9 A restricted one break Galois envelope
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We now isolate a finite Clifford setting in which the quotient projective action is defined on a multiplicity space. This avoids assigning a quotient action directly to arbitrary representatives of \(G/N\).
Definition 9.1: One break Clifford datum
A one break Clifford datum consists of finite groups and representations
with the following properties.
The quotient \(A\) is a finite elementary abelian \(p\) group.
The representations \(\Pi\) and \(\rho\) are irreducible.
The isomorphism class of \(\rho\) is fixed by \(G\).
The restriction of \(\Pi\) to \(N\) is \(\rho\) isotypic.
The quotient records one declared ramification break.
The multiplicity space is
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The isotypic condition gives an \(N\) equivariant evaluation isomorphism
Choose a section \(s:A\to G\). For every \(a\in A\), choose an intertwiner \(I_a\) from \(\rho\) to its conjugate by \(s(a)\). Under the evaluation isomorphism, there is a unique operator \(T_a\) on \(M_\rho\) such that
The operators \(T_a\) form a projective representation of \(A\).
Proposition 9.2: Clifford multiplicity action
The projective equivalence class of the action \((T_a)_{a\in A}\) is independent of the section, the intertwiners, and the chosen isotypic evaluation isomorphism. The projective action is irreducible.
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Proof of Proposition 9.2
Changing an intertwiner \(I_a\) by a scalar changes \(T_a\) by the inverse scalar. Changing the isotypic evaluation isomorphism conjugates all \(T_a\) by one element of \({\operatorname{GL}}(M_\rho)\). Changing the section multiplies \(s(a)\) by an element of \(N\). That factor is absorbed by the \(\rho\) action on \(W\), leaving a scalar change on the multiplicity operator by Schur's lemma. Thus all choices give projectively equivalent actions.
If a nonzero proper subspace of \(M_\rho\) were stable under all projective operators \(T_a\), then its tensor product with \(W\) would give a nonzero proper \(G\) stable subspace of \({\mathcal X}\). This contradicts irreducibility of \(\Pi\).
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Definition 9.3: Clifford cocycle class
Choose scalars \(c(a,b)\) such that
The cohomology class
is the Clifford cocycle class of the datum.
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Proposition 9.4: Intrinsic commutator pairing
The pairing
is independent of every choice in Definition 9.1. It is an alternating bicharacter
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Proof of Proposition 9.4
The projective equivalence class is intrinsic by Proposition 9.2. Conjugation does not change scalar commutators, and scalar changes alter the cocycle by a coboundary. The commutator pairing is therefore unchanged by Proposition 8.1. Its alternating and bilinear properties follow from Proposition 8.2. Since every element of \(A\) has order \(p\), every value has order dividing \(p\).
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Definition 9.5: One break symplectic envelope
The one break symplectic envelope is
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Theorem 9.6: Nondegenerate one break packet
If \(b_\rho\) is nondegenerate, then
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Proof of Theorem 9.6
The projective action gives a nonzero representation of the twisted group algebra on \(M_\rho\). The algebra is a full matrix algebra by Theorem 8.3. Its unique irreducible module has dimension \(\sqrt{{\left\lvert A\right\rvert}}\). The projective multiplicity action is irreducible by Proposition 9.2, so it is that module.
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Proposition 9.7: Isomorphism invariance
The cocycle class and the equivariant isometry class of \(V_\rho\) depend only on the isomorphism class of the one break Clifford datum.
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Proof of Proposition 9.7
An isomorphism of data transports the restriction, the multiplicity space, and its projective quotient action. It therefore transports the cocycle class, the commutator pairing, its radical, and the quotient.
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Proposition 9.8: Tame stabilizer action
Suppose a finite group \(C\) acts by automorphisms of the one break Clifford datum. Then \(C\) acts on \(V_\rho\) by symplectic automorphisms.
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Proof of Proposition 9.8
The automorphism action transports the intrinsic projective multiplicity representation. It therefore preserves the scalar commutator pairing and its radical. The induced action on the quotient is symplectic.
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Corollary 9.9: Heisenberg representation attached to one break
Fix a nontrivial central character. The envelope \(V_\rho\) determines one irreducible finite Heisenberg representation up to isomorphism.
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Proof of Corollary 9.9
Apply the finite Stone von Neumann theorem to the nondegenerate symplectic quotient.
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Remark 9.10
The group cohomology class records the extension obstruction for the multiplicity action. General background on projective representations and group cohomology can be found in [22], [3]. The theorem remains restricted because the finite quotient and the one break isotypic datum are assumed rather than constructed from arbitrary wild inertia.