The normalized principal series I(epsilon, nu)

Assume the Axiom of Choice (The Axiom of Choice) and use the data of Iwasawa and minimal-parabolic data for SL2(R). For $\epsilon \in \{0,1\}$ and $\nu \in \mathbb{C}$, extend ${\sigma }_{\epsilon }(m)=(\pm 1{)}^{\epsilon }$ on $m=\pm I$ and $e^{\nu }(a_t)=e^{\nu t\mathrm{/}2}$ to characters of $P=MAN$ that are trivial on the other factors. The normalized inducing character is ${\chi }_{\epsilon ,\nu }(p)={\delta }_P(p{)}^{1\mathrm{/}2}{\sigma }_{\epsilon }(m_p)e^{\nu }(p)=\mathrm{\mid }\alpha (p){\mathrm{\mid }}^{1+\nu }{\sigma }_{\epsilon }(m_p),p=m_p{a}_t{n}_x,$ where $\mathrm{\mid }\alpha (m_p{a}_t{n}_x)\mathrm{\mid }=e^{t\mathrm{/}2}$ and ${\delta }_P$ is the parabolic modular character of the preceding definition.

The normalized smooth principal series $I_{\epsilon ,\nu }$ is the space of smooth functions $\phi :G\to \mathbb{C}$ satisfying $\phi (pg)={\chi }_{\epsilon ,\nu }(p)\phi (g)(p\in P,g\in G),$ with the right-translation action $({\mathrm{\Pi }}_{\nu }(g_0)\phi )(g)=\phi (g{g}_0)\mathrm{.}$ It preserves covariance since $({\mathrm{\Pi }}_{\nu }(g_0)\phi )(pg)=\phi (pg{g}_0)={\chi }_{\epsilon ,\nu }(p)\phi (g{g}_0)$, and $({\mathrm{\Pi }}_{\nu }(g_1){\mathrm{\Pi }}_{\nu }(g_2)\phi )(g)=\phi (g{g}_1{g}_2)=({\mathrm{\Pi }}_{\nu }(g_1{g}_2)\phi )(g)$. The inducing character ${\sigma }_{\epsilon }{e}^{\nu }$ is unitary exactly when $\nu \in i\mathbb{R}$, since $\mathrm{\mid }{\sigma }_{\epsilon }(m)e^{\nu }(a_t)\mathrm{\mid }=e^{(\mathrm{Re}\nu )t\mathrm{/}2}$ for all $t\in \mathbb{R}$; ${\delta }_P^{1\mathrm{/}2}$ is the separate half-modular normalization.

The $K$-finite subspace $I_{\epsilon ,\nu }^K$ consists of those $\phi$ for which the right $K$-translates span a finite-dimensional space. It is the associated $({\mathfrak{s}\mathfrak{l}}_2(\mathbb{C}),K)$-module, but need not be stable under the full noncompact group $G$; the ambient smooth space above carries that action. This is Kerr's distinction between the smooth globalization and its Harish-Chandra module. Under inversion $F(g)=\phi (g^{-1})$, write $B=P$ and let $t(b)$ be the signed top-left diagonal entry of $b\in B$. Then $F(gb)=\phi (b^{-1}{g}^{-1})={\chi }_{\epsilon ,\nu }(b^{-1})F(g)=\mathrm{\mid }t(b){\mathrm{\mid }}^{-1-\nu }\mathrm{sgn}(t(b){)}^{\epsilon }F(g)\mathrm{.}$ The left action $(g_0\cdot F)(g)=F(g_0^{-1}g)$ corresponds under inversion to ${\mathrm{\Pi }}_{\nu }(g_0)$, so this is Etingof's $V^{\epsilon }(s)$ model with $s=-\nu$.

The parity-$\epsilon$ parameter lattice used below is ${\mathcal{W}}_{\epsilon }=\{\nu \in \mathbb{Z}:\nu \equiv \epsilon +1(\mathrm{m}\mathrm{o}\mathrm{d}2)\},$ so ${\mathcal{W}}_0$ is the odd integers and ${\mathcal{W}}_1$ is the even integers. AC is inherited through the Iwasawa data and the AC${}_{\omega }$ hypotheses of its exponential suppliers; this definition makes no additional choice.