2 Weil Deligne chains
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Let \(W_K\) be the Weil group of \(K\), let \(I_K\) be inertia, let \(P_K\) be wild inertia, and choose a geometric Frobenius lift \(\Phi\). A Weil Deligne representation is a pair \((r,N)\) with
Standard fact 2.1: Indecomposable monodromy blocks
Every Frobenius semisimple Weil Deligne representation is a direct sum of blocks
where \(\tau\) is an irreducible Weil representation and the nilpotent operator is one Jordan chain of length \(m\).
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This standard decomposition reduces the construction of a local correspondence to two tasks. One must understand irreducible Weil atoms, then one must attach the correct generalized Steinberg chain. The second task is formal once the first is known.
Definition 2.2: Monodromy packet chain
Let \(V_0,\ldots,V_{m-1}\) be copies of the space of \(\tau\). Put
A monodromy packet chain is a nilpotent operator \(N\) with
for \(j>0\), with \(N(V_0)=0\), together with a Frobenius operator whose conjugation scales \(N\) by \(q^{-1}\).
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Proposition 2.3: Exact block reconstruction
Let \(P_j\) be the projector onto \(V_j\). Then
Every power \(N^r\) is the sum of the packet paths of length \(r\) along the chain.
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Proof of Proposition 2.3
The projectors resolve the identity. Since \(N\) maps only from \(V_j\) to \(V_{j-1}\), every block except the displayed nearest neighbor blocks vanishes. A product of \(r\) nonzero blocks moves down exactly \(r\) levels. No path of length \(m\) exists, so \(N^m=0\).
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Proposition 2.4: Scalar covariance model
Let
and let \(N_{j-1,j}=1\). Then
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Proof of Proposition 2.4
The only nonzero entry in column \(j\) and row \(j-1\) is multiplied by \(q^{j-1}q^{-j}=q^{-1}\) under conjugation.
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Theorem 2.5: Formal atomic reconstruction
Assume that every irreducible Weil representation \(\tau\) has been assigned a supercuspidal representation \(\sigma_\tau\), compatibly with isomorphism and unramified twist. Then the rules
and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.
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Proof of Theorem 2.5
Decompose the parameter into indecomposable monodromy blocks. Replace each Weil atom by its assigned supercuspidal atom and each monodromy chain by the corresponding generalized Steinberg block. Order the resulting essentially square integrable factors in Langlands order and take the unique irreducible quotient. The multiset of indecomposable Weil Deligne blocks is intrinsic, so the result does not depend on the chosen display.
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Remark 2.6
Theorem 2.5 is a reconstruction theorem. It does not produce the atomic assignment \(\tau\mapsto\sigma_\tau\), and it does not prove the required local factor identities.