Highest- and lowest-weight submodules at the exceptional parameters

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).