Root system for \(\mathrm{sl}(2)\)
Definition. The symplectic group \[\mathrm{Sp}_{2n}({\mathbb{K}}) = \left\lbrace X\in \mathrm{SL}_{2n}(\mathbb{K}) : X^T JX = J\right\rbrace ,\] where \(J=\begin{bmatrix} 0 & I_n \\ -I_n & 0\end{bmatrix}\). A block matrix \(X = \begin{bmatrix} A & B \\ C & D\end{bmatrix}\in \mathrm{Sp}_{2n}(\mathbb{K})\) if and only if \(A^T C= C^T A\), \(B^T D = D^T B\), and \(A^TD-C^TB=1\).
Note that this is equivalent to the orthogonal group, seen as automorphism group of the standard quadratic form, where this is the automorphism group of the standard symplectic form on \(\mathrm{K}^{2n}\) where \([(q,p),(q',p')]\mapsto p\cdot q'-q\cdot p'\), which has Gram matrix \(J\).
We can also define the group of similitudes
\[\mathrm{GSp}_{2n}({\mathbb{K}}) = \left\lbrace X\in \mathrm{SL}_{2n}(\mathbb{K}) : JX^T JX \in \mathbb{K} \right\rbrace ,\] so \(X\in \mathrm{GSp}_{2n}(\mathbb{K})\) if \(X^TJX = t(X)J\) where \(t(X)\in \mathbb{K}\). The map \(X\mapsto t(X)\) is the multiplier, it is a character : \(\mathrm{GSp}_{2n}\rightarrow\mathrm{GL}_1 =\ mathbb{G}_m\).
We get the corresponding Lie algebra : \[ \mathfrak{sp} (\mathbb{K}) = \left\lbrace x\in \mathfrak{sl}_{2n}(\mathbb{K}) : X^TJ+JX = 0 \right\rbrace.\] Note that the assumption \(x\in \mathfrak{sl}_{2n}(\mathbb{K})\) is equivalent to taking \(x\in \mathfrak{gl}_{2n}(\mathbb{K})\) when \(\mathrm{char}(\mathbb{K}) \neq 2\).
Recall that \[\mathfrak{sl}_{2n}(\mathbb{K}) = \left\lbrace X\in \mathfrak{gl}_n(\mathbb{K}) : \mathrm{Tr}(X) = 0\right\rbrace .\]
The unitary group is \[U(n) = \left\lbrace X\in \mathrm{GL}_n(\mathbb{C}) : U^\star U = I_n \right\rbrace, \] where \(U^\star = \overline{U}^{T}\) denotes the conjugate transpose. Again, the corresponding lie algebra is \[\mathfrak{u}(n) = \left\lbrace X\in \mathfrak{gl}_n(\mathbb{C}) : X^\star + X = 0 \right\rbrace. \]
The special orthogonal group is
\[\mathrm{SO}(n) = \left\lbrace X\in \mathrm{GL}_n(\mathbb{R}) : X^TX = I_n \right\rbrace, \] and its Lie algebra
\[\mathfrak{so}(n) = \left\lbrace X\in \mathfrak{gl}_n(\mathbb{R}) : X^T+X = 0 \right\rbrace. \]
One can define \(\mathrm{Sp}(n) = \mathrm{Sp}_{2n}(\mathbb{C})\cap U(2n)\) the compact symplectic group, and \(\mathfrak{sp}_{2n} = \mathfrak{sp}_{2n}(\mathbb{C})\cap \mathfrak{u}(n)\).
We are interested in studying semisimple Lie algebras, i.e. algebras with no nontrivial bilateral ideal.
Let \(G=\mathrm{Sp}_{2n}(\mathbb{C})\) and \(\mathfrak{g} = \mathfrak{sp}_{2n}(\mathbb{C})\).
Let \[\mathfrak{h} = \left\lbrace H\in \mathfrak{g} : H = \mathrm{diag}(h_1,\cdots, h_n,-h_1,\cdots,-h_n) \right\rbrace ,\] it is an abelian subalgebra of \(\mathfrak{g}\).
Let \(E_{i,j}\) denote the canonical basis element of \(\mathfrak{gl}_{2n}\). and \(e_i\in \mathfrak{h}^\star\) as \(e_i(H) = h_i\).
For \(H\in\mathfrak{h}\) we have a corresponding adjoint map \(\mathrm{ad}H : \mathfrak{g}\rightarrow \mathfrak{g} : X\mapsto [H,X]\) where \([,]\) denotes the usual Lie bracket \([A,B] = AB-BA\).
For \(1\leq i,j\leq n\) we have \(\mathrm{ad}H(E_{i,j}) = [H,E_{ij}] = (e_j(H) - e_j(H))E_{i,j}\) so \(E_{i,j}\) is a simultaneous eigenvector for all \(\mathrm{ad}H, \ H\in \mathfrak{h}\), with eigenvalues \((e_i-e_j)(H)\).
In general, if we let \(1i,j\leq 2n\) we get \(\mathrm{ad}H(E_{ij}) = (\pm e_i(H)\pm e_j(H))E_{ij}\).
Define the set of roots \(\Delta = \left\lbrace e_i-e_j : 1\leq i\neq j\leq n \right\rbrace \cup \left\lbrace 2e_i : 1\leq i\leq n\right\rbrace\).
We have a root space decomposition \[\mathfrak{g} = \mathfrak{h}\bigoplus \left(\bigoplus_{\alpha\in \Delta}\mathfrak{g}_\alpha\right) ,\] where \(\mathfrak{g} = \left\lbrace X\in \mathfrak{g} : (\mathrm{ad}H)(X) = \alpha(X)X \ \forall H\in\mathfrak{h} \right\rbrace.\)
One has
\[[g_\alpha,g_\beta] = \left\lbrace \begin{matrix} \mathfrak{g}_{\alpha+\beta} \text{ if }\alpha+\beta\text{ is a root}\\\quad 0 \text{ if }\alpha+\beta\text{ is not a root or }0\\ \subseteq \mathfrak{h}\text{ if }\alpha+\beta = 0\end{matrix} \right. \]
Root system for \(\mathrm{sl}(2)\)
Definition. A linear connected reductive group is a closed connected subgroup of real/complex matrices that are stable under conjugate transpose. This is equivalent to the triviality of the unipotent radical (not trivial).
Definition. A linear connected semisimple group is a linear connected reductive group with finite centre.
Examples. \(\mathrm{SL}_n\), \(\mathrm{SO}_n\), \(\mathrm{Sp}_{2n}\) are all connected semisimple linear groups.
Cartan involution. Let \(G\) be a linear connected semisimple group, define \[\Theta : \begin{matrix}G\rightarrow G \\ X \mapsto (X^\star)^{-1}\end{matrix}\]
Note that we have \(\Theta^2 = \mathrm{id}\).
The differential of \(\Theta\) at the identity is denoted by \(\theta\) and it gives an automorphism of \(\mathfrak{g}\), given by \(\theta(X) = -X^\star\). This \(\theta\) is the Cartan involution, and is indeed an involution since \(\theta(XY)=\theta(Y)\theta(X)\), \(\theta(\lambda X) = \overline{\lambda}\theta(X)\) and \(\theta^2 = \mathrm{id}\).
Clearly the polynomial \(t^2-1\) kills \(\theta\) so the eigenvalues of \(\theta\) are \(\pm1\), let \(\mathfrak{k},\mathfrak{p}\) denote the eigenspaces of \(1\) and \(-1\) respectively. We have the descriptions :
\(\mathfrak{k} =\) Skew-Hermitian matrices and \(\mathfrak{p}=\) Hermitian matrices.
This gives us the Cartan decomposition \[\mathfrak{g} = \mathfrak{k}\oplus \mathfrak{p} .\]
We have \([\mathfrak{k},\mathfrak{k}]\subset \mathfrak{k}\), \([\mathfrak{k},\mathfrak{p}] \subset \mathfrak{p}\) and \([\mathfrak{p},\mathfrak{p}]\subset \mathfrak{k}\). In particular, \(\mathfrak{k}\) is a Lie subalgebra of \(\mathfrak{g}\).
Proposition 1. If \(G\) is a linear connected semisimple group, then \(\mathfrak{g}\) is semisimple. If \(G\) is a linear connected reductive group then \[\mathfrak{g} = Z(\mathfrak{g})\oplus [\mathfrak{g},\mathfrak{g}], \] where \(Z(\mathfrak{g})\) denotes the centre of \(\mathfrak{g}\).
Proposition 2. If \(G\) is a real linear connected reductive group then \(K\) is compact and connected and it is a maximal compact subgroup of \(G\). Its Lie algebra is \(\mathfrak{k}\). The map \[ \begin{matrix} K\times \mathfrak{p} \rightarrow G \\ (k,x)\mapsto k\mathrm{exp}(X)\end{matrix}\] is a diffeomorphism onto \(G\).
Idea of the proof of connectedness of \(\mathrm{SL}_n\) : It acts transitively on the column vectors of \(\mathbb{C}^n\backslash\{0\}\). The subgroup fixing the alst standard basis vector is \(\begin{bmatrix}SL_{n-1}(\mathbb{C}) & 0 \\ \mathbb{C}^{n-1} & 1\end{bmatrix} \cong \mathrm{SL}_{n-1}(\mathbb{C})\ltimes \mathbb{C}^{n-1}\).
We have
\[\mathrm{SL}_n(\mathbb{C}) / ( \mathrm{SL}_{n-1}(\mathbb{C})\ltimes \mathbb{C}^{n-1}) \cong \mathbb{C}^n\backslash\{0\} .\] Then use induction.
We can consider two involutions on \(\mathfrak{sl}_{2n}(\mathbb{C})\), the fixed points under the involution \(X\mapsto \overline{X}\) is simply \(\mathfrak{sl}_{2n}(\mathbb{R})\).
The fixed vectors under the Cartan involution \(\theta\) are just \(\mathfrak{k} = \mathfrak{so}_{2}(\mathbb{R}).\)
Both \(\mathfrak{sl}_2(\mathbb{R})\) and \(\mathfrak{su}(2)\cong \mathfrak{so}_3(\mathbb{R})\) ``complexify'' to \(\mathfrak{sl_2}(\mathbb{C})\), we say that the former is split.
In \(\mathfrak{sp}_{2n}(\mathbb{R})\), the maximum compact subalgebra is isomorphic to \(\mathfrak{u}(n)\).
In \(\mathfrak{sp}_{2n}(\mathbb{R})\), take \(X = \begin{bmatrix}A&B\\ C&D\end{bmatrix}\), the condition \[\begin{bmatrix}A^T&C^T\\ B^T&D^T\end{bmatrix} \begin{bmatrix}0&I\\ -I&0\end{bmatrix} =\begin{bmatrix}0&-I\\ I&0\end{bmatrix} \begin{bmatrix}A&B\\ C&D\end{bmatrix}\] becomes \[ \begin{bmatrix}-C^T&A^T\\ D^T&B^T\end{bmatrix} = \begin{bmatrix}-C&-D\\ A&B\end{bmatrix},\] so we get \(C=C^T\), \(B=B^T\) and \(A^T = -D\).
And we have \[\begin{bmatrix}A&B\\ C&-A^T\end{bmatrix} = \begin{bmatrix}-A^T&-C^T\\ =B^T&A\end{bmatrix} ,\] so the matrix is of the form \(\begin{bmatrix}A&B\\ -B&A\end{bmatrix}\), \(A,B\) being real matrices, which by considering the embedding \(\mathbb{C}\) into \(\mathfrak{gl}_2(\mathbb{R})\), we identify \(\begin{bmatrix}A&B\\ -B&A\end{bmatrix} = A+iB\) which identifies our subalgebra of fixed points with \(\mathfrak{u}(n)\).
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Let \(G\rightarrow \mathfrak{g} = \mathfrak{k}\oplus\mathfrak{p}\) defined by \((X,Y) = \mathrm{Tr}_{\mathfrak{g}}(\mathrm{ad}_x\mathrm{ad}_Y)\).
Restriction to \(\mathfrak{k},\mathfrak{p}\) is definite (so \(K\subset \mathcal{O}_{(\dot) \text{ or }k}\)).
Restriction to \(\mathfrak{p}\) is \(K\)-invariant so \(\mathfrak{p}\) can be identified with the tangent space \(T_e(G/K)\) with \(K-\)invariant inner product. So \(G/K\) has a \(G\)-invariant metric. \(G= K\exp(\mathfrak{p})\) so \(G/K\cong \mathrm{exp}(\mathfrak{p})\).
We have \(\theta|_{\mathfrak{p}} = -\mathrm{id}_\mathfrak{p}\).
The map \(\Theta : G/K\rightarrow G/K\) is an isometry, it reverses geodesics through \(eK\).
\(G/K\) is the space of Cartan involutions of \(G\), it is a ``symmetric space''.
\(L\subset G\) is compact then \(L\) contains a fixed point \(gK\) so \(L\subset gKg^{-1}\), hence \(K\) is a maximal compact subgroup and all maximal compact subgroups are conjugate.
\[P_n=\left\lbrace Q\in M_n(\mathbb{R}) : Q\text{ summetric positive definite with } \det(Q) = 1 \right\rbrace.\] The space \(\mathrm{SL}_n(\mathbb{R})\) acts on \(P_n\) via \(g\cdot Q = gQg^T\). This action is transitive, the stabilizer of \(I_n\) is \(\mathrm{SO}(n)\) hence \(P_n\cong \mathrm{SL}_n(\mathbb{R})/\mathbb{SO}(n).\) With a \(Q\) we define \(g_Q(X,Y)= \mathrm{Tr}(Q^{-1}XQ^{-1}Y)\).
\(\mathrm{SL}_2(\mathbb{R})\) acts on \(\{x+iy>0\}\) via \(g\dot z =\frac{ax+b}{cz+d}.\) We have \(y\left(\frac{az+b}{cz+d}\right)=\frac{y(z)}{|cz+d|^2}>0\) where \(y(z) =\mathrm{Im}(z)\) and \(\mathrm{Stab}_{\mathrm{SL_2}(\mathbb{R})}(i) = \mathrm{SO}(2)\).
More generally, we can define this upper-half plane as \[ \mathbb{H}_n =\left\lbrace Z\in M_n(\mathbb{C}) : Z\text{ symmetric and }y(Z)=\mathrm{Im}(Z)\text{ is positive definite} \right\rbrace.\]
Let \(g=\begin{bmatrix} A&B\\ C&D \end{bmatrix}\in \mathrm{GL}_{2n}(\mathbb{R})\) try setting \(g\cdot Z= (AZ+B)(CZ+D)^{-1}\)
Check : This is \(\mathbb{H}_n\) if and only if \(g\in \mathrm{Sp}_{2n}(\mathbb{R})\). We get a transitive ation of \(\mathrm{Sp}_{2n}(\mathbb{B})\) on \(\mathbb{H}_n\), the stabilizer of \(iI_n\) is \(\mathrm{U}(n)\).
In the 2-dimensional case, we know that we can send the upper-half plane to a circle via the Cayley transform \(z\mapsto \frac{1+z}{1-z}\). Here again we have a corresponding bounded symmetric domain : Open bounded subset \(D\subset \mathbb{C}^n\) where the group of biholomorphism is ``large enough''. In the general case, it's also probably the map \(Z\mapsto (I+Z)(I-Z)^{-1}\).
On is there is a canonical metric called Bargmann metric, invariant by \(\mathrm{Aut}(D)\). This with Cauchy formula lets us ling purely geometric volumes to algbraic quantities.