# Siegel domains and the Baily-Borel compactification of $\mathcal{A}_2$

Consider the connected, almost simple, algebraic group $Sp_4$ over $\mathbb{Q}$ (embedded canonically in $GL_4$). For the following facts, I refer the reader to Murnaghan, Linear Algebraic Groups, Example 8.2. The notes can be downloaded at the following address:

http://www.math.toronto.edu/murnaghan/courses/algp.pdf

My question refers to a statement in Baily and Borel, Compactification of arithmetic quotients of bounded symmetric domains, which I will henceforth refer to by [BB].

Let $S$ denote the maximal (split) $\mathbb{Q}$-torus such that

$$S(\mathbb{R})=\{diag(a,b,b^{-1},a^{-1}):a,b\in\mathbb{R}^{\times}\}.$$ A set $\Delta$ of simple roots for $S$ is $\{\alpha_1,\alpha_2\}$, where $\alpha_1$ sends $diag(a,b,b^{-1},a^{-1})$ to $ab^{-1}$ whereas $\alpha_2$ sends it to $b^2$. This is the $\textit{canonical numbering}$ on $\Delta$ as defined in [BB, 2.8]. We define $A$ to be $S(\mathbb{R})^+$ and, for any real number $t>0$, we define

$$A_t:=\{s\in A:\alpha_1(s)\leq t,\ \alpha_2(s)\leq t\}.$$

Let $K$ denote the intersection of the orthogonal group $O(4)$ and $Sp_4(\mathbb{R})$ in $GL_4(\mathbb{R})$, which is a maximal compact subgroup of $Sp_4(\mathbb{R})$, and let $N$ denote the unipotent $\mathbb{Q}$-subgroup of $Sp_4$ such that $N(\mathbb{R})$ is the group of upper triangular matrices with $1$s along the diagonal that belong to $Sp_4(\mathbb{R})$. We let $\omega$ denote a compact subset of $N(\mathbb{R})$ and we fix $t>0$. Then

$$\mathfrak{S}:=K\cdot A_t\cdot\omega$$

is a Siegel domain in $Sp_4(\mathbb{R})$ with respect to $K$, $S$ and $N$, as defined in [BB, 4.2].

Now let $T$ denote the $\mathbb{Q}$-subtorus $ker(\alpha_2)^{\circ}$ of $S$ and consider the centraliser of $T$ in $Sp_4$; it is isomorphic to $SL_2\times\mathbb{G}_m$ and is a Levi subgroup of a maximal standard parabolic $\mathbb{Q}$-subgroup of $Sp_4$. I am interested in the subgroup $L$ isomorphic to $SL_2\times\{1\}$. In the notation of [BB, 4.4], $L(\mathbb{R})$ is the group $L(F_1)$ associated with the rational boundary component $F_1$. The claim in [BB, 4.4] is that

$$\mathfrak{S}\cap L(\mathbb{R})$$

is a Siegel domain in $L(\mathbb{R})$ with respect to $K\cap L(\mathbb{R})$, $S\cap L$ and $N\cap L$. However,

$$A_t\cap L(\mathbb{R})=\{diag(1,b,b^{-1},1):b^{-1}\leq t,\ b^2\leq t\},$$

which is a compact set, whereas no Siegel domain in $SL_2(\mathbb{R})$ is compact.

Therefore, are my calculations incorrect?

• Read [BB,4.2] again. It doesn't make the statement you say it does. (in fact, that subsection is mainly a sequence of definitions). – Peter McNamara Mar 11 '16 at 12:51
• My apologies Peter; I intended to refer to section 4.4. I have edited the post accordingly. – Christopher Daw Mar 11 '16 at 13:49
• I believe that the above calculations are correct. However, as pointed out to me by Martin Orr, if one intersects $\mathfrak{S}$ with the Levi subgroup isomorphic to $SL_2\times\mathbb{G}_m$ and then takes the image of this set in $L$ under the natural projection, then this is indeed a Siegel domain in $L$. This is essentially the last statement of the penultimate paragraph of section 4.4. Subsequently, one can choose the element $a$ from section 4.5 to be an element of the aforementioned intersection. – Christopher Daw Mar 11 '16 at 14:00