10 The type side symplectic quotient
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Let \(\theta\) be a maximal simple character for \({\operatorname{GL}}_n(K)\), with compact open subgroups
The quotient
is regarded as a finite vector space over \({\mathbb F}_p\), where \(p\) is the residue characteristic. Fix a nontrivial additive character \(\psi_0:{\mathbb F}_p\to\mu_p\subset{\mathbb C}^\times\), where \(\mu_p\) is the group of \(p\)-th roots of unity. The simple-character commutator takes values in \(\mu_p\). Since \(\psi_0\) is injective, the equation
uniquely defines the \({\mathbb F}_p\)-valued commutator pairing \(h_\theta\). We use the same identification of \(\mu_p\) with the additive group of \({\mathbb F}_p\) on the Galois side when comparing the two symplectic spaces.
Standard fact 10.1: Type side Heisenberg quotient
The pairing \(h_\theta\) is nondegenerate and alternating. There is one irreducible representation \(\eta_\theta\) of \(J_\theta^1\) containing \(\theta\), and its dimension is
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This is part of the simple character and Heisenberg representation theory used in the construction of types [11]. The extension of \(\eta_\theta\) from \(J_\theta^1\) to \(J_\theta\) is not unique before a normalization is chosen.
Standard fact 10.2: Canonical beta extension
The canonical beta extension is obtained from a primary extension together with finite symplectic and tame orientation corrections. The construction is functorial in the declared simple character data and removes the remaining extension ambiguity.
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The normalization and its relation to finite Weil representations are developed in [2], [14]. The present article uses this theorem as type side input. It does not rederive simple character theory.
Proposition 10.3: Type side projective commutator
The projective action of \(J_\theta/J_\theta^1\) on the Heisenberg representation preserves the commutator form on \(V_\theta\). Any two choices of projective lifts give the same alternating form.
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Proof of Proposition 10.3
Conjugation by \(J_\theta\) preserves \(\theta\), so it preserves the commutator pairing defined from \(\theta([x,y])\). Two choices of projective lifts differ by scalars. The scalar commutator is therefore unchanged by Proposition 8.1.
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