Remember last time we had the diagram $$ \begin{CD} \mathcal{H}^n @>{\tilde{\varphi}}>> G/M \br @V{e}VV @V{p}VV \br (\Delta^\ast)^n @>{\varphi}>> S_{\Gamma,G,M}, \end{CD} $$ where $e(z) = e^{2 \pi i z}$. We had these Siegel sets $$ \mathfrak{S}_ \mathcal{H} = \lbrace x + iy : 0 \le x \le 1, 1 \le y \rbrace. $$ What we are left to show is that the images $\tilde{\varphi}(\mathfrak{S}_ \mathcal{H}^n)$ are contained in finite unions of Siegel sets.
Roughly polynomial functions#
Definition 1. We say that a function $f(\bar{x}, \bar{y})$ in the variables $x_1, \dotsc, x_n, y_1, \dotsc, y_n$ is roughly monomial when we can write it as $$ f(\bar{x}, \bar{y}) = \varphi(\bar{x}, \bar{y}) y_1^{s_1} \dotsm y_n^{s_n} $$ where $s_i \in \mathbb{Z}$ and $\varphi$ is a bounded function. We say that $f$ is roughly polynomial if $f(\bar{x}, \bar{y}) = \varphi(\bar{x}, \bar{y}) P(\bar{x}, \bar{y})$ where $P(\bar{x}, \bar{y}) \in \mathbb{C}[\bar{x}, \bar{y}, \bar{y}^{-1}]$.
Lemma 2. Let $f$ be roughly polynomial and $g$ be roughly monomial. Suppose that for all $1 \le n_0 \le n$, $\alpha_1, \dotsc, \alpha_{n_0} \in \mathbb{Q}_ +$, $\beta_1, \dotsc, \beta_{n_0} \in \mathbb{R}$, and $\zeta_{n_0+1}, \dotsc, \zeta_n \in \mathfrak{H}$, we have $\lvert f \rvert \ll \lvert g \rvert$ on the set $$ \lbrace (z_1, \dotsc, z_n) : \alpha_1 z_1 + \beta_1 = \dotsb = \alpha_{n_0} z_{n_0} + \beta_{n_0}, z_{n_0+1} = \zeta_{n_0+1}, \dotsc, z_n = \zeta_n \rbrace. $$ Then $\lvert f \rvert \ll \lvert g \rvert$.
Proof.
We replace $f$ by $f/g$, so that we can assume $g = 1$. We now want to show that $f$ is bounded, and induct on $n$. When $n = 1$, this can be checked. In general, we may assume that $f$ is a polynomial and write $$ f(\bar{x}, \bar{y}) = \sum_{j_1, j_2} a_{j_1,j_2} x_1^{j_1} x_2^{j_2} $$ where we have $$ a_{j_1,j_2}(e(z_1), x_2, \dotsc, x_n, y_2, \dotsc, y_n) = a_{j_1,j_2}^0(\bar{x}^1, \bar{y}^1) + a_{j_1,j_2}^1(\bar{x}^1, \bar{y}^1) e(z_1) + \dotsb. $$ Since all these $e(z_1)$ are bounded, we may assume that $a_{j_1,j_2}$ only has the dependence on $\bar{x}^1, \bar{y}^1$.
It now suffices to show that $\lvert a_{j_1,j_2} \rvert y_1^{j_2}$ is bounded on $\Sigma^n$. We note that $j_2 \le 0$, because otherwise restricting to $z_2 = z_3 = \dotsb = z_n = 37i$ contradicts our assumption. Then $\lvert a_{j_1,j_2} \rvert y_1^{j_2}$ is non-increasing in $y_1$, and because $y_1 \ge y_2$ on $\Sigma^n$, we may assume $y_1 = y_2$. By induction, if we restrict $f$ to $z_1 = m z_2 + c$ for $m \in \mathbb{N}$ and $c \in \mathbb{R}$, the function $$ f_{m,c}(\bar{x}^1, \bar{y}^1) = \sum a_{j_1,j_2} (mx_2 + c)^{j_1} (my_2)^{j_2} $$ is bounded. Take $(r_1, r_2)$ to be the largest term in $f$ so that $$ F_m = \frac{1}{r_1!} \sum_{i=0}^{r_1} \binom{r_1}{i} (-1)^i f{m,i} = \Delta_{x_2}^{r_1} f_{0,m}(x_2, \dotsc). $$ Then $F_m$ is a bounded function for every $m$, and we can isolate the term $a_{r_1,r_2} y_2^{r_2}$.
How are we going to use this? Recall that $\Sigma^n$ is simply connected, so we can trivialize the underlying Hodge structure $V_\mathbb{Z}$. We also have the polarization form $Q(u, v)$ that is $(-1)^w$-symmetric, and $h(u, v) = Q(Cu, v)$ that is bilinear symmetric, and $B(u, v) = Q(u, \bar{v})$ that is hermitian.
Proposition 3. If we consider $h(u, v)$ as a function on $z_1, \dotsc, z_n$ for fixed $u, v$, then this is roughly polynomial.
The theorem of Kashiwara#
Recall the nilpotent orbit theorem that allows us to write $$ \tilde{\varphi}(z) = \exp(\sum z_i N_i) \psi(e(z_1), \dotsc, e(z_n)) F $$ where $\psi$ is a holomorphic function on $\Delta^n$ valued in $\mathbb{G}(\mathbb{C})$ and $F \in \vee{D}$. We now consider the matrix $$ \gamma(\bar{z}) = \exp(\sum z_i N_i) \psi(e(z_1), \dotsc, e(z_n)) $$ where we think of $\psi$ as a matrix with entries in $\mathcal{O}(\bar{x}, \bar{y})$.
Also recall the Jacobson–Morozov theorem, which says that for every Lie algebra $\mathfrak{g}/\mathbb{C}$ and a nilpotent element $n \in \mathfrak{g}$, there exists a map $\mathfrak{sl}_ 2 \to \mathfrak{g}$ sending $e$ to $n$. Using this, we obtain the weight filtration $W_{\bullet,j}$ corresponding to $N_1 + \dotsb + N_j$.
Theorem 4 (Kashiwara, 1985). There is a splitting $V = \bigoplus_{p, q_1, \dotsc, q_r} I^{p_1, q_1, \dotsc, q_r}$ such that $F^s = \bigoplus_{p \ge s} I^{p_1, q_1, \dotsc, q_r}$ and $W_{s,j} \le \bigoplus_{p+q_j \le s} I$.
Using this, we see that if $u \in I^\alpha$ and $v \in I^\beta$, then we have $$ h(u) \sim \Bigl(\frac{y_1}{y_2}\Bigr)^{s_1} \dotsm \Bigl(\frac{y_{n-1}}{y_n}\Bigr)^{s_{n-1}} y_n^{s_n}. $$ Now using that $h$ is symmetric bilinear, we can write $u = u_1 + \dotsb + u_k$ with $u_i \in I^{\alpha_i}$ and then write $h(u)$ as the sum of $h(u_i)$ and $h(u_i, u_j)$.
So how does it relate to Siegel sets? For the natural embedding $\mathbb{G}(\mathbb{Q}) \subseteq \mathrm{SL}(V_\mathbb{Q})$, there is the induced map $$ \iota \colon D \to X $$ where $X$ is the moduli space of positive-definite symmetric forms on $V_\mathbb{R}$.
Definition 5. For a positive-definite symmetric form $B$, a $\mathbb{Z}$-basis $\bar{e}$, and a real number $C \gt 0$, we say that $B$ is $(\bar{e}, C)$-reduced when
- $\lvert B(e_i, e_j) \rvert \lt C B(e_i)$ for all $i, j$,
- $B(e_i) \lt C B(e_j)$ for all $i \lt j$.
Now we can define $T_{\bar{e},C}$ a set of $(\bar{e}, C)$-reduced forms such that a set $S$ of forms is contained in a Siegel set if there exist $\bar{e}, C$ such that all $B \in S$ are $(\bar{e}, C)$-reduced.