Definition (Lie algebra representation) A representation of a Lie algebra \(\mathfrak{g}\) is a \(\mathbb{C}\)-vector space \(V\) together with a map of Lie algebras \(\rho : \mathfrak{g}\rightarrow \mathfrak{gl}(V)\).
Let \(H=\begin{pmatrix} 1&0\\ 0& -1\end{pmatrix} , X=\begin{pmatrix} 0&1\\ 0& 0\end{pmatrix}\), and \(Y=\begin{pmatrix}0&0\\ 1& 0\end{pmatrix}\). We have \([H,X]=2X\) ,\([H,Y=-2Y]\), and \([X,Y]=H\).
If \(V\) is an irrep of \(\mathfrak{sl}_2(\mathbb{C})\), on which \(H\) acts diagonally via the character decomposition \(V=\bigoplus_{\alpha\in \mathbb{C}}V_\alpha\). In other words, on each \(V_\alpha\), \(H\) acts as \(Hv=\alpha v\).
If \(v\in V_\alpha\), then \(H(X(v)) = XH(v) + [H,X]v = (\alpha+2)X(v)\) hence \(X\) corresponds to a map \(V_{\alpha}\rightarrow V_{\alpha+2}\).
Likewise, one can see that \(Y\) corresponds to a mpa \(V_{\alpha}\rightarrow V_{\alpha-2}\).
If we take \(V\) finite-dimensional, there is only finitely many \(\alpha\) such that \(V_\alpha\neq 0\), So given any \(\alpha\) we have a finite chain \[ 0\leftarrow\cdots\overset{Y}{\leftarrow} V_{\alpha-2}\overset{Y}{\leftarrow}V_{\alpha} {\overset{Y}{\leftarrow}} V_{\alpha+2} \overset{Y}{\leftarrow} V_{\alpha+4} \leftarrow \cdots \leftarrow 0\]
There are also arrows in the reverse direction given by \(X\).
Fact : By irreducibility of \(V\), all \(V_\alpha\) are \(1\)-dimensional.
Such a chain give a subrepresentation so by irreducibility of \(V\), there is only one chain. We call the highest \(\beta\) such that \(V_\beta\neq 0\) the highest weight of \(V\).
Fact : \(\beta\in \mathbb{Z}_{\geq 0}\). Conversely, for any \(n\in \mathbb{Z}_{\geq 0}\) there exists a unique irreducible representation of \(\mathfrak{sl}_2(\mathbb{C})\) for which \(n\) is the highest weight. We denote this representation by \(V^{(n)}\), it is \(n+1\) dimensional (look at the chain from \(V^{(n)}_{-n}\) to \(V^{(n)}_n\)).
Representation of \(\mathfrak{sl}_2(\mathbb{C})\) on itself via the adjoint representation. Consider the adjoing operator \(\mathrm{ad} : \mathfrak{sl}_2\rightarrow \mathfrak{gl}(\mathfrak{sl}_2(\mathbb{C}))\), then as seen at the start of the section, one has \(\mathrm{ad}H(X) = [H,X] = 2X\), \(\mathrm{ad}H(Y)= -2Y\) and \(\mathrm{ad}H(H)=0\). So \[V^{(2)}\cong \mathfrak{sl}_2(\mathbb{C}) = \left\langle H\right\rangle \oplus \left\langle X\right\rangle \oplus \left\langle Y\right\rangle.\]
The standard representation. \(\mathfrak{sl}_2(\mathbb{C})\rightarrow \mathfrak{gl}_2(\mathbb{C})\). Take \(e_1=\begin{pmatrix}1\\0\end{pmatrix}, e_2=\begin{pmatrix}0\\1\end{pmatrix}\) the standard basis. Then \(He_1=e_1\) and #He_2=-e_2$ so \(\mathbb{C}^2\cong V^{(1)}\).
Fact : \(V=\bigoplus_{\alpha\in \mathfrak{h}^\star}V_\alpha\), where \(\mathfrak{h}^\star\) denotes the dual of \(\mathfrak{h}\).
Let \(v\in V_\alpha\), we have \(Hv=\alpha(H) v\) for \(H\in \mathfrak{h}\).
We have the decomposition \(\mathfrak{g}=\mathfrak{h}\oplus \bigoplus_{\alpha\in \mathfrak{h}^\star}\mathfrak{g}_\alpha\). The subset of \(\alpha \in \mathfrak{h}^\star\) such that \(\mathfrak{g}_\alpha\neq 0\) is called the root system of \(\mathfrak{g}\), we denote it by \(R\).
For the case of \(\mathfrak{sp}_4(\mathbb{C})\) we have \(\mathfrak{h} = \left\langle E_{1,1}-E_{3,3},E_{2,2}-E_{4,4} \right\rangle\), the character lattice of \(\mathfrak{h}\) is \(\mathbb{Z}^2\). The choice of highest weight here is not unique, since we can choose any element with maximum length.
Let \(e_i\in\mathfrak{h}^\star\) be such that $e_i(H) $ is the \(i\)th diagonal entry of \(H\), with \(H\in \mathfrak{H}\) for \(i\in\{1,2\}\).
The roots are \(R = \{\pm e_1,\pm e_2,\pm e_1\pm e_2 \}\). Identify \(e_1\) with \((1,0)\) and \(e_2\) with \((0,1)\) in \(\mathbb{R}^2\), call the Weyl chamber the cone between \(2e_1\) and \(e_1+e_2\). Each point on the lattice generated by \(2e_1,e_1+e_2\) inside this chamber corresponds to a unique finite-dimensional irreducible representation of \(\mathrm{sp}_4(\mathbb{C})\) on which this point corresponds to the highest weight. This sublattice generated by \(2e_1\) and \(e_1+e_2\) is the root lattice, here it is an index \(2\) sublattice of the weight lattice which is \(\mathbb{Z}\) seens as the character of \(\mathfrak{h}\).
Denote this representation by \(\Gamma_{a,b} = \Gamma_{a(2e_1)+b(e_1+e_2)}\).
\(\Gamma_{0,0}\) is the trivial representation.
** Definition** A manifold (Riemannian, complex) is homogeneous if its automorphism group acts transitively on \(M\).
We say \(M\) is symmetric if it is homogeneous and there is a point \(p\in M\) and an automorphism \(s_p:M\rightarrow M\) such that
Definition A Hermitian metric on a complex manifold \(M\) is a Riemann metric \(g\) together with a complex structure \(J\) (acts as complex structure, i.e. \(J^2=-1\) on the tangent spaces, defines a \(J\)-integrable notion) such that \(g_p(Jx,Jy) = g_p(x,y)\) for all tangent vectors.
A Hermitian manifold \((M,g)\) is a complex manifold with Hermitian metric \(g\).
Fact For any Hermitian symmetric space (Hermitian manifold, symmetric as a complex manifold) \(M\), we can write \(M=M_e\times M_c\times M_{nc}\) where \(M_e\) is of Euclidean type (zero curvature), of the form \(\mathbb{C}^n/\Lambda\) for some lattice, \(M_c\) is of compact type (nonnegative curvature), e.g. \(\mathbb{P}^1(\mathbb{C})\), and \(M_{nc}\) of non-compact type (nonpositive curvature).
** Main example : ** Siegel upper-half plane.
Define the upper-half plane by \[\mathbb{H}_g=\left\lbrace Z\in M_{n}(\mathbb{C}) : Z^T=Z, \mathrm{Im}(Z)>0 \right\rbrace\subset \mathbb{C}^{g(g+1)/2}.\] We define the transitive action of \(\mathrm{Sp}_{2g}(\mathbb{R})\) on \(\mathbb{H}_g\) by \[M= \begin{pmatrix} A&B\\C&D\end{pmatrix}\in \mathrm{Sp}_{2g}:Z\mapsto (AZ+B)(CZ+D)^{-1}\]
Fact. \(\mathrm{Aut}(\mathbb{H}_g) = \mathrm{Sp}_{2g+1}(\mathbb{R})/\{\pm 1\}\). As a \(\mathbb{C}\)-manifold, \(\mathbb{H}_g\) is homogeneous. The matrix \(\begin{pmatrix} 0& -I_g\\ I_g&0\end{pmatrix}\) is an involution of \(\mathbb{H}_g\) with \(iI_g\) as isolated fixed point, this gives us that \(\mathbb{H}_g\) is symmetric.
We have a map \(\mathbb{H}_g\rightarrow \mathbb{D}_g\), the open unit ball in \(\mathbb{C}^{g(g+1)/2}\), is is a bounded symmetric domain. It has a canonical hermitian metric called the Bergman metric.
There is a diffeomorphism \(\mathrm{Sp}_{2g}/\mathcal{U}(g)\rightarrow \mathbb{H}_g\) where \(\mathcal{U}(g)\) is the stabilizer of \(iI_g\).
Goal : Give \(\mathrm{Sp}_{2g}(\mathbb{R})/\mathcal{U}(g)\) the structure of a Hermitian Symmetric Domain such that this diffeomorphism is a holomorphism. We want to do it without just pulling back the structure of \(\mathbb{H}_g\), but define it intrinsically.
\(\mathrm{Sp}_{2g}\) is a symmetric space as a Riemannian manifold with the Poincaré metric.
Cartan Decomposition \(\mathfrak{sl}_{2g}(\mathbb{R})=\mathfrak{h}\oplus\mathfrak{p}\) where \(\mathfrak{p} = T_e(\mathrm{Sp}_{2g}(\mathbb{R})/\mathcal{U}(g))\). We want \(J_e : \mathfrak{p}\rightarrow \mathfrak{p}\) st \(\mathrm{J}_e^2=-1\). Consider the homomorphism \(u: S^1 \rightarrow \mathrm{Sp}_{2g}(\mathbb{R})\) defined by \(x+iy \mapsto \begin{pmatrix}xI_g& -yI_g\\ y I_g & x I_g\end{pmatrix}\), we have \(U(i) = \begin{pmatrix} 0&-I_g\\ I_g& 0\end{pmatrix}\) is the multiplication by \(i\).