A nonreduced bc root system from a real form

On ${\mathfrak{s}\mathfrak{l}}_m(\mathbb{C})$ define $\sigma (Z)=-I{Z}^{\ast }I$. It is a conjugate-linear involutive Lie automorphism: adjoint reverses products, so the minus sign preserves the commutator, and $I^2=1$. Its fixed space is exactly the trace-zero algebra in the statement. Every $Z$ decomposes uniquely as $X+iY$, where $X=(Z+\sigma Z)\mathrm{/}2$ and $Y=(Z-\sigma Z)\mathrm{/}(2i)$ are fixed by $\sigma$. Thus this fixed real algebra has complexification ${\mathfrak{s}\mathfrak{l}}_m(\mathbb{C})$ and real dimension $m^2-1$. A real basis of it is a complex basis of the complexification; the adjoint matrices of real elements in that basis have the same real and complex traces. Consequently its real Killing form is the restriction $B(X,Y)=2m\mathrm{tr}(XY)$ by [L1]. This establishes the real-form assertion rather than attributing it to the compact unitary-group example.

Solving $X^{\ast }I+IX=0$ gives the stated skew-Hermitian blocks $A,E$ and the off-diagonal pair $C,C^{\ast }$, with the single imaginary trace constraint. The map $\theta (X)=-X^{\ast }$ preserves this algebra, squares to the identity, and preserves brackets by the same adjoint calculation as in step 1.1. Moreover $B_{\theta }(X,Y)=2m\mathrm{tr}(X{Y}^{\ast })$ on this real space, and $B_{\theta }(X,X)=2m{\sum }_{i,j}\mathrm{\mid }{X}_{ij}{\mathrm{\mid }}^2>0$ for $X\mathrm{\ne }0$. The form is real by step 1.1 and symmetric by conjugate symmetry of the displayed trace. Hence $B$ is nondegenerate: if $B(X,Y)=0$ for all $Y$, take $Y=\theta X$. By [L1] the algebra is semisimple, and by [L2] $\theta$ is a Cartan involution with exactly the displayed ${\mathfrak{k}}_0,{\mathfrak{p}}_0$.

Simultaneously diagonalize the matrices $H_D$ on ${\mathbb{C}}^m$ using the basis $v_i^{+}=e_i+e_{p+i}$, $v_i^{-}=e_i-e_{p+i}$ for $1\le i\le p$, and $z_k=e_{2p+k}$ for $1\le k\le r$. These have respective weights $f_i,-f_i,0$. Their independence follows separately on each two-dimensional plane and on the remaining coordinates. In the corresponding matrix-unit basis of $\mathrm{End}({\mathbb{C}}^m)$, the operator taking a basis vector of weight $\nu$ to one of weight $\mu$ has adjoint weight $\mu -\nu$. These units form a simultaneous eigenbasis. Every nonzero-weight unit is traceless, while the zero-weight space in ${\mathfrak{s}\mathfrak{l}}_m$ is the trace-zero part of its zero-weight endomorphism space.

The $H_D$ commute, since both products have diagonal blocks $D{D}^{\mathrm{\prime }}$ and $\mathrm{diag}(D{D}^{\mathrm{\prime }},0_r)$. To compute their centralizer in ${\mathfrak{p}}_0$, put $C=[C_1{C}_2]$. Vanishing of $[H_D,Y]$ for all real diagonal $D$ gives $D{C}_1^{\ast }=C_{1}D$, $D{C}_1=C_1^{\ast }D$, and $D{C}_2=0$. Taking $D=1$ in the last equation gives $C_2=0$. In the first equation the $(i,j)$ entry reads $d_i\overline{(C_1{)}_{ji}}=(C_1{)}_{ij}{d}_j$. Independent $d_i,d_j$ force off-diagonal entries to vanish, and the diagonal entries are real. Conversely every real diagonal $C_1$ satisfies all equations. Thus this centralizer is exactly $\mathfrak{a}$, proving maximality: any abelian subspace containing it lies in that centralizer. Its dimension is $p$, so the real rank is $p$.

Counting the units in step 2.2 gives the complete nonzero weight list and complex dimensions. For distinct $i,j$, the weight $f_i-f_j$ has the two ordered pairs $(f_i,f_j)$ and $(-f_j,-f_i)$; $f_i+f_j$ has $(f_i,-f_j)$ and $(f_j,-f_i)$. Reversing pairs gives the negatives, each also of dimension two. Weight $f_i$ has $r$ pairs $(f_i,0)$ and $r$ pairs $(0,-f_i)$, giving dimension $2r$; its negative has the same dimension. Weight $2f_i$ has only the pair $(f_i,-f_i)$, giving dimension one, and similarly for its negative. No other differences occur. The zero-weight endomorphisms have dimension $2p+r^2$, from the $2p$ separate nonzero-weight lines and the full endomorphisms of the $r$-dimensional zero space; trace zero imposes one independent condition, giving $2p+r^2-1$.

These complex dimensions equal the required real multiplicities. Indeed $\sigma$ fixes every $H_D$ and commutes with their adjoint action on a weight space of real weight $\lambda$. That complex weight space is therefore $\sigma$-stable. Its real fixed space is precisely ${\mathfrak{g}}_0^{\lambda }$, and every vector decomposes as $X+iY$ with both $X,Y$ in that fixed space by the formulas of step 1.1. A real basis of the fixed space is a complex basis of the weight space, so the dimensions agree. This applies also to weight zero, and proves a complete real simultaneous decomposition without dividing by $\lambda (H_D)$, which may vanish at particular $D$.

For clarity, the zero space in the original blocks has $C_1$ real diagonal, $C_2=0$, $E_{12}=E_{21}=0$, and $A=E_{11}$ diagonal and purely imaginary, while $E_{22}$ is an arbitrary skew-Hermitian $r$-by-$r$ matrix subject to $2\mathrm{tr}A+\mathrm{tr}{E}_{22}=0$. The $C$ equations follow as in step 3.1; the other equations are $D{E}_{11}=AD$, $D{E}_{12}=0$ and $E_{21}D=0$ for every $D$, which give exactly these conditions. The $C_1$ part is $\mathfrak{a}$ of dimension $p$; the other part is $Z_{{\mathfrak{k}}_0}(\mathfrak{a})$ of dimension $p+r^2-1$. In particular the zero space contains every $H_D$, as it must.

The real dimensions sum to $(2p+r^2-1)+4p(p-1)+4pr+2p=(2p+r{)}^2-1=m^2-1$, agreeing with step 1.1. Also $B(H_D,H_{D^{\mathrm{\prime }}})=4m{\sum }_i{d}_i{d}_i^{\mathrm{\prime }}$, so the dual inner product gives the $f_i$ equal lengths and mutual orthogonality. Reflections in $f_i$ or $2f_i$ negate one coordinate and reflections in $f_i\pm f_j$ are signed coordinate swaps; all preserve the displayed set. For denominator root $f_i$, $2f_i$, or $f_i\pm f_j$, the Cartan integer is respectively $2{\beta }_i$, ${\beta }_i$, or ${\beta }_i\pm {\beta }_j$ in these coordinates, always integral. The set is finite and spans, and it is exactly the standard $B{C}_p$ set in [L4]. It is nonreduced because both $f_i$ and $2f_i$ occur with positive multiplicities.

At $p=1<q$ the mixed-root family is empty and the roots are $\pm f_1,\pm 2f_1$ of multiplicities $2(q-1)$ and one. The zero space has dimension $(q-1{)}^2+1$, so the same count gives $q^2+2q$. The hypotheses exclude $p=0$ and $p=q$; in particular $q-p>0$ guarantees that the short roots counted above actually occur. This proves all assertions, retaining the standing AC assumption [A1] but using only finite matrix calculations.