# Aaron Pollack, Talk 2 

:::{.remark}
Last time: modular forms on $\G_2$.
Note that $\G_2$ over $K$ does not have a $\G_2\dash$invariant complex structure, while $\SL_2(\RR)/\SO_2 = \mfh$ has an $\SL_2(\RR)\dash$invariant complex structure.
:::

:::{.remark}
Today: let $f(z) = \sum_{k\geq 0} a_f(k)q^k$ of weight $\ell$ where $\phi_f(g) = j(g, i)^{-\ell} f(gi)$ where $\phi_f: \SL_2(\RR) \to \CC$.
Define 
\[
W_n: \SL_2(\RR) &\to \CC \\
g &\mapsto j(g, i)^{-\ell} \exp(2\pi i n (gi))
.\]

Some properties:

- $W_n\qty{\matt 1 * 0 1 g} = e^{2\pi i n x}W_n(g)$
- $W_n(gk_0) = e^{-i\ell \theta} W_n(g)$ where $k_\theta = \matt{\cos(\theta)}{-\sin(\theta)}{\sin(\theta)}{\cos(\theta)}$,
- $X_- W_n = 0$
- $W_n \diag(y^{1\over 2}, y^{-{1\over 2}}) = y^{\ell\over 2} e^{2\pi i ny}$ is complete explicit
- $\phi_f(g) = \sum a_f(k) W_k(g)$ is the Fourier expansion.

:::

:::{.remark}
What will happen: we'll define $\phi: \dcosetl{\Gamma}{\G_2}\to V_\ell$ where $V_\ell = \Sym^2(\CC^2)\tensor \one$ which admits an action by $K = \SU_2\times \SU_2/\pm I$.
In this case, we'll essentially have $\phi \approx \sum_{f\in ?} a_\phi(f) W_f(g)$ where the $a_\pi(f) \in \CC$ are Fourier coefficients and $W_f$ satisfies similar properties.
:::

:::{.remark}
Recall that $\lieg_2 = \liesl_3 + V_3 + V_3\dual$, spanned by $\ts{E_{ij}}, \ts{v_1,v_2,v_3}, \ts{\delta_1, \delta_2, \delta_3}$ respectively.

Note that

- $\G_2$ has 2 conjugacy classes of maximal parabolics,
- $P$ will be the parabolic where $\Lie(P)$ yields the top 3 layers of the root diagram
- $P = MN$ where $M\cong \GL_2$ and $N\contains Z = [N, N]$ with $N/Z$ abelian.

We want to define a Fourier expansion along the unipotent radical of $P$.
:::

:::{.remark}
Some facts:

- $Z = \exp(\RR E_{13})$ 
- $W = \RR E_{12} + \RR v_1 + \RR \delta_3 + \RR E_{23}$
- $N/Z = \exp(W)$
- $M\actson Z$ by the determinant
- $M\actson N/Z$ as $\Sym^3(V_3) \tensor \det(V_3)\inv$
- There is a symplectic form on $W$ where $[w, w'] = \inp{w}{w'} E_{13}$
- Explicitly, one can write $w = \sum a E_{12} + {b\over 3} v_1 + {c\over 3} \delta_3 + d E_{13}$ and $w'$ similarly, then $\inp{w}{w'} = ad' - {bc'\over 3} + {cb'\over 3}- da'$, and $\inp{mw}{mw'} = \det(m) \inp{w}{w'}$.

:::

:::{.remark}
What are the characters of $N$?
Suppose

- $\phi$ is an automorphic form on $\G_2(\AA)$
- $\psi: \dcosetl{\QQ}{\AA}\to \CC\units$ is a fixed adelic character
- $w\in W(\QQ)$

Define
\[
\phi_w(g) \da \int_{[N]} \psi\inv(\inp{w}{\bar n}) \phi(ng) \dn
,\]
where $\bar n$ is the image of $n$ in $N/Z$ which we identify with $W$ via the exponential.
Similarly define
\[
\phi_Z(g) = \int_{[Z]} \phi(zg)\dz,\qquad
\phi_N(g) = \int_{[N]} \phi(zg)\dz
.\]
Then
\[
\phi_Z(g) = \phi_N(g) + \sum_{w\in W(\QQ)} \phi_w(g)
,\]
and we'll produce a refinement.
:::

:::{.proposition title="?"}
\[
\phi_Z(g) \equiv 0 \implies \phi(g) \equiv 0
.\]
:::

## Generalized Whittaker Functions

:::{.definition title="Generalized Whittaker functions"}
Suppose $\phi: \dcosetl{\G_2(\QQ)}{\G_2(\AA)} \to V_\ell$ is a modular form of weight $\ell$.
These satisfy

- $\phi_w(g)$ is of moderate growth.
- $\phi_w(g) = \psi(\inp{w}{\bar n}) \psi_W(g)$ (equivariant for the Heisenberg parabolic)
- $\phi_w(gk) = k\inv \phi_w(g)$ (equivariant for $K$)
- $D_\ell \phi_w = 0$.

Call such functions satisfying these properties **general Whittaker functions of type $(w ,\ell)$**.
:::

:::{.remark}
We'll show that such functions are uniquely determined up to a scalar multiple, i.e. for some explicit $W_w$,
\[
\phi_w(g) = \lambda W_w(g)
.\]
From this, we'll obtain a Fourier expansion for $\phi$ a modular form of weight $\ell$:
\[
\phi_Z(g) = \phi_N(g) + \sum_{w\neq 0} a_{\phi}(w) W_w(g)
.\]
:::

:::{.remark}
Identify $W$ as a space $B$ of binary cubics under
\[
W &\to B \\
w \da a E_{12} + {b\over 3} v_1 + {c\over 3} \delta_3 + d E_{23}
&\mapsto
f_w\da
au^3 + bu^2v + cuv^2 + dv^3
.\]
For $w\in W(\RR)\smz$, for $m\in \GL_2(\RR)$ define
\[
\beta_w(m) \da \inp{w}{m\cdot (u-? v)^3}
,\]
which will appear in Fourier expansions.
:::

:::{.proposition title="?"}
TFAE:

- $\beta_w(m) \neq 0$ for all $m\in \GL_2(\RR)$,
- $f_w(z, 1) \neq 0$ for $z\in \mfh$,
- $f_w$ splits into linear factors over $\RR$.

:::

:::{.definition title="PSD"}
If $w$ satisfies these properties, say $w$ is **positive semidefinite** and write $w\geq 0$.
:::

:::{.example title="of PSD binary cubics"}
\envlist

- $f_w(u, v) \da au^3 \geq 0$.
- $f_w(u, v) \da -u^3 + uv^2 = u(v-u)(v+u)\geq 0$.
- $f_w(u, v) \da u^3 + v^2 \not\geq 0$.

:::

:::{.definition title="?"}
For $m\in \GL_2(\RR) = M(\RR)$ and $w\geq 0$ PSD,
\[
W_w(m) = \abs{\det w}\det(w)^\ell \sum_{-\ell \leq v\leq 0}
\qty{\abs{\beta_w(m)} \over \beta_w(m) }^v K_v(\abs{\beta_w(m)}) {x^{\ell+v} y^{\ell -v} \over (\ell+v)! (\ell-v)!}
.\]
where $x,y$ are a fixed basis of $V_\ell$, and $K_v$ is a classical $K\dash$Bessel function
\[
K_v(y) \da {1\over ?}\int_0^N e^{-{y(t+t\inv)\over 2 }} t^v {\dt\over t}
,\]
which diverges at $y=0$.
:::

:::{.remark}
These functions $W_w: M(\RR) \to V_\ell$ extend uniquely to $\G_2\to V_\ell$, viz

- $W_w(ng) = e^{2\pi i \inp w {\bar n}} W_w(g)$ for all $n\in N(\RR)$
- $W_w(gk) = k\inv W_w(g)$ for all $k\in K$.

:::

:::{.theorem title="?"}
Suppose $w\neq 0$ and $F$ is a generalized Whittaker function of type $(w, \ell)$.
Then

- $w\not\geq 0\implies F = 0$
- $w\geq 0\implies F(g) = \lambda W_w(g)$ for some $\lambda \in \CC$

Consequently, if $\phi$ is a modular form on $\G_2$ of weight $\ell$, there exist $a_\phi(w)\in \CC$ with
\[
\phi_Z(g) = \phi_N(g) + \sum_{w\geq 0 \text{ integral}} a_\phi(w) W_w(g)
.\]
Moreover, $\phi_N$ can be explicitly described in terms of modular forms of weight $3\ell$ on $\GL_2$.
:::

:::{.definition title="?"}
The terms $a_\phi(w)$ are by definition the Fourier coefficients of $\phi$.
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:::{.remark}
Gan-Gross-Savim used a multiplicity 1 result of Wallach to define the Fourier coefficients without using the explicit function $W_w(s)$.
:::