SL2(R): Discrete Series and the Unitary Dual

The curve $t\mapsto F_v({\gamma }_X(t))$ is smooth by [F1] and [F2], so its derivative at $0$ exists and is $L_{X}v$. For every $g\in G$, bounded linearity of $\pi (g)$ gives $\pi (g)L_{X}v={\frac{d}{dt}\mid }_0{F}_v(g{\gamma }_X(t))=(X^L{F}_v)(g)$. In local coordinates $X^L={\sum }_i{a}_i{\mathrm{\partial }}_i$ with smooth coefficients, so $X^L{F}_v={\sum }_i{a}_i{\mathrm{\partial }}_i{F}_v$ is smooth as an $H$-valued map. Hence the orbit map of $L_{X}v$ is smooth and $L_{X}v\in H^{\mathrm{\infty }}$.

Applying the preceding identity twice gives $L_X{L}_{Y}v=(X^L(Y^L{F}_v))(e)$ and $L_Y{L}_{X}v=(Y^L(X^L{F}_v))(e)$. For an $H$-valued smooth map the commutator identity follows by applying every continuous linear functional on $H$ to it; these functionals separate points, and differentiation commutes with them. Therefore $L_X{L}_{Y}v-L_Y{L}_{X}v=([X^L,Y^L]F_v)(e)=([X,Y{]}^L{F}_v)(e)=L_{[X,Y]}v$ by [F3].

For $k\in K$, the orbit map of $\pi (k)v$ is $g\mapsto F_v(gk)$, so $K$ preserves $H^{\mathrm{\infty }}$. The curve $k{\exp }_G(tX)k^{-1}$ is a one-parameter subgroup with tangent $kX{k}^{-1}$, and hence equals ${\exp }_G(t\mathrm{Ad}(k)X)$ by [F2]. Differentiating gives $\pi (k)L_{X}v=L_{\mathrm{Ad}(k)X}\pi (k)v$ for real $X$, and complex-linearity gives it for complex $X$. If $v\in V$, let $E=\mathrm{span}\{\pi (k)v:k\in K\}$; it is a finite-dimensional $K$-invariant subspace of $H^{\mathrm{\infty }}$. For a basis $X_1,X_2,X_3$ of ${\mathfrak{g}}_{\mathbb{C}}$, the finite-dimensional span of all $L_{X_i}u$, $u\in E$, is $K$-invariant by this covariance identity and contains every $L_{X}v$. Thus $L_{X}v$ is $K$-finite as well as smooth, and $L_X$ preserves $V$.

Complex-linearity extends this bracket identity to ${\mathfrak{g}}_{\mathbb{C}}$, so $X\mapsto L_X$ is a Lie-algebra action on $H^{\mathrm{\infty }}$. The universal property of the enveloping algebra, Lie algebra actions extend to unital actions of the enveloping algebra, then gives the unique unital action of $U({\mathfrak{g}}_{\mathbb{C}})$ extending it.

The $K$-action preserves $V$, since translating a finite-dimensional $K$-orbit span leaves it unchanged. Its restriction to each such span is smooth: all its vectors are smooth and coordinate functionals on the finite-dimensional span recover smooth matrix entries from their orbit maps. Differentiating $\pi (k_{\theta })v$ gives $L_{J}v$ because $k_{\theta }={\exp }_G(\theta J)$; this identity follows from [F2] since both curves have tangent $J$. Hence the infinitesimal $K$-action and the restricted Lie-algebra action agree. Together with step 2.2 this proves all stated $(\mathfrak{g},K)$ compatibilities. Stability under each $L_X$ also makes $V$ stable under their finite products and sums, so the action from step 3.1 restricts to $V$.

A local smooth probability kernel exists in every identity neighborhood $U$. In the matrix chart $\mathrm{\Psi }(a,b,c)=\left(\begin{array}{cc}1+a& b\\ c& (1+bc)\mathrm{/}(1+a)\end{array}\right)$ around $I$, choose $r>0$ with $\mathrm{\Psi }({\bar{B}}_r)\subset U$ and ${\bar{B}}_r\subset (-1,1{)}^3$. Pull back the function $b_r(x)=\exp (-1\mathrm{/}(r^2-\mathrm{\parallel }x{\mathrm{\parallel }}^2))$ for $\mathrm{\parallel }x\mathrm{\parallel }<r$, extended by zero for $\mathrm{\parallel }x\mathrm{\parallel }\ge r$, and extend it by zero off the chart. This gives a nonnegative smooth compactly supported function $q_U$ in $U$, positive on a nonempty open set. By [F2], $0<{\int }_G{q}_{U}dg<\mathrm{\infty }$; hence $f_U=q_U\mathrm{/}{\int }_G{q}_{U}dg$ has $\int f_U=1$.

For smooth compactly supported $f$, put $f_h(x)=f(h^{-1}x)$. Left invariance and the pairing formula [F1] give $\pi (h)\pi (f)=\pi (f_h)$. Near any fixed $h_0$, choose a relatively compact coordinate neighborhood; the supports of $f_h$ there lie in one compact set $C$, and all parameter derivatives of $f(h^{-1}x)$ are jointly continuous on the product of a compact neighborhood with $C$. Uniform continuity of each derivative shows, by the difference-quotient definition and induction on its order, that $h\mapsto f_h$ is $C^{\mathrm{\infty }}$ in $L^1$ norm (the $L^1$ error is bounded by the uniform error on $C$ times its finite Haar measure). The bound in [F1] makes $h\mapsto \pi (f_h)v$ norm-$C^{\mathrm{\infty }}$. Thus $\pi (f)v\in H^{\mathrm{\infty }}$.

Every irreducible unitary representation $\sigma$ of $K=\mathrm{S}\mathrm{O}(2)$ is one dimensional. By [F4] it is finite dimensional; since $K$ is abelian, each $\sigma (k)$ is a self-intertwiner and hence scalar by Schur's lemma. Irreducibility then forces dimension one. Writing $\chi (k_{\theta })=q(\theta )$ for this character, [F5] makes $q$ differentiable. Its homomorphism law gives $q^{\mathrm{\prime }}(\theta )=q(\theta )q^{\mathrm{\prime }}(0)$; since $\mathrm{\mid }q(\theta )\mathrm{\mid }=1$, write $q^{\mathrm{\prime }}(0)=ia$ with $a\in \mathbb{R}$, and solve to get $q(\theta )=e^{ia\theta }$. The period $k_{\theta +2\pi }=k_{\theta }$ forces $e^{2\pi ia}=1$, so $a\in \mathbb{Z}$. Conversely each ${\sigma }_n(k_{\theta })=e^{in\theta }$ is an irreducible unitary character. Thus these are exactly the irreducibles of $K$.

For each $n$, a nonzero vector in $H_n$ spans a copy of ${\sigma }_n$, while every ${\sigma }_n$-copy lies in $H_n$; since $H_n$ is closed, it is exactly the ${\sigma }_n$-isotypic subspace. By [F3], $H={\widehat{⨁}}_{n\in \mathbb{Z}}{H}_n$, with orthogonal projections $P_n$ onto $H_n$.

Given $v\in H$ and $ϵ>0$, strong continuity supplies an identity neighborhood $U$ with $\mathrm{\parallel }\pi (g)v-v\mathrm{\parallel }<ϵ$ for all $g\in U$. Choose $f_U$ from step 1.1. The pairing formula [F1] and $f_U\ge 0$, $\int f_U=1$ yield $\mathrm{\mid }⟨\pi (f_U)v-v,w⟩\mathrm{\mid }\le ϵ\mathrm{\parallel }w\mathrm{\parallel }$ for every $w\in H$, hence $\mathrm{\parallel }\pi (f_U)v-v\mathrm{\parallel }\le ϵ$. By step 1.2, $\pi (f_U)v\in H^{\mathrm{\infty }}$, so $H^{\mathrm{\infty }}$ is dense in $H$.

If $w\in H^{\mathrm{\infty }}$, then $\pi (g)P_{n}w={\int }_K\overline{{\chi }_n(k)}{F}_w(gk)dk$, where ${\chi }_n(k_{\theta })=e^{in\theta }$ and $dk$ is normalized Haar probability. On each compact coordinate neighborhood in $G$, every derivative in $g$ of the integrand is continuous and uniformly bounded over compact $K$. The Bochner norm inequality [F7] bounds the integral of a uniform difference-quotient error by that same error, so induction on derivative order lets every derivative pass through the integral. Hence $P_{n}w\in H^{\mathrm{\infty }}$, and step 2.1 gives $P_{n}w\in H_n$, so $P_{n}w\in V$. If also $w\in M$, let $Q$ be the orthogonal projection onto $M^{\perp }$. It is bounded, and $Q\pi (k)w=0$ for every $k\in K$; bounded maps commute with the Bochner integral [F8], so $Q{P}_{n}w=0$ and $P_{n}w\in \ker Q=M$.

By step 2.2, $H^{\mathrm{\infty }}$ is dense in $H$; for $w\in H^{\mathrm{\infty }}$, the finite partial sums of ${\sum }_n{P}_{n}w$ converge to $w$ by step 2.1, and each term is in $V$ by step 3.1. Thus $V$ is dense in $H$. For $v\in V$, its finite-dimensional $K$-orbit span is a unitary representation of $K$; applying the compact-group decomposition and step 1.3 shows that it has only finitely many weights $n$. Therefore $V={⨁}_{n\in \mathbb{Z}}{V}_n$.

Let $M\mathrm{\ne }\{0\}$ be closed and $\pi (G)$-invariant. Choose $v\in M$ nonzero and an identity neighborhood $U$ with ${\sup }_{g\in U}\mathrm{\parallel }\pi (g)v-v\mathrm{\parallel }<\mathrm{\parallel }v\mathrm{\parallel }\mathrm{/}2$; use step 1.1 to choose $f_U$. Then $w=\pi (f_U)v$ lies in $H^{\mathrm{\infty }}$ by step 1.2 and satisfies $\mathrm{\parallel }w-v\mathrm{\parallel }<\mathrm{\parallel }v\mathrm{\parallel }\mathrm{/}2$ by the pairing estimate of step 2.2, so $w\mathrm{\ne }0$. For every $z\in M^{\perp }$, $⟨w,z⟩={\int }_G{f}_U(g)⟨\pi (g)v,z⟩dg=0$, since $\pi (g)v\in M$; hence $w\in (M^{\perp }{)}^{\perp }=M$ by [F6]. Since $H={\widehat{⨁}}_n{H}_n$, some $P_{n}w\mathrm{\ne }0$, and step 3.1 gives $P_{n}w\in M\cap V$.

The definition Smooth and K-finite vectors for SL2(R), and the (g,K)-module already proves that $H^{\mathrm{\infty }}$ is stable under every $L_X$ and is a $U({\mathfrak{g}}_{\mathbb{C}})$-module. Within $V$, stability under $K$ follows because right translation of a smooth orbit map is smooth and the $K$-orbit span of $\pi (k)v$ equals that of $v$. It is stable under $L_X$ as well. For $k\in K$, conjugation gives $\pi (k)L_{X}v=L_{\mathrm{Ad}(k)X}\pi (k)v$. If $v\in V$, choose finite bases of ${\mathfrak{g}}_{\mathbb{C}}$ and of $\mathrm{span}\pi (K)v$; the right side lies in the span of the finitely many vectors $L_{X_i}{w}_j$, so $L_{X}v$ is $K$-finite, and it is smooth by the same definition supplier. Thus $V$ is a ${\mathfrak{g}}_{\mathbb{C}}$- and $K$-stable algebraic direct sum of the $V_n$, and the enveloping-algebra action restricts to it. In particular the algebra element $XY$ acts as $L_X{L}_Y$.

Steps 1.2, 2.1, 2.2, 3.1, 4.1, 4.2, and 5.1 establish claims 1–4, including the nonzero-subspace qualification and the zero-Hilbert-space case stated above. No endpoint or parameter case occurs in this lemma.

For $X=J,H,S$, define $L_{X}f={\frac{d}{ds}\mid }_0{\mathrm{\Pi }}_{\nu }({\exp }_G(sX))f$ using [F6]; the matrix curves are $\exp (sJ)=k_s$, $\exp (sH)=\mathrm{diag}(e^s,e^{-s})$, and $\exp (sS)=\left(\begin{array}{cc}\cosh s& \sinh s\\ \sinh s& \cosh s\end{array}\right)$. If $b(s)$ is the bottom row of $k_{\theta }\exp (sX)$, then the $ANK$ factorization in [F1] gives $b(s)=e^{-u(s,\theta )\mathrm{/}2}(-\sin \beta (s,\theta ),\cos \beta (s,\theta ))$, so $e^{u\mathrm{/}2}=1\mathrm{/}\mathrm{\parallel }b\mathrm{\parallel }$ and ${\mathrm{\partial }}_s\beta {\mathrm{\mid }}_0=\det (b,b^{\mathrm{\prime }})\mathrm{/}\mathrm{\parallel }b{\mathrm{\parallel }}^2$. At $s=0$, $b=(-\sin \theta ,\cos \theta )$; for $J$, ${\mathrm{\partial }}_s\log (e^{u\mathrm{/}2})=0$ and ${\beta }^{\mathrm{\prime }}=1$; for $H$, $b^{\mathrm{\prime }}=(-\sin \theta ,-\cos \theta )$, giving ${\mathrm{\partial }}_s\log (e^{u\mathrm{/}2})=\cos 2\theta$ and ${\beta }^{\mathrm{\prime }}=\sin 2\theta$; for $S$, $b^{\mathrm{\prime }}=(\cos \theta ,-\sin \theta )$, giving ${\mathrm{\partial }}_s\log (e^{u\mathrm{/}2})=\sin 2\theta$ and ${\beta }^{\mathrm{\prime }}=-\cos 2\theta$. Differentiating [F1] yields $L_J={\mathrm{\partial }}_{\theta }$, $L_H=(1+\nu )\cos 2\theta +\sin 2\theta {\mathrm{\partial }}_{\theta }$, and $L_S=(1+\nu )\sin 2\theta -\cos 2\theta {\mathrm{\partial }}_{\theta }$. Hence $L_W=-i{\mathrm{\partial }}_{\theta }$, $L_{E_{\pm }}=\frac{1}{2}{e}^{\pm 2i\theta }((1+\nu )\mp i{\mathrm{\partial }}_{\theta })$, and on each basis vector $L_W{f}_m=m{f}_m$, $L_{E_{+}}{f}_m=\frac{1+\nu +m}{2}{f}_{m+2}$, and $L_{E_{-}}{f}_m=\frac{1+\nu -m}{2}{f}_{m-2}$.

If $v={\sum }_{m\in S}{c}_m{f}_m$ is a finite K-type sum and $M$ is a K-stable submodule containing $v$, then $P_{m}v:={\int }_K{e}^{-im\varphi }{\mathrm{\Pi }}_{\nu }(k_{\varphi })vdk=c_m{f}_m$. The orbit span of $v$ is finite dimensional and lies in $M$, hence is closed; its integral also lies in $M$. Thus every nonzero submodule contains some weight vector $f_m$.

The operators in step 1.1 satisfy $[L_W,L_{E_{\pm }}]f_m=\pm 2L_{E_{\pm }}{f}_m$. Also $L_{E_{+}}{L}_{E_{-}}{f}_m=\frac{{\nu }^2-(m-1{)}^2}{4}{f}_m$ and $L_{E_{-}}{L}_{E_{+}}{f}_m=\frac{{\nu }^2-(m+1{)}^2}{4}{f}_m$, so $[L_{E_{+}},L_{E_{-}}]f_m=m{f}_m=L_W{f}_m$. Since the $f_m$ span $I_{\epsilon ,\nu }^K$, these relations make $X\mapsto L_X$ a Lie-algebra action there; [F5] gives its enveloping-algebra action.

In part (a), at $\nu =n$ the coefficients in step 1.1 vanish at $L_{E_{-}}{f}_{n+1}$ and $L_{E_{+}}{f}_{-(n+1)}$, so the positive and negative tails displayed in the Statement are stable under $K,E_{+},E_{-}$. On the positive tail, all upward coefficients are nonzero and every downward coefficient except the boundary one is nonzero, so step 1.2 shows any nonzero submodule contains a weight vector from which both directions generate the whole tail; the negative tail has the same property with the roles reversed. Hence both tails are irreducible, and their disjoint weights make their sum direct.

In part (b), at $\nu =n-1$ the coefficients are $L_{E_{-}}{f}_m=\frac{n-m}{2}{f}_{m-2}$ and $L_{E_{+}}{f}_m=\frac{n+m}{2}{f}_{m+2}$; hence $L_{E_{-}}{f}_n=0$, $L_{E_{+}}{f}_{-n}=0$, and all other coefficients along the two tails are nonzero, so [F2] and step 1.2 give the stated irreducible submodules. In part (c), at $\nu =0$ and odd weights, $L_{E_{-}}{f}_1=L_{E_{+}}{f}_{-1}=0$; the other coefficients along the positive and negative tails are nonzero, and these tails partition the entire odd basis, giving the stated direct sum of irreducible submodules.

For every weight $m$, [F3] and step 1.1 give $E_{+}{E}_{-}{f}_m=\frac{(1+\nu -m)(\nu +m-1)}{4}{f}_m=\frac{{\nu }^2-(m-1{)}^2}{4}{f}_m$. Therefore $\mathrm{\Omega }{f}_m=(\frac{m^2}{8}-\frac{m}{4}+\frac{{\nu }^2-(m-1{)}^2}{8})f_m=\frac{{\nu }^2-1}{8}{f}_m$. At $\nu =n-1$ this is $\frac{1}{8}((n-1{)}^2-1)$, and at $\nu =0$ it is $-\frac{1}{8}$.

The quotient by the two tails has basis ${\bar{f}}_{n-1},{\bar{f}}_{n-3},\dots ,{\bar{f}}_{-(n-1)}$, so it has dimension $n$. The class ${\bar{f}}_{n-1}$ is a highest-weight vector of weight $n-1$, since $L_{E_{+}}{f}_{n-1}$ lies in the positive tail. The positive root is determined by $[W,E_{+}]=2E_{+}$, so its coroot is $W$ and its fundamental weight $\omega$ satisfies $\omega (W)=1$; the highest weight is $(n-1)\omega$. Every nonzero ${\mathfrak{s}\mathfrak{l}}_2$-submodule of the quotient is $W$-stable, so a Lagrange interpolation polynomial in $W$ isolates a nonzero weight line; all ladder coefficients between consecutive weights in the finite range are nonzero, so the ladder operators generate every line. Thus the quotient is a finite-dimensional simple highest-weight module, hence $L_{n-1}$ by [F4].

If a proper submodule of $I_{\epsilon ,n}^K$ contained any central weight $f_m$ with $-(n-1)\le m\le n-1$, step 1.1 shows the nonzero ladder coefficients move it through every weight of the parity class, making the submodule all of $I_{\epsilon ,n}^K$. By step 1.2 every submodule decomposes into its weight lines, so every proper submodule lies in the sum of the two tails. Since the quotient in step 3.1 is simple, that sum is the unique maximal proper submodule.

Steps 2.2–4.1 establish part (a), step 2.3 establishes the irreducible submodules and direct sum in parts (b) and (c), and step 2.4 establishes both stated Casimir scalars. The case $n=1$ in part (a) has the one-dimensional quotient $\mathbb{C}{\bar{f}}_0$; no $n=0$ case is asserted there, and its separate odd-parameter case is part (c).