Iwasawa and minimal-parabolic data for SL2(R)

Assume the Axiom of Choice (The Axiom of Choice). Let $G={\mathrm{S}\mathrm{L}}_2(\mathbb{R})=\{g\in M_2(\mathbb{R}):\det g=1\}$, the embedded matrix Lie group of General and special linear Lie groups, with the real Lie-group conventions of Lie group and Real and complex Lie groups. Set

$K=\mathrm{S}\mathrm{O}(2)=\{k_{\theta }=\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right):\theta \in \mathbb{R}\mathrm{/}2\pi \mathbb{Z}\}\mathrm{.}$

This is a closed embedded Lie subgroup (Orthogonal and special orthogonal Lie groups), isomorphic to the circle group $\mathbb{T}=\mathbb{R}\mathrm{/}2\pi \mathbb{Z}$; the isomorphism with the quotient $\mathbb{R}\mathrm{/}\mathbb{Z}$ of The one-dimensional torus and its normalized Haar integral is $[\theta {]}_{2\pi }\mapsto [\theta \mathrm{/}(2\pi ){]}_1$. Write $dk$ for the normalized Haar probability measure on $K$ (Normalized Haar measure on a compact Lie group). Then $k_{\theta +\pi }=-k_{\theta }$.

Let $A=\{a_t=\mathrm{diag}(e^{t\mathrm{/}2},e^{-t\mathrm{/}2}):t\in \mathbb{R}\},N=\{n_x=\left(\begin{array}{cc}1& x\\ 0& 1\end{array}\right):x\in \mathbb{R}\},M=\{\pm I\}\mathrm{.}$ In fact $M=Z(G)$: commuting with $a_1$ forces a central matrix to be diagonal, and commuting with $n_1$ forces its diagonal entries to agree; determinant one then gives precisely $I$ and $-I$.

The coordinates $a_t\leftrightarrow t$ and $n_x\leftrightarrow x$ identify $A$ and $N$ with the additive group $\mathbb{R}$; explicitly, $a_t{a}_s=a_{t+s}$, $n_x{n}_y=n_{x+y}$, and $a_t{n}_x{a}_{-t}=n_{e^{t}x}$. The standard minimal parabolic subgroup is $P=MAN=\{\left(\begin{array}{cc}u& b\\ 0& u^{-1}\end{array}\right):u\in {\mathbb{R}}^{\times },b\in \mathbb{R}\},$ the closed stabilizer of the line $\mathbb{R}(1,0)$ in the standard action on ${\mathbb{R}}^2$: preservation of that line is exactly the vanishing of the lower-left entry. The factorization is unique: for the displayed matrix, set $m=\mathrm{sgn}(u)I$, $t=2\log \mathrm{\mid }u\mathrm{\mid }$, and $x=b\mathrm{/}u$; then $p=m{a}_t{n}_x$, and the diagonal and top-right entries force these same values from any such factorization. The displayed coordinates identify $P$ with ${\mathbb{R}}^{\times }\times \mathbb{R}$, so its identity component is the $u>0$ component $AN$. A unipotent element has both eigenvalues equal to $1$, forcing $u=1$; hence the unipotent elements of $P$ are exactly $N$, which is connected and normal. Thus $N$ is the unipotent radical. Directly, $K\cap P=\{\pm I\}=M$.

Put $H=\mathrm{diag}(1,-1)$, $e_0=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, $\mathfrak{a}=\mathbb{R}H$, and $\mathfrak{n}=\mathbb{R}{e}_0$. By General and special linear Lie groups, the Lie algebra of this real $G$ is the traceless real matrices; $H,e_0$ lie in it, and direct multiplication gives $[H,e_0]=2e_0$, the same matrix relation used in The special linear Lie algebra sl_2. The explicit matrix exponential gives $a_t={\exp }_G(tH\mathrm{/}2)$ and $n_x={\exp }_G(x{e}_0)$, so their coordinate maps are one-parameter subgroups (Exponential map of a Lie group, One-parameter subgroup of a Lie group, One-parameter subgroups are exactly exponentials). In particular, $\mathrm{Ad}(a_t)e_0=e^t{e}_0$.

For $p\in P$, define the parabolic modular character by ${\delta }_P(p)=\mid \det \text{\hspace{0.17em}⁣}(\mathrm{Ad}(p){\mathrm{\mid }}_{\mathfrak{n}})\mid \mathrm{.}$ Then ${\delta }_P{\mathrm{\mid }}_{MN}=1$ and ${\delta }_P(a_t)=e^t$. To distinguish this character from the group modular function of Modular function of a locally compact group, write the latter as ${\mathrm{\Delta }}_P$: with the convention ${\int }_{P}f(x{p}^{-1})d\mu (x)={\mathrm{\Delta }}_P(p){\int }_{P}fd\mu$, one has ${\mathrm{\Delta }}_P={\delta }_P^{-1}$ and hence ${\mathrm{\Delta }}_P(a_t)=e^{-t}$. Indeed, on $AN$ the coordinates $a_t{n}_x$ have left Haar measure $dtdx$; right translation by $a_{-s}$ is $(t,x)\mapsto (t-s,e^{s}x)$ and scales the integral by $e^{-s}$. Kerr denotes the parabolic modular character ${\delta }_P$ by ${\mathrm{\Delta }}_P$.

For $\epsilon \in \{0,1\}$ and $\nu \in \mathbb{C}$, let ${\sigma }_{\epsilon }(\pm I)=(\pm 1{)}^{\epsilon }$, extend it trivially over $AN$, and define the character $e^{\nu }$ to be trivial on $MN$ and satisfy $e^{\nu }(a_t)=e^{\nu t\mathrm{/}2}$. Set $\mathrm{\mid }\alpha (a_t)\mathrm{\mid }=e^{t\mathrm{/}2}$. The normalized inducing character is ${\sigma }_{\epsilon }{\delta }_P^{1\mathrm{/}2}{e}^{\nu }$; on $A$ it is $\mathrm{\mid }\alpha (a_t){\mathrm{\mid }}^{1+\nu }=e^{(1+\nu )t\mathrm{/}2}={\delta }_P(a_t{)}^{1\mathrm{/}2}{e}^{\nu }(a_t)\mathrm{.}$

All principal-series parameters on this page use the letter $\nu$ with this normalization, and the half-modular shift is applied exactly once. AC is used to obtain $dk$ through Normalized Haar measure on a compact Lie group. A countable family of nonempty sets is a family to which AC applies, so AC supplies the ${\mathrm{A}\mathrm{C}}_{\omega }$ hypothesis stated in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) and required by the cited exponential-map and one-parameter-subgroup results. The groups and coordinates above are otherwise explicit.