Talk 4 : Automorphic Forms and Vector Bundles

The goal is to interpret automorphic forms as algebraic sections of algebraic vector bundles on Shimura varieties \(\Gamma\backslash \mathbf{G}(\mathbb{R})/K\).

We focus on the case \(\mathbb{G}=\mathrm{Sp}_{2g}\).

Motivation : Use tools from algebraic geometry (sheaves+cohomology) to study automorphic forms (dimension formulas via Riemann-Roch).

Let \(f\) be a classical weight \(K\) modular form \(\Gamma\subset \mathrm{SL}_2(\mathbb{Z})\). Define \(F: \mathrm{SL}_2(\mathbb{R})\rightarrow \mathbb{C}\) by \(F(g)= j(g,i)^{-K}f(g\cdot i)\).

We check how \(F\) behaves under a representation of \(\mathrm{SO}(2)\) on the right. We get \[F(\gamma g \cdot K_\theta) = \underset{\text{rep of }\mathrm{SO}(2)}{\underbrace{e^{iK\theta}}}\underset{\Gamma-\text{invariance on the left}}{\underbrace{F(g)}}\] for all \(\gamma\in \Gamma, g\in \mathrm{SL})2(\mathbb{R})\) and \(K_\theta\in \mathrm{SO}(2)\).

Let \(G\) be a Lie group, \(K\subset G\) a closed Lie subgroup (assume \(K\) connected). The map \(G\rightarrow G/K\) has the structure of a principal \(K\)-bundle i.e.

• 1. Free right \(K\)-action on \(G\).
• 2. Fibers are diffeomorphisms to \(K\).
• 3. Locally trivial.

Let \((V,\sigma)\) be an \(r\)-dimensional complex representation of \(K\). Define \(E_K:= G\times_K V = G\times V/K\). The right action of \(K\) on \(G\times V\) is given by \(k\cdot(g,v)\mapsto (gk,\sigma(k^{-1})v)\).

Fact : \(\Pi : E_V \rightarrow G/K\) is a rank \(r\) smooth (complex) vector bundle (it is an associated vector bundle).

Definition. We say \(\Pi : E\rightarrow X\) is a smooth complex rank \(r\) vector bundle if \(\Pi\) is a smooth surjection between smooth manifolds such that

• 1. \(\Pi^{-1}(x) \cong \mathbb{C}^r\)
• 2. There is an open cover \(\{U_i\}\) of \(X\) with diffeomorphisms \(\phi_i : \Pi^{-1}(U_i)\rightarrow U_i\times \mathbb{C}^r\) with compatibility condition \(g_{i,j} = \phi_i\circ \phi_j^{-1} : U_i\cap U_j \rightarrow \mathrm{GL}_r(\mathbb{C})\), this map is called a transition function. Those maps must satisfy Cocycle condition. \(g_{ij}=g_{ik}g_{kj}\) on \(U_i\cap U_j\cap U_k\). Also we need the composition of \(\phi_i\) with projection on the first coordinate to be equal to \(\Pi\) on \(Pi^{-1}(U_i)\).

For associated bundles, the triangle commutes \[\mathrm{GL}(V)\overset{\sigma}{\leftarrow}K\overset{t_{ij}}{\leftarrow} U_i\cap U_j \overset{g_{ij}}{\rightarrow} \mathrm{GL}(V) .\]

Remarks. If \(X\) is a complex manifold or algebraic variety, a holomorphic (resp. algebraic bundle) is such that the transition functions \(g_{ij}\) are holomorphic (res. algebraic) + likewise for \(\Pi\).

Definition. A smooth section of a vector bundle \(\Pi: E\rightarrow X\) is a smooth map \(s:X\rightarrow E\) such that \(\Pi\circ s = \mathrm{Id}_X\).

A holomorphic (res. algebraic) section of a holomorphic (resp. algebraic) bundle is a section where \(s\) is also holomorphic (resp. algebraic). The set of smooth sections of a vector bundles is usually denoted \(\Gamma_{C^\infty}(E)\).

Key Lemma. We have the bijection \[\left\lbrace F:G\rightarrow V | F(gk)=\sigma(k^{-1})F(g) \ \forall g\in G,\ \forall k\in K \right\rbrace \overset{\cong}{\longrightarrow}\Gamma_{C^\infty}(E_v),\] \[F\mapsto s\text{, where } s(gK)=[(g,F(g))]\] \[\text{conversely} s \mapsto F\text{, defined by } F(g)=v,\text{ where }s(gK)=[(g,v)]. \]

Main example. Let \(\mathbf{G}\) be a connected semisimple algebraic group and \(K\subset \mathbf{G}(\mathbb{R})\) denote a maximal compact subgroup.

Suppose that \(X=\mathbf{G}(\mathbb{R})/K\) is a Hermitian symmetric domain.

Let \((V,\sigma)\) be a representation of \(K\). Let \(E_v = \mathbf{G}(\mathbb{R})\times_K V \rightarrow X\) the associated smooth complex vector bundle over \(X\). Let \(\Gamma\subset \mathbf{G}(\mathbb{R})\) a torsion-free lattice. Define a \(\Gamma\)-action on \(\mathbf{G}(\mathbb(R)\times V\) by \(\gamma \cdot (g,v)\mapsto (\gamma g,v)\).

Let \(E_{V,\Gamma} = \Gamma\backslash \mathbf{G}(\mathbb{R})\times_K V \rightarrow \Gamma\backslash X\) is a smooth complex vector bundle on \(\Gamma\backslash X\).

\[\Gamma_{C^\infty}(E_{V,\Gamma}) = \left\lbrace F:\mathbf{G}(\mathbb{R})\rightarrow V | F(\gamma\cdot g\cdot k) = \sigma(k^{-1})F(g)\ \forall \gamma\in \Gamma,\ g\in \mathbf{G}(\mathbb{R}),\ k\in K \right\rbrace.\]

These ``formally'' look like what we would call vector-valued automorphic forms, except \(E_{v,\Gamma}\) and its section are apriori smooth, not algebraic.

Borel Embedding.

A Hermitian symmetric domain \(X=\mathbf{G}(\mathbb{R})/K\) embeds into its ``compact dual'' \(\hat{X}=\mathbf{G}(\mathbb{C})/P\). The map \(\beta X\rightarrow \hat{X}\) is the Borel embedding.

Here \(P\) is some parabolic subgroup of \(\mathbf{G}(\mathbb{C})\) (i.e. \(\mathbf{G}(\mathbb{C})/P\) is a smooth projective variety, ``generalize flag variety'').

Example. Take \(\mathbf{G}=\mathrm{SL}_2\), we have \(\mathbb{H}= \mathrm{SL}_2(\mathbb{R})/\mathrm{SO}(2).\)

Let \(K = \mathrm{SO}(2) = \mathrm{Stab}_{\mathrm{SL}_2(\mathbb{R})}(i)\). In \(\mathrm{SL}_2(\mathbb{C})\), we have that \(\begin{pmatrix}0 & -1 \\ 1& -i\end{pmatrix}\in \mathrm{SL}_2(\mathbb{C})\) takes \(i\) to \(\infty\).

\[\mathrm{Stab}_{\mathrm{SL}_2(\mathbb{C})}(\infty) = P=\left\lbrace \begin{pmatrix}\star &\star\\ 0& \star\end{pmatrix} \right\rbrace .\]

Flags. A flag is a sequence of vector spaces \(\{0\}=V_0\subset V_1\subset \cdots \subset V_s=\mathbb{C}^n.\) Let \(n_i = \dim V_i/V_{i-1}\), \(n=n_1+\cdots+n_s\). The stabilizer of this flag is a block triangular matrix with \(s\) diagonal blocks (in the appropriate basis), where the \(i\)th is of size \(n_i\). This is the parabolic subgroup stabilizing the flag.

Let \(Y_1=\{\{O\}\subset L\subset \mathbb{C}^2 | \dim(L)=1\}\). Then \(\mathrm{SL}_2(\mathbb{C})\) acts transitively on \(Y_1\).

The standard flag : \(\{0\}\subset\mathrm{span}_\mathbb{C}(e_1)\subset \mathbb{C}\). Its stabilizer is \(P==\left\lbrace \begin{pmatrix}\star &\star\\ 0& \star\end{pmatrix} \right\rbrace\). We have \(Y_1 \cong \mathrm{SL}_2(\mathbb{C})/P \cong \mathbb{P}^1(\mathbb{C}),\) the compact dual of \(\mathbb{H}\).

Example. \(\mathbf{G}=\mathrm{Sp}_{2g}\), \(X=\mathbb{H}_g=\mathrm{Sp}_{2g}(\mathbb{R})/{U}(g)\)

Consider \((\mathbb{Z}^{2g},\langle \cdot, \cdot \rangle)\) the symplectic lattice with basis \(\{e_1,\dots, e_g,f_1,\dots f_g\}\), where \(\langle e_i,e_j\rangle = \langle f_i,f_j\rangle\) and \(\langle e_i,f_j\rangle = \delta_{ij}\).

We have \((\mathbb{C}^{2g}),\langle\cdot,\cdot\rangle) \cong (\mathbb{Z}^{2g},\langle\cdot,\cdot,\rangle)\otimes \mathbb{C}\).

Let

\[Y_g = \left\lbrace \{0\}\subset L\subset \mathbb{C}^{2g} | \dim(L)=g, \ \langle x,y\rangle = 0\ \forall x,y\in L \right\rbrace .\] \(L\) here is a Lagrangian, i.e. maximal totally isotropic subspace.

\(Y_g\) is the Grassmannian of Lagrangian subspaces, and \(\mathrm{Sp}_{2g}(\mathbb{C})\) acts transitively on \(Y_g\).

Standard flag. \(0\subset \langle e_1,\dots,e_g \rangle\subset \mathbb{C}^{2g}\). The stabilizer of the standard flag is \[P=\left\lbrace \begin{bmatrix}A& \star \\ 0& (A^{-1})^T\end{bmatrix}\in \mathrm{Sp}_{2g} \right\rbrace . \] Therefore, \(Y_g \cong \mathrm{Sp}_{2g}/P\).

We have an open set \(Y_g^+ = \{L\in Y_g | -i\langle x,\overline{x}\rangle > 0 \ \text{ for all }x\in L\}\).

\(\mathrm{Sp}_{2g}(\mathbb{R})\) acts transitively on \(Y_g^+\),

\[ \left\langle x,\overline{x}\right\rangle \underset{g\in \mathrm{Sp}_{2g}(\mathbb{C})}{=} \langle gx, g\overline{x}\rangle\underset{g\in\mathrm{Sp}_{2g}(\mathbb{R})}{=} \langle gx,\overline{gx} ,\rangle.\]

\(-i\langle \overline{x} , x\rangle= i \langle x,\overline{x}\rangle\), it is a hermitian form. The stabilizer in \(\mathrm{Sp}_{2g}(\mathbb{R})\) of a point in \(Y_g^+\) is \(U(g)\).

Proposition. We have a \(1-1\) correspondence \[\left\lbrace \text{Smooth bundles }\mathbf{G}(\mathbb{R})\times_KV\rightarrow X\right\rbrace \leftrightarrow \left\lbrace \text{Algebraic bundles }\mathbf{G}(\mathbb{C})\times_P V\rightarrow \hat{X} \right\rbrace \]

• Left-to-right : Extend \((V,\sigma)\) to \((V,\hat{\sigma})\).

• Right-to left : Restrict via \(\beta : X\rightarrow \hat{X}\).

Proof. Given a representation \((V,\sigma)\) of \(K\), by the Weyl unitary trick we get a unique unitary representation of \(K_\mathbb{C}\). We use the Levi decomposition \(P=U\rtimes M\) where \(M\cong K_\mathbb{C}\) (the particular levi factor here is the complexification of \(K\)). We can pullback the representation on \(M\) to \(P\) (equiv define a representation of \(P\) where \(U\) acts trivially). So we get \(\mathbf{G}(\mathbb{C})\times_P V\rightarrow \hat{X}\) as an algebraic bundle.

Corollary. Pulling back via \(\beta\), we realize \(\mathbf{G}(\mathbb{R})\times_K V\rightarrow X=\mathbf{G}(\mathbb{R})/K\) to be holomorphic.

It was immediate previously that since \(\mathbf{G}(\mathbb{R})\times_K V\rightarrow X\) is a holomorphic bundle, then \(E_{v,\Gamma} \rightarrow \Gamma\backslash X\) is one as well (we just quotient by \(\Gamma\), which is discrete so it stays holomorphic).

Bailey-Borel Theorem. \(\Gamma\backslash X\) is a quasi-projective algebraci variety (called a Shimura variety), and \(E_{V,\Gamma}\) are algebraic bundles.