Paper XI: Finite Heisenberg packets

7 Finite Heisenberg packets

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Let \(p\) be odd. Let \(X\) be an \(r\) dimensional vector space over \({\mathbb F}_p\), and put

\[ V=X\oplus X^*. \]

Define the alternating form

\[ b\bigl((x,\xi),(y,\eta)\bigr) = \eta(x)-\xi(y). \]

Let \(Z={\mathbb F}_p\), and define the Heisenberg group

\[ {\mathfrak H}(V)=V\times Z \]

with multiplication

\[ (v,z)(w,t) = \left(v+w,z+t+\frac12 b(v,w)\right). \]

Fix the additive character

\[ \psi(z)={\mathrm e}^{2\pi iz/p}. \]

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Definition 7.1: Schrödinger operators

On \({\mathcal H}={\mathbb C}^X\), define

\[ \bigl(\rho(x,\xi,z)f\bigr)(u) = \psi\left(z+\xi(u)+\frac12\xi(x)\right)f(u+x). \]

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Theorem 7.2: Finite Schrödinger representation

The map \(\rho:{\mathfrak H}(V)\to{\operatorname{GL}}({\mathcal H})\) is a representation. It is irreducible, its center acts by \(\psi\), and

\[ \dim{\mathcal H}=p^r=\sqrt{{\left\lvert V\right\rvert}}. \]

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Proof of Theorem 7.2

A direct calculation gives

\[ \rho(v,z)\rho(w,t) = \rho\left(v+w,z+t+\frac12b(v,w)\right). \]

Translations by \(X\) and multiplications by characters from \(X^*\) generate the full matrix algebra on \({\mathbb C}^X\). Indeed, the character multipliers give all diagonal projectors by finite Fourier inversion, and translations move each coordinate line to every other coordinate line. The commutant is therefore scalar, so the representation is irreducible. The dimension is \({\left\lvert X\right\rvert}=p^r\).

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Proposition 7.3: Character support

The character of the Schrödinger representation satisfies

\[ \boxed{ {\operatorname{Tr}}\rho(v,z) = \begin{cases} p^r\psi(z),&v=0,\\ 0,&v\ne0. \end{cases} } \]

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Proof of Proposition 7.3

If the translation component \(x\) is nonzero, the operator permutes the standard basis without a fixed point, so its trace is zero. If \(x=0\) but \(\xi\ne0\), the trace is

\[ \psi(z)\sum_{u\in X}\psi(\xi(u))=0. \]

When \(x=\xi=0\), the operator is the scalar \(\psi(z)\).

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Standard fact 7.4: Finite Stone von Neumann theorem

For a finite Heisenberg group with nondegenerate commutator form and a fixed nontrivial central character, there is one irreducible representation up to isomorphism. Its dimension is the square root of the order of the symplectic quotient.

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The standard fact follows from the twisted group algebra argument in Section 8. It is also the finite form used in the classical Weil representation literature [15], [26].

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7.1 Maximal isotropic selectors

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Let \(L\subseteq V\) be maximal isotropic. Choose a subgroup lift \(\widetilde L\subseteq{\mathfrak H}(V)\) on which the central coordinate is zero.

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Proposition 7.5: Isotropic averaging projector

The operator

\[ P_L = \frac1{{\left\lvert L\right\rvert}} \sum_{\ell\in\widetilde L} \rho(\ell) \]

is an orthogonal projector. In the Schrödinger model with \(L=X\), it has rank one and selects the constant functions.

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Proof of Proposition 7.5

The lifted isotropic subgroup is abelian, so group averaging is an idempotent. Every operator in the average is unitary, and the subgroup is closed under inversion, so the average is self adjoint. For \(L=X\), the average is the translation average

\[ (P_Lf)(u) = \frac1{{\left\lvert X\right\rvert}} \sum_{x\in X}f(u+x), \]

which is the constant function with value equal to the mean of \(f\). Its range is one dimensional.

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Corollary 7.6: Primitive line selector

Every maximal isotropic subgroup selects a one dimensional fixed line in the irreducible Heisenberg representation.

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Proof of Corollary 7.6

The projector rank equals its trace. The identity element contributes \(p^r\), while every nonidentity element of the lifted isotropic subgroup is noncentral and has trace zero by Proposition 7.3. Since \({\left\lvert L\right\rvert}=p^r\), we get

\[ {\operatorname{Tr}} P_L = \frac{p^r}{p^r}=1. \]

Thus the projector has rank one.

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The finite Heisenberg packet and its maximal isotropic selector.

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