$\def\FF{\mathbb{F}}\def\CC{\mathbb{C}}\def\Sp{\text{Sp}}\def\SL{\text{SL}}\def\GL{\text{GL}}\def\PGL{\text{PGL}}$Let $p$ be an odd prime. The Weil representation is a $p^g$-dimensional complex representation of $\Sp_{2g}(\FF_p)$. If you read most descriptions of the Weil representation, they hit a key technical point: "linearizing the projective representation". They then usually apologize for how difficult this is and either (1) cite the details to some one else (2) perform intricate computations with generators and relations of $\Sp_{2g}(\FF_p)$ (3) perform computations in group cohomology or (4) invoke tools from algebraic geometry.

I think I have a way to linearize the projective representation which is completely elementary, and which allows me to write down the matrices of $\Sp_{2g}(\FF_p)$ in a completely elementary way. So I'm writing to ask if anyone has seen this, or if they know any reason it can't work.

The standard description: Let $L$ be a $g$-dimensional vector space over $\FF_p$, let $L^{\vee}$ be the dual space and let $V = L \oplus L^{\vee}$, equipped with a symplectic form $( \ , \ )$ in the usual way. Let $H$ be the Heisenberg group, which is a certain extension $1 \to \FF^+_p \to H \to V \to 1$. I'll denote $\FF^+_p$ as $Z$ when I am thinking of it as the center of $H$. The Heisenberg group comes with a natural action of $\Sp(V)$; I'll write it as $h \mapsto h^g$ for $h \in H$ and $g \in \Sp(V)$. It is important to know that $z^g=z$ for all $z \in Z$ and $g \in \Sp(V)$.

Let $S$ be the $p^g$-dimensional vector space of $\CC$ valued functions on $L$. There is a natural action of $H$ on $S$ called the "Schrodinger representation", which is characterized as the unique irreducible representation of $H$ where $c \in \FF_p \subset H$ acts by $\zeta^c$. We'll write $\rho_S : H \to \GL(S)$ for the Schrodinger representation. I'll give explicit matrix formulas for $\rho_S$ below.

Our group $\Gamma$ will be the collection of $\gamma(R,q)$, for $R$ and $q$ obeying the condition of the lemma.

$\def\FF{\mathbb{F}}\def\CC{\mathbb{C}}\def\QQ{\mathbb{Q}}\def\Sp{\text{Sp}}\def\SL{\text{SL}}\def\GL{\text{GL}}\def\PGL{\text{PGL}}$Let $p$ be an odd prime. The Weil representation is a $p^k$-dimensional complex representation of $\Sp_{2k}(\FF_p)$. If you read most descriptions of the Weil representation, they hit a key technical point: "linearizing the projective representation". They then usually apologize for how difficult this is and either (1) cite the details to some one else (2) perform intricate computations with generators and relations of $\Sp_{2k}(\FF_p)$ (3) perform computations in group cohomology or (4) invoke tools from algebraic geometry.

I think I have a way to linearize the projective representation which is completely elementary, and which allows me to write down the matrices of $\Sp_{2k}(\FF_p)$ in a completely elementary way. So I'm writing to ask if anyone has seen this, or if they know any reason it can't work.

The standard description: Let $L$ be a $k$-dimensional vector space over $\FF_p$, let $L^{\vee}$ be the dual space and let $V = L \oplus L^{\vee}$, equipped with a symplectic form $( \ , \ )$ in the usual way. Let $H$ be the Heisenberg group, which is a certain extension $1 \to \FF^+_p \to H \to V \to 1$. I'll denote $\FF^+_p$ as $Z$ when I am thinking of it as the center of $H$. The Heisenberg group comes with a natural action of $\Sp(V)$; I'll write it as $h \mapsto h^g$ for $h \in H$ and $g \in \Sp(V)$. It is important to know that $z^g=z$ for all $z \in Z$ and $g \in \Sp(V)$.

Let $S$ be the $p^k$-dimensional vector space of $\CC$ valued functions on $L$. There is a natural action of $H$ on $S$ called the "Schrodinger representation", which is characterized as the unique irreducible representation of $H$ where $c \in \FF_p \subset H$ acts by $\zeta^c$. We'll write $\rho_S : H \to \GL(S)$ for the Schrodinger representation. I'll give explicit matrix formulas for $\rho_S$ below.

Our group $\Gamma$ will be the collection of $\gamma(R,q)$, for $R$ and $q$ obeying the condition of the lemma.

Incidentally, it is easy to describe the group $\Gamma H$: One just allows $R$ to be an affine linear space and takes $q$ an inhomogenous polynomial function of degree $\leq 2$ on $R$. When $k=1$, I believe this is the full normalizer of $H$ in $\SL_p(\CC)$; when $k>1$, I think it is the normalizer of $H$ in $\SL_{p^k}(\QQ(\zeta))$.

The Weil representation I will now give a similar description of a list of matrices normalizing $H$. Each matrix will be indexed by the following data: (1) A vector space $R \subseteq L \oplus L$ and (2) a quadratic form $q$ on $R$. These will obey conditions, to be described later. Define $$K(R,q)_{xy} = \begin{cases} \zeta^{q((x,y))} & (x,y) \in R \\ 0 & \text{otherwise} . \end{cases}.$$

The Weil representation I will now give a similar description of a list of matrices normalizing $H$. Each matrix will be indexed by the following data: (1) A vector space $R \subseteq L \oplus L$ and (2) a quadratic form $q$ on $R$. These will obey conditions, to be described later. Define $$K(R,q)_{yx} = \begin{cases} \zeta^{q((x,y))} & (x,y) \in R \\ 0 & \text{otherwise} . \end{cases}.$$