Plancherel support for SL2(R)

Write $d{g}_H$ for the Haar measure used in the cited Harish-Chandra identity and ${\mathrm{\Theta }}^H$ for its integrated traces. The authorized fact [F1] is $2\pi f(e)={\sum }_{n=2}^{\mathrm{\infty }}(n-1){\mathrm{\Theta }}_n^H(f)+\frac{1}{4}{\int }_{\mathbb{R}}{\mathrm{\Theta }}_{0,i\nu }^H(f)\nu \tanh \text{\hspace{0.17em}⁣}\left(\frac{\pi \nu }{2}\right)d\nu +\frac{1}{4}{\int }_{\mathbb{R}}{\mathrm{\Theta }}_{1,i\nu }^H(f)\nu \coth \text{\hspace{0.17em}⁣}\left(\frac{\pi \nu }{2}\right)d\nu$ for $f\in C_c^{\mathrm{\infty }}(G)$. This identity, including its source normalization and constants, is cited without a claimed reading or local derivation of the original paper.

Let $f\in C_c^{\mathrm{\infty }}(G)$ and $h=f^{\ast }\ast f$ for the source Haar measure. Unimodularity and [F12] give $h(e)=\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(d{g}_H)}^2$ and $\pi (h)=\pi (f{)}^{\ast }\pi (f)$. Applying [F1] to $h$ yields $\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(d{g}_H)}^2={\sum }_{n=2}^{\mathrm{\infty }}\frac{n-1}{2\pi }\mathrm{\parallel }{\pi }_n^H(f){\mathrm{\parallel }}_{\mathrm{H}\mathrm{S}}^2+\frac{1}{8\pi }{\int }_{\mathbb{R}}\mathrm{\parallel }{\pi }_{0,i\nu }^H(f){\mathrm{\parallel }}_{\mathrm{H}\mathrm{S}}^2\nu \tanh \text{\hspace{0.17em}⁣}\left(\frac{\pi \nu }{2}\right)d\nu +\frac{1}{8\pi }{\int }_{\mathbb{R}}\mathrm{\parallel }{\pi }_{1,i\nu }^H(f){\mathrm{\parallel }}_{\mathrm{H}\mathrm{S}}^2\nu \coth \text{\hspace{0.17em}⁣}\left(\frac{\pi \nu }{2}\right)d\nu ,$ where ${\pi }_n^H=D_n^{+}\oplus D_n^{-}$ in the source normalization. Each trace is an extended nonnegative trace and is finite almost everywhere because the left side is finite. This is the norm identity on the explicitly parameterized model family; no onto claim is made here.

Fix either the trivial representation or one spherical complementary representation $\pi =I_{0,\nu }$ with $0<\nu <1$, and let $v$ be its normalized $K$-fixed vector. The positive witness below may depend on this fixed $\pi$. Choose a nonnegative smooth approximate identity $\psi$ with integral $1$ and support sufficiently close to $e$ that $\mathrm{\parallel }\pi (\psi )v-v\mathrm{\parallel }<1\mathrm{/}2$; strong continuity gives such a choice. Put $\varphi =e_K\ast \psi \ast e_K$, where $e_K$ is normalized Haar probability on $K$. The formula $\varphi (g)={\int }_{K\times K}\psi (k^{-1}g{\mathrm{\ell }}^{-1})dkd\mathrm{\ell }$ shows that $\varphi \in C_c^{\mathrm{\infty }}(G)$; $\lambda (\varphi )=P_K\lambda (\psi )P_K$ has range in the left $K$-fixed subspace, and $\mathrm{\parallel }\pi (\varphi )v-v\mathrm{\parallel }=\mathrm{\parallel }{P}_K\pi (\psi )v-v\mathrm{\parallel }<1\mathrm{/}2$, so $\pi (\varphi )v\mathrm{\ne }0$. The infinitesimal Möbius fields for $J,D,S$ are respectively $(1+x^2-y^2){\mathrm{\partial }}_x+2xy{\mathrm{\partial }}_y$, $2x{\mathrm{\partial }}_x+2y{\mathrm{\partial }}_y$, and $(1-x^2+y^2){\mathrm{\partial }}_x-2xy{\mathrm{\partial }}_y$. At $i\in \mathbb{H}$, these fields are $0,2{\mathrm{\partial }}_y,2{\mathrm{\partial }}_x$; inserting $W=-iJ$ and $E_{\pm }=(D\pm iS)\mathrm{/}2$ into $\mathrm{\Omega }=W^2\mathrm{/}8-W\mathrm{/}4+E_{+}{E}_{-}\mathrm{/}2$ fixes the second-order coefficient of the invariant operator on $K\mathrm{\backslash }G\cong \mathbb{H}$. There is no invariant first-order term because $K$ has no fixed cotangent vector, and the zero-order term vanishes since both operators annihilate constants; hence $-d\lambda (\mathrm{\Omega })=\mathrm{\Delta }\mathrm{/}2$, where $\mathrm{\Delta }=-y^2({\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2)$ on $L^2(\mathbb{H},dxdy\mathrm{/}{y}^2)$. For $u\in C_c^{\mathrm{\infty }}(\mathbb{H})$, expansion and integration by parts give ${\int }_{\mathbb{H}}{\mid {\mathrm{\partial }}_{y}u-\frac{u}{2y}\mid }^{2}dxdy={\int }_{\mathbb{H}}\mathrm{\mid }{\mathrm{\partial }}_{y}u{\mathrm{\mid }}^{2}dxdy-\frac{1}{4}{\int }_{\mathbb{H}}\mathrm{\mid }u{\mathrm{\mid }}^2\frac{dxdy}{y^2}\ge 0,$ so $\mathrm{\Delta }\ge \frac{1}{4}$ and $-\mathrm{\Omega }-\frac{1}{8}=\frac{1}{2}(\mathrm{\Delta }-\frac{1}{4})\ge 0$ on this subspace. Interpret $D={\varphi }^{\ast }(-\mathrm{\Omega }-\frac{1}{8})\varphi$ as the convolution-differential product: applying the invariant differential operator to the smooth compactly supported kernel keeps it in $C_c^{\mathrm{\infty }}(G)$. For any strongly continuous unitary representation, integration against a smooth compact kernel gives smooth vectors: differentiating $\pi (\exp (sX))\pi (\varphi )$ differentiates the translated kernel in $L^1$, so every iterated derivative is the bounded integrated derivative of that kernel. On such vectors, $\lambda (D)=\lambda (\varphi {)}^{\ast }(-d\lambda (\mathrm{\Omega })-\frac{1}{8})\lambda (\varphi )$ and $\pi (D)=\pi (\varphi {)}^{\ast }(-d\pi (\mathrm{\Omega })-\frac{1}{8})\pi (\varphi )$. For $\xi \in C_c^{\mathrm{\infty }}(G)$, $\lambda (\varphi )\xi$ is smooth, compactly supported and left $K$-fixed, so the displayed estimate gives $⟨\lambda (D)\xi ,\xi ⟩\ge 0$; boundedness and density extend positivity to $L^2(G)$. The regular representation identifies $C_r^{\ast }(G)$ with its concrete image, so $D$ is positive in $C_r^{\ast }(G)$. The Casimir scalar of [F5] holds on these smooth vectors as well: it commutes with every $K$-projection, each projected vector is a smooth finite $K$-type, and the dense $K$-decomposition forces the difference from the claimed scalar to vanish. On $I_{0,\nu }$, [F5] gives $\pi (D)=-\frac{{\nu }^2}{8}\pi (\varphi {)}^{\ast }\pi (\varphi )$, a nonzero negative operator; on the trivial representation it is $-\frac{1}{8}\pi (\varphi {)}^{\ast }\pi (\varphi )$, also nonzero negative. If either representation factored through $C_r^{\ast }(G)$, it would send this positive element to a positive operator, a contradiction. Thus neither is weakly contained in the regular representation.

In the global smooth coordinates $K\times {\mathbb{R}}^2$ of [F2], write a compactly supported continuous function as $F(\theta ,t,x)$, periodic in $\theta$. Periodize a nonnegative compact smooth Euclidean mollifier in the first coordinate and convolve in all three coordinates. The resulting functions are smooth and have support in one slightly enlarged compact cylinder. Uniform continuity makes them converge uniformly to $F$: the error is bounded by the supremum of $\mathrm{\mid }F(z-h)-F(z)\mathrm{\mid }$ for small shifts $h$. The smooth positive Haar weight $e^t\mathrm{/}(2\pi )$ is bounded on that cylinder, whose Haar mass is finite, so convergence holds in both $L^1$ and $L^2$. Together with [F13], this proves $C_c^{\mathrm{\infty }}(G)$ dense in both spaces. For each member of the countable $C_c$-dense family of [F10], choose one such smooth approximant within $1\mathrm{/}k$ in $L^1$ for every positive integer $k$; enumerating the pairs gives a countable smooth $L^1$-dense family $(u_j)$. The same local bumps, normalized by their positive Haar integral, supply the smooth approximate identities used for the positivity and nonvanishing arguments below.

Dividing step 1.1 by $2\pi$ gives source-scale mass $(n-1)\mathrm{/}(2\pi )$ on each $D_n^{\pm }$ and continuous densities $\nu \tanh (\pi \nu \mathrm{/}2)\mathrm{/}(8\pi )$ and $\nu \coth (\pi \nu \mathrm{/}2)\mathrm{/}(8\pi )$ in the redundant real parameter. Both densities are positive for $\nu \mathrm{\ne }0$; the even one tends to $0$ and the odd one to $1\mathrm{/}(4{\pi }^2)$ at zero. Thus these absolutely continuous parameter measures have no atom at zero, while every discrete mass is positive.

In the source normalization, $a_r^H=\mathrm{diag}(e^r,e^{-r})=a_{2r}$, and Hochs's KAK Haar density is $2\pi \sinh (2r)d{k}_{1}drd{k}_2$. Setting $\tau =2r$ gives $d{g}_H=\pi \sinh (\tau )d{k}_{1}d\tau d{k}_2$. By [F2] the native Haar has density $2\pi \sinh (\tau )$, so its uniqueness clause gives $dg=2d{g}_H$ and $\mathrm{\Theta }(f)=2{\mathrm{\Theta }}^H(f)$. Multiplying step 1.1 by $2$ gives the native formula in the Statement; the native Plancherel measure is half the source-scale measure in step 2.1 because the integrated traces double. Thus the native mass is $(n-1)\mathrm{/}(4\pi )$ for each $D_n^{\pm }$, and the redundant-parameter densities are $\nu \tanh (\pi \nu \mathrm{/}2)\mathrm{/}(16\pi )$ and $\nu \coth (\pi \nu \mathrm{/}2)\mathrm{/}(16\pi )$.

Let $X$ be the disjoint union of two copies of $(0,\mathrm{\infty })$, labelled $(\epsilon ,s)$, and the countable discrete labels $(n,+),(n,-)$ for $n\ge 2$. Use the explicit compact-picture representation ${\sigma }_x=I_{\epsilon ,is}$ or the indicated discrete model. The field has fixed countable orthonormal bases on each component. The principal coefficients are Borel in $s$ by the compact-picture continuity of [F7], with off-diagonal coefficients obtained by polarization; discrete components are countable. Thus [F14] gives a Borel class map $q:X\to \widehat{G}$, and [F3] makes it injective. Give $X$ the native positive-parameter densities $s\tanh (\pi s\mathrm{/}2)\mathrm{/}(8\pi )$ and $s\coth (\pi s\mathrm{/}2)\mathrm{/}(8\pi )$, and masses $(n-1)\mathrm{/}(4\pi )$ at each discrete label. Call this sigma-finite measure $w$. The doubling of the continuous densities follows from the sign equivalence of [F3]: the two redundant signs have equal Hilbert–Schmidt norms by unitary conjugation and equal density. Put $\mu =q_{\ast }w$. This measure is on the actual dual, with no separate fibre for each redundant sign.

The class transfer requires no global representative selector. Choose an equivalent probability $P$ on $X$ by positive summable weights on a finite-measure exhaustion. Its pushforward $\beta =q_{\ast }P$ is equivalent to $\mu$. The Borel graph relation $\{(b,x):q(x)=b\}$ has completion-measurable image by [F14], and that image has full $\beta$-measure. Choose a conull Borel $B\subseteq q(X)$ and apply the conull selection in [F14] to obtain a Borel $x(b)$ with $q(x(b))=b$ there. Injectivity makes this the inverse of $q$ on $q^{-1}B$. The sets $\{b\in B:x(b)\in E_j\}$ for a finite-$w$-measure exhaustion of $X$ show that $\mu$ is sigma-finite. Transport the representation and its basis to $B$, extending the fibre by zero off $B$. A matrix $(t_{ij})$ with square-summable entries defines a bounded operator: for a finite vector $v$, Cauchy–Schwarz gives $\mathrm{\parallel }Tv{\mathrm{\parallel }}^2\le {\sum }_{ij}\mathrm{\mid }{t}_{ij}{\mathrm{\mid }}^2\mathrm{\parallel }v{\mathrm{\parallel }}^2$. It extends to the whole carrier, and [F15] identifies its Hilbert–Schmidt norm with this square-sum. Hence $\mathrm{H}\mathrm{S}(H_b)$ is the Hilbert space ${\mathrm{\ell }}^2$ of the matrix entries, by [F15], with rank-one matrix units as an orthonormal basis. These units give a countable measurable fundamental family for the Hilbert–Schmidt field. Its direct integral is Hilbert by [F14]. The entries of ${\sigma }_b(f)$ are Borel for each smooth test $f$ by the integrated Borel correspondence, and their square-sum is measurable. No assertion that every test is Hilbert–Schmidt at every dual point is needed; the norm identity will supply almost-everywhere finiteness.

Rescale the norm identity in step 1.2 using $dg=2d{g}_H$, $\mu =\frac{1}{2}{\mu }_H$ in the redundant parameter and $\sigma (f)=2{\sigma }_H(f)$, then use the sign identification in step 4.1. It gives $\mathrm{\parallel }f{\mathrm{\parallel }}_2^2=\int \mathrm{\parallel }{\sigma }_b(f){\mathrm{\parallel }}_{\mathrm{H}\mathrm{S}}^{2}d\mu (b)$ on $C_c^{\mathrm{\infty }}(G)$. Thus the matrix field of step 5.1 is Hilbert–Schmidt almost everywhere and defines an isometry $\mathcal{F}$. Step 1.4 and Hilbert completeness extend it uniquely to an isometry $L^2(G,dg)\to {\mathcal{H}}_{\mu }:={\int }^{\oplus }\mathrm{H}\mathrm{S}(H_b)d\mu (b)$ with closed range. Left and right translations $f(x)\mapsto f(g^{-1}x)$ and $f(x)\mapsto f(xg)$ give $\mathcal{F}(\lambda (g)f{)}_b={\sigma }_b(g)\mathcal{F}(f{)}_b$ and $\mathcal{F}(\rho (g)f{)}_b=\mathcal{F}(f{)}_b{\sigma }_b(g{)}^{-1}$ by Haar change of variables. Both target actions are unitary by [F15] and strongly continuous: on a rank-one matrix this follows from strong continuity of ${\sigma }_b$, finite matrix truncation gives fibre continuity, and the bound $2\mathrm{\parallel }{T}_b{\mathrm{\parallel }}_{\mathrm{H}\mathrm{S}}$ gives continuity of the integrated action by dominated convergence [F16] for its squared norm, along any sequence $g_j\to g$; the matrix group is first countable. Density extends the intertwining identities to all $L^2$.

Every nonzero principal parameter has positive density in step 3.1, and each discrete class $D_n^{\pm }$ has positive atomic mass. By [F7], every Fell neighborhood of $[I_{\epsilon ,i{\nu }_0}]$ for ${\nu }_0\mathrm{\ne }0$ contains a parameter interval, so has positive measure. The even family likewise approaches $[I_{0,0}]$. For either odd limit, [F7] puts it in the closure of positive-parameter odd classes; a neighborhood contains such an interior class and, by Fell continuity there, an interval of positive density. Hence $I_{0,0}$ and both $D_1^{\pm }$ are in the closed support but have no atom. By [F3], $I_{1,0}$ is reducible and is not a point of $\widehat{G}$.

Put $A=C^{\ast }(G)$ and let $L$ denote the left action on ${\mathcal{H}}_{\mu }$, so $L(a{)}_{b}T={\sigma }_b(a)T$. For any closed ideal $J$ of $A$, its intrinsic support projection $P_J$ onto $\overline{L(J){\mathcal{H}}_{\mu }}$ commutes with $L(A)$ and its commutant: the ideal span reduces $L(A)$, and every commutant operator and its adjoint preserve that span. Therefore $P_J\in L(A{)}^{\mathrm{\prime }\mathrm{\prime }}$. It is precisely the scalar multiplier of the ideal-open $\{b:J\subseteq ̸\ker {\sigma }_b\}$. To verify this, take a countable dense sequence in $J$, apply its represented operators to the countable matrix fundamental family, and use [F14] for the measurable closed spans. In an irreducible fibre a nonzero ideal has full support; its approximate unit tends strongly to1 on $H_b$, hence left multiplication tends to1 on Hilbert–Schmidt matrices by finite-column truncation and the uniform norm bound. The fibre ideal span is therefore all of $\mathrm{H}\mathrm{S}(H_b)$ or0. Its global span equals the integral of those spans: bounded finite-measure scalar localizations of the generating sections are $L(j)$ of localized fundamental sections and lie in the global span, and conversely every $L(j)\xi$ takes values in the fibre spans; their density is [F14]. Countably many such ideal-opens generate all dual Borel sets by [F3,F14]. Their indicator multipliers generate the full scalar diagonal algebra by monotone strong limits of indicators and bounded simple approximation. This is the actual ideal-support argument of [F4] applied to the present field. If $Q$ is the closed-range projection of step 6.1, the left/right intertwining and inverse group actions make its range reducing, so $Q$ commutes with both target actions. In particular it commutes with $L(A{)}^{\mathrm{\prime }\mathrm{\prime }}$ and the intrinsic scalar diagonal. By [F14] it is decomposable, $Q={\int }^{\oplus }{Q}_{b}d\mu (b)$.

On a single conull set, the projection field $Q_b$ commutes with both target group actions: apply uniqueness of decomposable fields [F14] to each commutator at a fixed countable dense subset of $G$, remove the countable union of null exceptions, and extend by the fibre strong continuity proved in step 6.1. Here is the joint irreducibility calculation. On the Hilbert–Schmidt matrix space, view each column as a copy of $H_b$. Each column block of a bounded operator commuting with left ${\sigma }_b(G)$ is a bounded self-intertwiner of ${\sigma }_b$, hence scalar by [F9]. The scalar column matrix defines a bounded operator $C$ on the column ${\mathrm{\ell }}^2$ (test finite columns with one fixed unit row vector); thus the original operator is $I\otimes C$. Right multiplication by ${\sigma }_b(g{)}^{-1}$ acts on these columns by the conjugate representation. That representation is irreducible, because conjugating a closed invariant subspace gives one for ${\sigma }_b$. A second use of [F9] makes $C$ scalar. Applied to the projection $Q_b$, this gives $Q_b=0$ or $I$ almost everywhere.

Suppose the measurable zero-fibre set $E=\{b:Q_b=0\}$ has positive measure; measurability follows from its countable matrix coefficients. For every smooth member $u_j$ of the countable $L^1$-dense family in step 1.4, its transform lies in the range of $Q$, so ${\sigma }_b(u_j)=0$ almost everywhere on $E$. Remove the countable union of these exceptions and choose $b\in E$ where its irreducible carrier is nonzero. Contractivity $\mathrm{\parallel }{\sigma }_b(f)-{\sigma }_b(u_j)\mathrm{\parallel }\le \mathrm{\parallel }f-u_j{\mathrm{\parallel }}_1$ gives ${\sigma }_b(f)=0$ for every $f\in L^1(G)$. But for a nonzero vector $v$, a normalized smooth bump sufficiently near $e$ has $\mathrm{\parallel }{\sigma }_b(\eta )v-v\mathrm{\parallel }<\mathrm{\parallel }v\mathrm{\parallel }\mathrm{/}2$ by strong continuity, contradicting this vanishing. Thus $Q_b=I$ almost everywhere and $\mathcal{F}$ is onto. Left multiplication on the Hilbert–Schmidt field is the amplification of ${\sigma }_b$ with multiplicity $\dim H_b$, as seen by its columns. It is consequently an actual irreducible disintegration of the regular representation; the canonical central/refinement and uniqueness interfaces of [F4] apply to this constructed model.

Let $S$ be the closed support of $\mu$. The dual has a countable base by [F3,F14], so the complement of $S$ is a countable union of measure-zero basic opens; thus $\mu$ is carried by $S$. For $a\in A$, onto disintegration gives $a\in \ker \lambda$ exactly when ${\sigma }_b(a)=0$ almost everywhere, by uniqueness of decomposable fields: left multiplication by ${\sigma }_b(a)$ vanishes exactly when that operator vanishes, as testing rank-one matrices shows. Define the class norm $q_a(b)=\mathrm{\parallel }a+\ker b\mathrm{\parallel }$ on the entire dual; on the retained conull $B$ it equals $\mathrm{\parallel }{\sigma }_b(a)\mathrm{\parallel }$, while it still denotes the genuine irreducible class norm at null endpoint classes rather than the zero-extended field. Its strict superlevel $\{b:q_a(b)>r\}$ is open by [F3,F14]. If it met $S$ for $r>0$, it would have positive measure, contradicting this almost-everywhere vanishing. Conversely $q_a=0$ at every point of $S$ implies almost-everywhere vanishing. Hence $\ker \lambda ={\bigcap }_{b\in S}\ker b$, using genuine class kernels also at null endpoints. By [F8], an irreducible is weakly contained in $\lambda$ exactly when it is in the Fell closure of $S$, which is $S$ itself. This proves the support/tempered bridge locally for the constructed measure.

The discrete mass $d_n=(n-1)\mathrm{/}(4\pi )$ is its formal degree in the coefficient convention. Fix $\sigma =D_n^{\pm }$ and vectors $v,w$; let $R_{v,w}z=⟨z,w⟩v$ be its Hilbert–Schmidt rank-one operator, and take the target field equal to $R_{v,w}$ at the atom and zero elsewhere. Onto isometry gives an inverse $h\in L^2(G)$ with $\mathrm{\parallel }h{\mathrm{\parallel }}_2^2=d_n\mathrm{\parallel }v{\mathrm{\parallel }}^2\mathrm{\parallel }w{\mathrm{\parallel }}^2$. For every smooth compact test $f$, the transform pairing gives $⟨f,h⟩=d_n⟨\sigma (f)w,v⟩$. Consequently $h(g)=d_n⟨\sigma (g^{-1})v,w⟩$ almost everywhere: both sides are locally integrable, and their difference has zero pairing with every compact smooth test; in coordinates multiply that difference by the positive smooth Haar density, convolve it locally with the kernels of [F16], and then pass to its local $L^1$ limit; division by the positive density yields the asserted equality. Thus every coefficient has squared $L^2$ norm $d_n^{-1}\mathrm{\parallel }v{\mathrm{\parallel }}^2\mathrm{\parallel }w{\mathrm{\parallel }}^2$. Pairing two rank-one target fields gives the full coefficient orthogonality by the same identity and polarization. This proves the claimed formal degree locally, rather than merely naming the trace-inversion coefficient.

The measure in steps 4.1 and 3.1 is carried by the stated principal/discrete classes; step 7.1 puts every such class and precisely the stated endpoint candidates in its closed support. Step 1.3 excludes every positive-parameter spherical complementary class and the trivial class from that support by the positive reduced-algebra witness and step 10.1, rather than by zero mass. Classification [F3] leaves no further classes. The endpoint fibres have no atoms, and the odd zero direct sum is not a dual point. Steps 9.1 and 10.1 establish the full regular disintegration, onto transform, canonical measure-class/multiplicity and exact weak-containment support. The Haar conversion and every field, range, endpoint and exclusion argument are local; the original trace identity [F1] is the only cited exception.