Square integrability of discrete-series matrix coefficients

By [F1], a finite-dimensional K-invariant span decomposes into finitely many characters of $K=\mathrm{S}\mathrm{O}(2)$, and the corresponding character spaces in $D_n^{-}$ are precisely the lines $\mathbb{C}{e}_j$. Hence every K-finite $v,w\in {\mathcal{H}}_n^{+}$ has finite expansions in this basis.

In the Cayley coordinate $w=(z-i)\mathrm{/}(z+i)$, put $F_f(w)=(z+i{)}^{n}f(z)$, so $F_{f_{n,j}}=w^j$. Substitution of $z=i(1+w)\mathrm{/}(1-w)$ in the weighted integral gives $\mathrm{\parallel }f{\mathrm{\parallel }}_n^2=2^{2-2n}{\int }_{\mathbb{D}}\mathrm{\mid }{F}_f(w){\mathrm{\mid }}^2(1-\mathrm{\mid }w{\mathrm{\mid }}^2{)}^{n-2}dA(w)$: the factors are $z+i=2i\mathrm{/}(1-w)$, $y=(1-\mathrm{\mid }w{\mathrm{\mid }}^2)\mathrm{/}\mathrm{\mid }1-w{\mathrm{\mid }}^2$, and $\mathrm{\mid }dz\mathrm{/}dw{\mathrm{\mid }}^2=4\mathrm{/}\mathrm{\mid }1-w{\mathrm{\mid }}^4$. Polar integration therefore gives $N_j^2=2^{2-2n}\pi B_j$, where $B_j={\int }_0^1{x}^j(1-x{)}^{n-2}dx$. For every real $t$, write $r=\tanh (t\mathrm{/}2)$. The inverse-action formula in [F1] gives $F_{{\pi }_n(a_t)f_{n,j}}(w)=\cosh (t\mathrm{/}2{)}^{-n}(w-r{)}^j(1-rw{)}^{-n-j}$. Expand the polynomial numerator and the denominator by the geometric-series derivatives; since $\mathrm{\mid }r\mathrm{\mid }<1$, the resulting power series converges absolutely and uniformly on $\mathrm{\mid }w\mathrm{\mid }\le 1$. Its coefficient of $w^k$ is $\cosh (t\mathrm{/}2{)}^{-n}{Q}_{jk}(r)$, with $Q_{jk}(r)={\sum }_{p=0}^{\min (j,k)}(-1{)}^{j-p}\left(\genfrac{}{}{0px}{}{j}{p}\right)\left(\genfrac{}{}{0px}{}{n+j+k-p-1}{k-p}\right)r^{j+k-2p}$. Uniform convergence permits integration against ${\bar{w}}^k(1-\mathrm{\mid }w{\mathrm{\mid }}^2{)}^{n-2}$; angular orthogonality leaves precisely that coefficient times $N_k^2$. Hence $⟨{\pi }_n(a_t)f_{n,j},f_{n,k}⟩=\cosh (t\mathrm{/}2{)}^{-n}{Q}_{jk}(r)N_k^2$, and $Q_{j0}(r)=(-r{)}^j$. At $j=k=0$ this agrees with [F2].

For basis vectors $e_j,e_k$, step 1.2 and boundedness of the fixed polynomial $Q_{jk}$ on $[-1,1]$ give $\mathrm{\mid }⟨{\pi }_n(k_1{a}_t{k}_2)e_j,e_k⟩\mathrm{\mid }\le C_{jk}\cosh (t\mathrm{/}2{)}^{-n}\le 2^n{C}_{jk}{e}^{-nt\mathrm{/}2}$ for $k_1,k_2\in K$ and $t\ge 0$, since each K-factor acts on its weight vector by a scalar of modulus one. Thus $\mathrm{\mid }⟨{\pi }_n(g)e_j,e_k⟩{\mathrm{\mid }}^2$ is a continuous K-bi-invariant function and [F3] gives its integral at most $2\pi (2^n{C}_{jk}{)}^2{\int }_0^{\mathrm{\infty }}{e}^{-nt}\sinh (t)dt\le \pi (2^n{C}_{jk}{)}^2\mathrm{/}(n-1)<\mathrm{\infty }$. For finite expansions $v={\sum }_j{v}_j{e}_j$ and $w={\sum }_k{w}_k{e}_k$, the coefficient is the finite sum ${\sum }_{j,k}{v}_j\overline{w_k}⟨{\pi }_n(g)e_j,e_k⟩$; the pointwise Cauchy–Schwarz inequality bounds its squared modulus by $\left({\sum }_{j,k}\mathrm{\mid }{v}_j\overline{w_k}{\mathrm{\mid }}^2\right){\sum }_{j,k}\mathrm{\mid }⟨{\pi }_n(g)e_j,e_k⟩{\mathrm{\mid }}^2$. The latter is integrable as a finite sum, proving the assertion for all K-finite $v,w$.

Put $e_0=f_{n,0}\mathrm{/}{N}_0$ and define $\mathrm{\Phi }(v)(g):=⟨{\pi }_n(g^{-1})v,e_0⟩$ for K-finite $v$. Unitarity gives $\mathrm{\Phi }(v)(g)=\overline{⟨{\pi }_n(g)e_0,v⟩}$, so step 2.1 shows $\mathrm{\Phi }(v)\in L^2(G)$; linearity follows from the first-variable-linear pairing. For $t\ge 0$, step 1.2 gives $\mathrm{\Phi }(e_j)(a_t)=⟨{\pi }_n(a_{-t})e_j,e_0⟩=\frac{N_0}{N_j}\tanh (t\mathrm{/}2{)}^j\cosh (t\mathrm{/}2{)}^{-n}$. For $k_1,k_2\in K$, the K-eigenvector identities give $\mathrm{\Phi }(e_j)(k_{1}g{k}_2)={\chi }_j(k_1{)}^{-1}\overline{{\chi }_0(k_2)}\mathrm{\Phi }(e_j)(g)$, so this basis coefficient's modulus is K-bi-invariant.

Applying [F3] to $\mathrm{\mid }\mathrm{\Phi }(e_j){\mathrm{\mid }}^2$ and setting $r=\tanh (t\mathrm{/}2)$ yields $\mathrm{\parallel }\mathrm{\Phi }(e_j){\mathrm{\parallel }}_2^2=2\pi \frac{B_0}{B_j}{\int }_0^{\mathrm{\infty }}\tanh (t\mathrm{/}2{)}^{2j}\cosh (t\mathrm{/}2{)}^{-2n}\sinh (t)dt=2\pi \frac{B_0}{B_j}2B_j=4\pi B_0=\frac{4\pi }{n-1}$. Indeed, $\sinh (t)dt=4r(1-r^2{)}^{-2}dr$, $\cosh (t\mathrm{/}2{)}^{-2n}=(1-r^2{)}^n$, and $x=r^2$ reduces the radial integral to $2{\int }_0^1{x}^j(1-x{)}^{n-2}dx=2B_j$. Each $B_j$ is finite and positive, and $B_0=1\mathrm{/}(n-1)$. Also, $(\lambda (k_{\theta })\mathrm{\Phi }(e_j))(g)=\mathrm{\Phi }(e_j)(k_{-\theta }g)=⟨{\pi }_n(g^{-1}{k}_{\theta })e_j,e_0⟩={\chi }_j(k_{\theta })\mathrm{\Phi }(e_j)(g)$. If $j\mathrm{\ne }k$, choose $\theta$ with ${\chi }_j(k_{\theta })\mathrm{\ne }{\chi }_k(k_{\theta })$; unitarity of $\lambda$ then gives $⟨\mathrm{\Phi }(e_j),\mathrm{\Phi }(e_k)⟩={\chi }_j(k_{\theta })\overline{{\chi }_k(k_{\theta })}⟨\mathrm{\Phi }(e_j),\mathrm{\Phi }(e_k)⟩$, so this inner product is zero. Thus, for $C_n:=4\pi \mathrm{/}(n-1)$ and every K-finite $v={\sum }_j{v}_j{e}_j$, $\mathrm{\parallel }\mathrm{\Phi }(v){\mathrm{\parallel }}_2^2=C_n{\sum }_j\mathrm{\mid }{v}_j{\mathrm{\mid }}^2=C_n\mathrm{\parallel }v{\mathrm{\parallel }}^2$.

The K-finite span is dense in ${\mathcal{H}}_n^{+}$ by [F1]. Therefore $C_n^{-1\mathrm{/}2}\mathrm{\Phi }$ extends uniquely by continuity to a linear isometry $J_n^{-}:{\mathcal{H}}_n^{+}\to L^2(G)$. Its image is closed: if $J_n^{-}{v}_m$ converges, the isometry identity makes $(v_m)$ Cauchy, and completeness of ${\mathcal{H}}_n^{+}$ gives a limit whose image is the stated range limit.

For any $v\in {\mathcal{H}}_n^{+}$, choose K-finite $v_m\to v$. Then $J_n^{-}{v}_m\to J_n^{-}v$ in $L^2(G)$, so [F5] gives a subsequence converging almost everywhere to a representative of $J_n^{-}v$. At every $g\in G$, unitarity gives $⟨{\pi }_n(g^{-1})v_m,e_0⟩\to ⟨{\pi }_n(g^{-1})v,e_0⟩$; hence $J_n^{-}v$ is almost everywhere equal to $C_n^{-1\mathrm{/}2}⟨{\pi }_n(g^{-1})v,e_0⟩$. For $h\in G$, the coefficient identity $C_n^{-1\mathrm{/}2}⟨{\pi }_n(g^{-1}){\pi }_n(h)v,e_0⟩=C_n^{-1\mathrm{/}2}⟨{\pi }_n((h^{-1}g{)}^{-1})v,e_0⟩=(\lambda (h)J_n^{-}v)(g)$ holds almost everywhere, using preservation of null sets by left translation. Thus $J_n^{-}{\pi }_n(h)=\lambda (h)J_n^{-}$; since ${\pi }_n(h)$ is onto, the closed range of $J_n^{-}$ is G-invariant.

Complex conjugation $C:{\mathcal{H}}_n^{+}\to {\mathcal{H}}_n^{-}$ is antiunitary and satisfies $C{\pi }_n(h)={\pi }_n^{-}(h)C$ by the model definition. Complex conjugation $C_G$ on $L^2(G)$ is antiunitary and commutes with $\lambda (h)$, because $\lambda (h)$ acts by real-variable translation. The complex-linear map $J_n^{+}:=C_G{J}_n^{-}{C}^{-1}$ is therefore an isometric intertwiner of $D_n^{+}$ with $\lambda$, with closed G-invariant range. The same conjugation identity shows that every K-finite matrix coefficient of $D_n^{+}$ is the complex conjugate of one for $D_n^{-}$, so it too lies in $L^2(G)$.