# Hyperbolic Geometry

## Hyperbolic lattices

\dzg{Note: some of this mixes conventions, need to fix later.}

:::{.warning}
There is a significant gap in the AG literature vs the physics literature for the terminology for hyperbolic spaces, and the traditional AG terminology can be "wrong" in some senses. For example, the AG literature will typically call $\ts{v\in L_\RR \mid v^2 > 0}$ a "light cone", but this is not quite correct: the actual \textit{light cone} in general relativity is $\ts{v\in L_\RR \mid v^2 = 0}$. The following picture is the usual mnemonic:

\begin{figure}[H]
    \centering
    \includegraphics[width=0.7\textwidth]{figures/timelike_lightlike.jpg}
    \caption{$\ts{v^2=0}$ is the light cone, its interior is timelike and exterior spacelike.}
    \label{fig:enter-label}
\end{figure}
:::

:::{.definition title="Hyperbolic lattices"}
An indefinite lattice $L$ is a **hyperbolic lattice**
\footnote{Also called a **Lorentzian lattice**.} if $\signature(L) = (1, n_-)$ or $(n_+, 1)$ for some $n_-, n_+ \geq 1$. By convention, by twisting $L$ to $L(-1)$ if necessary, we assume hyperbolic lattices have signature $(1, n)$. In this convention, the single positive-definite direction is referred to as **timelike**, and the remaining directions are **spacelike**.
:::

:::{.definition title="Time/light/spacelike vectors"}
Let $L$ be a hyperbolic lattice of signature $(1, n)$.
We say a vector $v\in L_\RR$ is

- **timelike** if $v^2 < 0$,
- **lightlike** or **isotropic** if $v^2=0$.
- **spacelike** if $v^2 > 0$,
More generally, a subspace $W \subseteq \EE^{1, n}$ with the restricted form $({-}, {-})_W$ is

- **timelike** if $({-}, {-})_W$ is negative-definite, or is indefinite and non-degenerate, 
- **lightlike** or **isotropic** if $({-}, {-})_W$ is degenerate, or
- **spacelike** if $({-}, {-})_W$ is positive-definite.

Define
\begin{align*}
    L^{< 0} \da \ts{v\in L_\RR \mid v^2 < 0} &\quad \text{The timelike regime} \\
    L^{= 0} \da \ts{v\in L_\RR \mid v^2 = 0} &\quad \text{The lightlike regime} \\
    L^{> 0} \da \ts{v\in L_\RR \mid v^2 > 0} &\quad \text{The spacelike regime}
\end{align*}
:::

:::{.remark}
\cite{AE22nonsympinv} refers to the non-spacelike regime $L^{\geq 0} \da \ts{v\in L_\RR \mid v^2 \geq 0}$ as the **round cone**; this is used for a model over $\overline{\HH^n}$ with ideal points included, and is often used as the support of a semifan for a semitoroidal compactification.
:::

:::{.definition title="Past and future light cones"}
\label{def:past-future-lightcones}
Let $L$ be a hyperbolic lattice of signature $(1, n)$. The spacelike regime  $L^{> 0}$ of $L$ has an irreducible component decomposition
\[
L^{> 0 } \da \ts{v\in L_\RR \mid v^2 > 0} = C_L^+ \amalg C_L^-
, \]
whose components we refer to as the **future light cone** and **past light cone** of $L$ respectively, and can be distinguished by the sign of the coordinate in the negative-definite direction:
\[
C^+_L \da \ts{v\in L^{ >0} \mid v_{0}  > 0 },\qquad 
C^-_L \da \ts{v\in L^{ >0} \mid v_{0}  < 0 }
.\]
We write their closures in $L_\RR$ as $\overline{C_L^+}$ and $\overline{C_L^-}$ respectively, and write \(C_L \da C_L^{+}\) for a fixed choice of a **future** light cone and $\overline{C_L}$ for its closure.
:::

## Models of hyperbolic space

:::{.definition title="Euclidean upper-half space"}
The upper-half space in $\EE^n$ is
\[
\EE^n_+ \da \ts{(x_1,\cdots, x_{n}) \in \EE^{n} \mid x_1 > 0}
.\]
:::

:::{.definition title="Minkowski space"}
\label{def:minkowski}
The **$n\dash$dimensional Minkowski space** $\EE^{1, n}$ is the real vector space $\RR^{n+1}$ equipped with a bilinear form of signature $(1, n)$ which can be explicitly written as 
\[
vw \da -v_0 w_0 + \sum_{i=1}^n v_i w_i
\]
with the associated quadratic form
\[
v^2 \da Q(v) \da -v_0^2 + \sum_{i=1}^n v_i^2
.\]
This induces a metric
\[
\rho(v, w) \da \mathrm{arccosh}(-vw)
.\]
:::

:::{.remark}
If $L$ is hyperbolic of signature $(1, n)$ then $L_\RR \cong \EE^{1, n}$ is a Minkowski space of dimension $n+1 = \rank_\ZZ L$.
:::

### Half-plane models

> https://arxiv.org/pdf/1908.01710.pdf#page=40&zoom=auto,-95,626

:::{.definition title="de Sitter space and light cone of a lattice"}
Let $L$ be a hyperbolic lattice and consider the squaring functional 
\begin{align*}
f_L: L_\RR &\to \RR \\
v & \mapsto v^2
.\end{align*}
One can show that $\pm 1$ are regular values of $f_L$ and thus define two canonical "hyperbolic unit spheres" which are regular surfaces. We define the **de Sitter space of $L$** as 
\[
\mathrm{dS}_L \da f_L\inv(1) = \ts{v\in L_\RR \mid v^2 = 1 } \subseteq L^{ > 0}
\]
in the spacelike regime
and the **unit hyperboloid of $L$** as the two-sheeted hyperboloid
\[
H_L \da f_L\inv(-1) = \ts{v\in L_\RR \mid v^2 = -1} \subseteq L^{< 0}
\]
in the timelike regime.
:::


:::{.example}
\Cref{fig:deSitter} shows the de Sitter space and unit hyperboloid for a lattice $L$ of signature $(2, 1)$ in $\EE^{2, 1}$, visualized in $\RR^3$.
\begin{figure}[H]
    \centering
    \includegraphics[width=0.9\textwidth]{figures/deSitterHyperboloid.png}
    \caption{The hyperbolic unit spheres: the de Sitter space and light cone for $\EE^{2, 1}$.}
    \label{fig:deSitter}
\end{figure}
:::

:::{.definition title="Half-plane model/Lobachevsky space of 
a lattice"}
Let $L$ be a hyperbolic lattice. The **half-plane model of $\HH^n$ associated to $L$** or **Lobachevsky space of $L$** is the unit hyperboloid of $L$ intersected with its future light cone,
\[
\bL^n_L \da H_L \intersect C_L \da \ts{v \in L_\RR \mid v^2 = -1, v_{0} > 0} 
,\]
given the metric restricted from $L_\RR \cong \EE^{n,1}$.
This more simply be described as the future sheet of the unit hyperboloid $H_L$, using the irreducible component decomposition
\[
H_L = H^+_L \amalg H^-_L = \ts{v\in H_L \mid v_0 > 0} \amalg \ts{v\in H_L \mid v_0 < 0}
\]
and setting $\bL^n_L \da H_L^+$.
:::

:::{.remark}
Note that $H_L^+$ is in the timelike regime.
This gives a model of the hyperbolic space $\HH^n$ which we often denote $\HH_L$ when we do not fix a specific choice of model, or simply by $\HH^n$ when the dependence on $L$ is not important.
:::

:::{.remark title="The isometry group of hyperbolic spaces"}
It can be shown that the isometries of the timelike regime $L^{< 0}$ are restrictions of isometries of the ambient Minkowski space $\EE^{1, n}$, and thus 
\[
\Isom(L^{ < 0}) \cong \Isom(\EE^{1, n}) \cong \Orth_{1, n}(\RR)
.\]
Using the half-plane model, we can thus naturally identify
\[
\Isom(\bL^n) \cong \Orth^+_{1, n}(\RR) \da \Stab_{\Orth_{1, n}(\RR)}(C_L)
,\] 
the index 2 subgroup which stabilizes the future light cone $C^+_L$ of $L$. These are precisely the isometries of $\EE^{1, n}$ of positive spinor norm.
:::

### Ball models

:::{.definition title="The Poincar\'e ball model"}
Let $L$ be a hyperbolic lattice. The **Poincar\'e ball model of $\HH^n$ associated to $L$** is defined as 
\[
\bB^n_L \da \PP (L^{ < 0})
,\]
the projectivization of the timelike regime of $L$, where
\begin{align*}
\PP(\wait): \EE^{1, n}\sm\ts{x_{n}\neq 0} &\to \EE^{n} \\
(x_0,\cdots, x_{n-1}, x_n) &\mapsto \qty{ {x_0\over x_n}, \cdots, { x_{n-1} \over x_n} }
.\end{align*}
In this model, there is a natural compactification $\overline{\HH^n}$ in $\PP(S^n)$ such that the interior is given by $\bB^n_L$ as above and the boundary by $\partial \overline{\HH^n} = \PP(L^{=0})$, i.e. ideal points correspond to (the projectivization of) the lightlike regime.
:::

:::{.remark title="An alternative construction"}
It can be explicitly constructed by considering the future light cone $C_L$ described in \autoref{def:past-future-lightcones}.
Letting $\RR_{> 0}$ act on $L_\RR\cong \EE^{1, n}$ by scaling along the timelike direction (i.e. in the coordinate $v_0$), the ball model can be formed as the quotient
\[
\BB^n_L \cong C_L/\RR_{> 0} \subset \PP(\SS^n)
.\]
:::

:::{.remark}
The advantage of $\bB^n_L$ over $\bL^n_L$ is that the former provides a natural compactification in $\PP(\SS^n)$. Moreover, it can be easier to work with hyperplanes in the ball model: let $\pi: \EE^{1, n}\ \to \PP(\SS^n)$ be the natural projection, then every hyperplane $H_v \da v^\perp$ for $v\in \bB^n_L$ is of the form
\[
H_v = \ts{\pi(x) \mid x\in C_L,\, xv = 0}
\]
One can also concretely interpret the bilinear form geometrically in the ball model in the following way:
\begin{align*}
    H_v \transverse H_w \implies \abs{vw} < 1 &
    \implies -vw = \cos(\angle(H_v, H_w)) \\
    H_v \parallel H_w \implies \abs{vw} = 1 
    &\implies -vw = \cos(\angle(H_v, H_w)) \\
    H_v \diverge H_w \implies \abs{vw} > 1 
    &\implies -vw = \cosh(\rho(H_v, H_w)),
\end{align*}
where $\rho$ is the hyperbolic metric described in \Cref{def:minkowski}.
:::

:::{.remark}
$\Isom(\bB^n) = \PP\Orth_{1, n}(\RR)$. 
:::

### Ideal points

:::{.remark}
Let $H_L \cong \HH^n$ be a model of hyperbolic space associated to a hyperbolic lattice $L$ of signature $(1, n)$.
Boundary points $\partial\overline{H_L}$ correspond to ideal points in $\HH^n$, i.e. points "at infinity", which in turn correspond to 1-dimensional isotropic subspaces of $L$. 

\includegraphics{figures/hyperboloid.png}

In this model, points in $\HH^n$ are points in the interior of the cone and on the hyperboloid. Moreover points on $\partial{\overline{\HH^n}}$ correspond to points on the surface of the cone:
\[
\partial\overline{\HH^n} \cong \ts{v = (v_0,\cdots, v_{n+1}) \in L_{\RR} \mid v_0 > 0} \intersect \ts{v\in L_\RR \mid v^2 = 0}
.\]
We interpret $uv = -\cos(\angle(H_u H_v))$, so $uv = -1$ means $H_u \intersect H_v \in \partial{\overline \HH^n}$, i.e. they are "parallel" planes. Hyperplanes in $\HH^n$ correspond to branches of hyperbolas obtained by slicing the hyperboloid by a plane in $L_\RR$. 
:::


:::{.remark}
Define Minkowski space as $\EE^{1, n}$, which is $\RR^n$ with the form $vw = v_0w_0 -\sum v_i w_i$. Define Lobachevsky space $\bL^n$ as the hyperboloid model of hyperbolic space, a certain "hyperbolic unit sphere":
\[
\bL^n \da \ts{v\in \EE^{1, n} \mid v^2 = 1, v_0 > 1}
.\]
The geodesic curves are precisely intersections of the form $H_2 \intersect \bL^n$ where $H_2\in \Gr_2(\RR^{n+1})$ is a standard 2-plane passing through the origin in the ambient space. The hyperbolic metric on $\bL^n$ is gotten by computing the length in the standard metric in $\RR^{n+1}$ of any geodesic curve between two points. The associated Poincare ball model is contained in the standard Euclidean ball $\bB^n \subset \RR^{n+1}$ and is the projection of $\bL^n$ onto the hyperplane $\ts{x_0 = 0} \subset \RR^{n+1}$ using rays passing through $(-1, 0, 0,\cdots, 0)$. Explicitly, the projection is 
\begin{align*}
\phi: \bL^n &\to \bB^n \\
(v_0, \cdots, v_n) &\mapsto {1\over 1 + v_0}(v_1,\cdots, v_n)
\end{align*}
Geodesics are now straight lines through the origin or arcs of Euclidean circles intersecting $\partial \bB^n$ orthogonally. Define the hyperbolic upper-half-space as
\[
\HH^n \da \ts{x = (x_1,\cdots, x_n) \in \RR^n \mid  x_1 > 0 }
\]
which is obtained by taking inversions through certain spheres centered on $\partial \bB^n$. Geodesics are now straight lines orthogonal to $\partial \HH^n$ or half-circles centered on $\partial \HH^n$.
:::

## Root Systems

:::{.definition title="Primitive vectors"}
    Let $L$ be any lattice. A finite set $S \da \ts{s_1,\cdots, s_n}\subseteq L$ of elements in $L$ is **primitive** if $S$ is $\RR\dash$linearly independent and $L\intersect \RR S = \bigoplus_{i=1}^n L s_i$, i.e. no $s_i$ can be replaced with a small vector in the same 1-dimensional subspace which is also in $L$. A primitive set of size one is called a **primitive element**, and we write $L_{\prim}$ for the set of such.
:::

:::{.definition title="Roots and $k$-roots in lattices"}
Let $L$ be any lattice. For $k \in \ZZ_{> 0}$, define the set of **$k\dash$roots in $L$** as
\[
\Phi_k{L} \da\ts{v\in L_{\prim} \mid v^2 = k,\, 2(v, L) \subseteq k\ZZ}
\]
A **root** is by definition a $2\dash$root. We write the set of roots in $L$ as $\Phi(L)$, and the complete set of roots as
\[
\Phi_\infty{L} \da \Union_{k\geq 1} \Phi_k{L}
.\] 
:::

:::{.remark}
In the theory of 2-elementary lattices, the roots consist of all $(-2)\dash$vectors along with any $(-4)\dash$vector $v$ with $\operatorname{div}(v) = 2$.
:::

:::{.definition title="Reflections"}
Let $L$ be any lattice and $L_\RR$ its associated $\RR\dash$module. An element $s\in \GL(L_\RR)$ is a **reflection** if there exists a vector $v\in L_\RR$ and an $\RR\dash$linear functional $f\in \Hom_\RR(L_\RR, \RR)$, both depending on $s$, 
\[
s(x) = x - f(x)v \quad \forall x\in L_\RR,\qquad f(v) = 2
.\]
Concretely, $s$ is an isometry of $L_\RR$ which pointwise fixes a hyperplane and is an involution satisfying $\det(s) = -1$. Every reflection can be written in the form
\[
s(u) = s_v(u) = u - {uv \over v^2/2 }v
\]
for some $v^2\neq 0$ in $L_\RR$ determined up to scaling. The reflection in $v$ is only well-defined when $2 \operatorname{div}(v) \in v^2 \ZZ$ where $\operatorname{div}(v)$ is the divisibility of $v$ defined in \autoref{def:divisibility}.
The **reflection hyperplane** associated to $s$ is the fixed subspace
\[
H_v \da \ker(f) = \ker(\id - s) \cong v^\perp
.\]

:::

:::{.remark}
Alternatively: $s\in \GL_n(\CC)$ is a quasi-reflection if it 1 eigenvalue $\lambda \neq 1$ with an eigenspace of dimension 1 and the remaining eigenvalues all 1. It is a reflection if $\lambda = -1$.
\end{remark}

:::{.definition title="Mirrors in hyperbolic lattices"}
Any root $v\in \Phi(L)$ defines a reflection $s_v$ through the mirror 
\[
H_v \da v^\perp \da \ts{x\in C^+_L \mid xv = 0}
.\] 
If $H_v$ is the reflection hyperplane of a root, we say it is a **mirror** in $L$. Note that $H_v$ is nonempty if and only if $v^2 < 0$.
:::

:::{.definition title="Weyl group"}
Let $L$ be any lattice. The **Weyl group of $L$** is defined as the group generated by reflections in $2\dash$roots,
\[
\weylgroup{L} \da \ts{s_v \mid v\in \Phi(L) } \leq \Orth_L(\RR)
\]
:::

:::{.definition title="The discriminant locus"}
For $L$ a hyperbolic lattice, define the **discriminant locus of $L$** as the union of all mirrors of $2\dash$roots,
\[
\Delta(L)
\da \Union_{v\in \Phi(L)} v^\perp
\da \Union_{v\in \Phi(L)} H_v
.\]
:::

:::{.definition title="Weyl chambers"}
The **chamber decomposition** of $C_L$\dzg{Forgot to write down what is $C_L$.} is defined as
\[
C_L^\circ \da C_L \setminus \Delta(L) = C_L \setminus \qty{ \Union_{\delta \in \Phi(L)} \delta^\perp}
,\]
the complement of all mirrors.
This further decomposes into connected components called **Weyl chambers**: fixing a chamber $P$, there is a decomposition into orbits
\[
C_L^\circ = \amalg_{s_v \in \weylgroup{L} } s_v(P)
.\]
:::

:::{.remark}
Any Weyl chamber $P$ is a simplicial cone, so the orbit decomposition yields a decomposition of $C_L^\circ$ into simplicial cones. Since $W$ acts on the set of Weyl chambers $\pi_0 C_L^\circ$ simply transitively and the closure $\overline P$ of any chamber is a fundamental domain for this action, there is a homeomorphism $\overline P \cong C_L^\circ/W$.
:::

> https://www.ms.u-tokyo.ac.jp/preprint/pdf/2007-12.pdf#page=7&zoom=100,88,601

:::{.definition title="Fundamental chamber"}
Let $P$ be a Weyl chamber of $L$, define
\begin{align*}
\Phi(L)^+ &\da \ts{v \in \Phi(L) \mid (v, P) > 0 } \\
\Phi(L)^- &\da \ts{v\in \Phi(L) \mid (v, P) < 0 } = -\Phi(L)^+
\end{align*}
which induces a decomposition
\[
\Phi(L) = \Phi(L)^+ \amalg  \Phi(L)^-
.\]
:::

:::{.remark}
Thus $P$ can be written as 
\[
P = \ts{v\in C_L \mid (v,\Phi(L)^+ ) > 0 } = \ts{}
.\]
This realizes $P$ as an intersection of positive half-spaces and thus as a polytope.
:::

:::{.definition title="Walls"}
Let $\overline P$ be the closure in $L_\RR$ of $P$.
We say a mirror $H_v \subseteq L_\RR$ for $v\in \Phi(L)^+$ is a **wall of $P$** if $\codim_{L_\RR}(H_v\intersect \overline{P}) = 1$.
:::

:::{.definition title="Simple systems"}
Let $P$ be a Weyl chamber of $P$ and let 
\[ 
\Pi(L, P) \da\ts{v\in \Phi(L) \mid H_v \text{ is a wall of } P}
\]
be the set of walls of $P$. We can more economically define $P$ by
\[
P = \ts{v\in C_L \mid \qty{ v, \Pi(L, P) } > 0}
,\]
\dzg{Todo, messed up notation here a bit.}
where no inequality is redundant. Moreover, $(P, \Pi(L, P))$ forms a Coxeter system, and $\overline{P}$ is a fundamental domain for $W(L) \actson L_\RR$.
:::

:::{.definition title="Chambers and $\Orth^+(L)$"}
The connected components of
\[
V^+_L \da \ts{x\in L^\pm \mid (\Phi(L), x)\neq 0}
\]
are called **chambers** of $L$. Any positive isometry preserves $L^+$ and $L^-$ set-wise, motivating the definition of the **group of positive isometries of $L$**
\[
\Orth^+(L) \da \ts{\gamma\in \Orth(L) \mid \gamma(L^+) = L^+, \gamma(L^-) = L^-}
\]
:::

:::{.definition title="Positive isometries"}
    We say an isometry $\gamma \in \Orth(L)$ is **positive** if it preserves a chamber (i.e. a connected component of $V^+_L$)
:::


:::{.definition title="Roots, root systems, root lattices"}
A vector $v\in L$ is a **root** if $v^2 = - 2$\footnote{
One occasionally calls any time-like vector $v^2 < 0$ a "root", in which case distinguishes between e.g. $(-2)\dash$roots $\Phi_2(L)$ and $(-4)\dash$roots $\Phi_4(L)$.}, and we write $\Phi(L)$ for the set of roots in $L$. If $L$ is negative definite and $L = \ZZ \Phi(L)$\footnote{i.e. if the roots form a $\ZZ\dash$generating set for $L$}, we say $L$ is a **root lattice**. Any root lattice decomposes as a direct sum of root lattices of ADE type.
:::

:::{.definition title="Weyl group"}
The **Weyl group** of $L$ is the maximal subgroup of the orthogonal group of $L$ generated by hyperplane reflections in roots, 
\[
\weylgroup{L}^2 \da \bracket{s_v \mid v\in \Phi(L) }_\ZZ \leq \Orth(L)
.\]

One can similarly define the group of reflections in \textit{all} vectors,
\[
\weylgroup{L}\da \bracket{s_v \mid v\in L}_\ZZ \unlhd \Orth(L)
.\]
Since conjugating a reflection by any automorphism is again a reflection, this is a normal subgroup.
If $L$ is a hyperbolic lattice, we replace $\Orth(L)$ in the above definition by $\Orth^+(L)$, the isometries that preserve the future light cone.
:::

:::{.definition title="Mirrors/walls and chambers"}
The **mirror** or **wall** associated with a root $v\in \Phi(L)$ is the hyperplane $H_v \da v^\perp$. As $v$ ranges over $\Phi(L)$, these partition $L_\RR$ into subsets called **chambers**. The Weyl group acts on $L_\RR$ by isometries and acts simply transitively on chambers, and we often distinguish a fundamental domain for this action called the **fundamental chamber**. 
We write $D_L$ for the closure in $L_\RR$ of a fundamental chamber. A **cusp** of $L$ is a primitive isotropic lattice vector $e\in   D_L \intersect L$.
:::

## General Period Domains

:::{.remark}

Define $G^{L} \da G_{L^\perp}$ for $G$ any algebraic group determined by $L$.\dzg{Very useful convention: $\Omega^S$ involves $S^\perp$, while $\Omega_S$ involves just $S$.}

Let $L\leq \lkt$ be a sublattice of signature $(1, r-1)$ so $\signature(L^\perp) = (2, 19-r+1)$. One can always form the period domain corresponding to $L\dash$polarized K3 surfaces as 
\[
\Omega^{L} \da \Omega_{L^\perp} \da \ts{x\in (L^\perp)_\CC \mid x^2 = 0, x\overline{x} > 0},
\]
The period domain can be described as a Hermitian symmetric space:
\[
\Omega^{L} \cong 
{\SO^{L}(\RR) \over \SO_2(\RR) \times \SO_{20-r}(\RR) }
\]
> http://content.algebraicgeometry.nl/2020-5/2020-5-021.pdf#page=10&zoom=auto,-85,607

For any arithmetic subgroup $\Gamma\leq \Orth^{L}(\RR)$ there is a complex-analytic isomorphism
\[
\Gamma\backslash \Omega^L \cong 
\qty{\Gamma \intersect \SO^{L}(\RR)} \backslash \Omega^L
.\]
In particular, for $L$ a primitive sublattice of $\lkt$, letting $F_{L}$ be the stack of $L\dash$polarized K3 surfaces, the period map $\tau_L$ yields an open immersion
\[
\tau_L: F_{L}(\CC) \injects \widetilde{\SO^L}(\ZZ) \backslash \Omega^L
\]
where $\widetilde{\SO^L}$ are isometries of $L$ which extend to an isometry of $\lkt$ which fixes $L$.

For $\signature L = (2, n)$ let $G_{L} \da \SO_{L_\QQ}$ be its associated rational isometry group and let $\bX\da \Omega_L$ the associated Hermitian symmetric space as above, forming a Shimura datum $(\bX, G) \da (\Omega^L, \SO_{L_\QQ})$. We can then realize 
\[
\mathrm{Sh}_L(\CC) \da \mathrm{Sh}_{\bK_L}[G_L, \bX_L](\CC) \cong \widetilde{\SO_L}(\ZZ) \backslash \bX_L
\]
where $\bK_L \da \ker\qty{G_L \to \Aut(\discgroup{L}) }(\hat{\ZZ})$, the admissible morphism in $G_L(\bA_f)$; the stack $\mathrm{Sh}_{\bK}[G, \bX]$ is a certain well-known quotient stack attached to a Shimura datum $(G, \bX)$ and a choice of a compact open subgroup $\bK \leq G(\bA_f)$ of the finite adeles.

Defining the compact dual:
\[
\Omega^{L, \mathrm{cd}} \da \ts{x\in (L^\perp)_\CC \mid x^2 = 0}
.\]
    
:::