapsto (x,-y)$ (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$, Real and imaginary parts, complex conjugation, and modulus).

The integral of a complex function is defined by integrating its real and imaginary parts and combining the two real integrals (Integrable real and complex functions, and their integrals).

A continuous real-valued function is bounded on compact $K$; since $dk(K)=1$, a bounded measurable function on $K$ is integrable by monotonicity (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, Monotonicity and nonnegative homogeneity of the nonnegative integral, Normalized Haar measure on a compact Lie group).

AC supplies normalized Haar probability on compact $K$ and implies the countable-choice hypothesis for the torus measure; no vector is selected in this proof (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), Normalized Haar measure on a compact Lie group).