# Aaron Pollack: Modular forms on exceptional groups (Lecture 1) :::{.remark} Plans for lectures: 1. What is $\G_2$, what are modular forms on it? 2. Fourier expansions of modular forms on $\G_2$. 3. Examples and theorems about modular forms on $\G_2$. 4. Beyond $\G_2$, possibly $\E_8$. ::: :::{.remark} First generalize modular forms to modular functions: let $f:\mfh\to \CC$ be a modular form of level $\Gamma$ and weight $\ell>0$. Define \[ \phi_f: \SL_2(\RR) &\to \CC \\ \phi_f(g) &\da j(g, z)^{-\ell} f(gz) \\ \\ j\qty{ g = \matt a b c d,z} &\da cz+d .\] Some properties: 1. Growth: $\phi_f$ is of moderate growth. 2. Invariance: $\phi_f(\gamma g) = \phi_f(g)$ for all $\gamma\in \Gamma \leq \SL_2(\ZZ)$. 3. Equivariance on a compact: $k_\theta \da \matt{\cos \theta}{-\sin \theta}{\sin \theta}{\cos \theta}\in \SO_2(\RR)$ satisfies $\phi_f(g k_\theta) = e^{-i\ell \theta} \phi_f(g)$ 4. Operator equation: $D_{\CR} \phi_f \equiv 0$ where we decompose the complexified Lie algebra \[ \liesl_3(\RR) \tensor_\RR \CC \cong (k_0 \tensor \CC) + (\phi_0\tensor \CC) \] as antisymmetric and symmetric parts, then $p_0\tensor \CC = \CC X_+ + \CC X_-$ where $X_{\pm} = \matt{1}{\pm i}{\pm i}{-1}$, and $D_{\CR} \phi_f \da X_- f$. Conversely, if $\phi: \SL_2(\RR) \to \CC$ satisfies these properties, then $f(z) = j(g_z, w^\ell)\phi(g_z)$ where $g_z \cdot c = z$ is well-defined, holomorphic, weight $\ell$, level $\Gamma$ modular forms. ::: ## Modular forms on $\G_2$ :::{.remark} Recall that $\G_2$ is a simple noncompact Lie group of dimension 14, with maximal compact $K = (\SU_2\times \SU_2)/\gens{\pm I}$. Write the first factor as $\SU_2^l$ for "long" and the second as $\SU_2^s$ for "short", then the root system looks like the following: ![](figures/2022-03-05_14-36-01.png) There is an action of $K$ on $V_\ell \da \Sym^\ell(\CC^2)\tensor \one$, and the diagonal acts trivially. ::: :::{.definition title="Modular forms on $\mathsf G_2$"} Suppose \( \Gamma\leq \G_2 \) is a congruence subgroup, so $\Gamma = \G_2(\QQ) \intersect K_f$ where $K_f \subseteq \G_2(M_f)$, and let $\ell \in \ZZ_{>0}$, A **modular form** of weight $\ell$ and level $\Gamma$ is a map $\phi: \G_2 \to V_\ell$ such that 1. Growth: $\phi$ has moderate growth. 2. Invariance: $\phi( \gamma g) = \phi(g)$ for all \( \gamma\in \Gamma \). 3. Equivariance on a compact: $\phi(gk) = k\inv \phi(g)$ for all $k\in K$. 4. Operator equation: $D_\ell \phi = 0$. Equivalently, a map $\phi: \dcosetl{\G_2(\QQ)}{\G_2(\AA)}$ satisfying similar conditions. ::: :::{.remark} The upshot: modular forms on $\G_2$ have a classical Fourier expansion and Fourier coefficients, which appear very arithmetic. ::: ## What is $\G_2$? :::{.remark} Todos: - What is $\G_2$? - What is $D_\ell$? - What are some examples/theorems about modular forms on $\G_2$? ::: :::{.remark} We'll define a $C_3\dash$graded Lie algebra over $\QQ$: \[ \lieg_2 = \liesl_3[0] + V_g(\QQ)[1] + V_3\dual (\QQ)[2] ,\] where $\liesl_3$ are the traceless matrices as usual and $V_3$ is the 3-dimensional standard representation of $\liesl_3$. The grading will mean that $[x,y]$ will land in degree $\abs{x} + \abs{y}$. The bracket is defined as follows: \[ [\phi, \phi'] &\da \phi\phi' - \phi'\phi & \phi,\phi'\in \liesl_3 \\ [\phi, v] &\da \phi(v), & v\in V_3\\ [\phi, \delta] &\da \phi(\delta),&\delta\in V_3\dual .\] ::: :::{.observation title="Constructing $\mathsf G_2$"} \[ \Extalg^3 V_3= \one \implies \Extalg^2 V_3= V_3\dual \implies \Extalg^2 (V\dual) = V_3 .\] Fix a basis $V_3 = \gens{v_1, v_2, v_3}$ and $V_3\dual = \gens{\delta_1, \delta_2, \delta_2}$ its dual basis, then - $v_i \wedgeprod v_{i+1} = \delta_{i-1}$ - $\delta_i \wedgeprod \delta_{i+1} = \delta_{i-1}$ Moreover, \[ [v, v'] &= 2 v\wedgeprod v' \in \Extalg^2 V_3 \cong V_3\dual \\ [\delta, \delta'] &= 2\delta\wedgeprod \delta' \in \Extalg^2 V_3\dual\cong V_3 \\ [\delta, v] &= 3v\tensor \delta - \delta(v)\one \in \liesl_3 ,\] noting that the last is traceless and $3v\tensor \delta\in V_3\tensor V_3\dual \cong \Endo(V_3)$. All other brackets are determined by antisymmetry and linearity ::: :::{.proposition title="Construction of $\mathsf G_2$"} The algebra $\lieg_2$ as defined above is a simple Lie algebra, i.e. the Jacobi identity holds and there are no nontrivial ideals. Moreover \[ \Aut(\lieg_2) = \ts{g\in \GL(\lieg_2) \st [gx, gy] = g[x, y]\, \forall x, y \in \lieg_2} \] and $\G_2 \cong \Aut^0(\lieg_2)$ is the connected component. ::: :::{.remark} Note: a similar procedure can be used to define all of the exceptional groups, see notes. ::: :::{.remark} What is the root diagram for $\lieg_2$? Let $\lieh\leq \liesl_3$ be the diagonal elements, i.e. \[ \lieh = \ts{\sum_{1\leq i\leq 3} \alpha_i E_{ii} \st \sum \alpha_i = 0} ,\] and let $r_1, r_2, r_3: \lieh\to \QQ$ be such that \[ r_j \sum_{1\leq i\leq 3} \alpha_i E_{ii} = \alpha_j ,\] i.e. projection onto the $j$th component. Note that $\sum r_i = 0$. What are the weights of $\lieh$ on $\lieg_2$? Since $\lieg_2 = \liesl_3 + V_3 + V_3\dual$, the actions are: - On $V_3$ it acts by $r_1, r_2, r_3$. - On $V_3\dual$ it acts by $-r_1, -r_2, -r_3$. - On $\liesl_3$ it acts by $\ts{r_i - r_j \st i\neq j}$. This yields a root diagram: ![](figures/2022-03-05_15-01-32.png) ::: :::{.remark} On the differential operator: take the Cartan involution \[ \Theta: \lieg_2\tensor \RR \to \lieg_2 \tensor \RR .\] Explicitly, - On $\liesl_3$, this acts as $X\mapsto {}^{-t}X$ - On $V_3$, it's $V_3\mapsto V_3\dual$ by $v_j\mapsto \delta_j$. Define - $k_0 = (\lieg_2\tensor \RR)^{G=\id}$ - $p_0 = (\lieg_2\tensor \RR)^{G=-\id}$ - $K = \ts{g\in \G_2 \st \Ad_g\circ \Theta = \Theta \circ \Ad_g}$ - $k = k_0 \tensor \CC$, something about $\liesl_3 + \liesl_2$ - $p = p_0 \tensor \CC$, something about $V_2\tensor \Sym^3(V_2)$. - $D_\ell = \pr \tilde D_\ell$, which we'll define. Suppose $\phi: \G_2\to V_\ell = \Sym^{2\ell}(\CC^2)\tensor \one$ such that $\phi(gk) = k\inv \phi(g)$ for all $k\in K$. Let $\ts{X_\alpha}$ be a basis of $p$, $\ts{X_\alpha\dual}$ basis of $p\dual$, then \[ \tilde D_\ell \phi = \sum_\alpha X_\alpha p \tensor X_\alpha\dual \in V_\ell \tensor p\dual .\] where $X_\alpha \phi$ is the derivative of the right regular action, i.e. if $X\in p_0$, \[ (X_p)(g) = \dd{}{t} \phi(g\exp(tx))\evalfrom_{t=0} .\] Then \[ V_\ell \tensor \phi\dual &= (S^{2\ell} \tensor \one) \times V_\ell \boxtensor \Sym^3(V_2) \\ &= (S^{2\ell + 1} + S^{2\ell - 1})\boxtensor S^3(V_2) \\ &\mapsvia{\pr} S^{2\ell -1}(V_\ell) \boxtensor S^3(V_3) .\] This relates - $\G_2\leadsto \SL_2$ - ? :::