Temperedness at ramified places for exceptional groups

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Temperedness at ramified places for exceptional groups

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Take a ramified place of a globally generic cuspidal representation of G2, F4, E6, E7 or E8 over a function field, and explain why the Weil part of the parameter can't decide temperedness on its own, and what does.

Why ramified places are hard

At an unramified place the Satake class determines everything, and a companion theorem already makes those components tempered. At a ramified place the parameter is a Weil–Deligne pair (r, N), and Lafforgue / Genestier–Lafforgue local–global compatibility only identifies r. [0:12]

The 2×2 example [0:50]: r(Fr) = diag(Q^{-1/2}, Q^{1/2}) with N e2 = e1 is pure of weight zero, since degrees ±1 match eigenvalue sizes Q^{±1/2}. Keep r the same but set N = 0, and purity fails: the L-factor picks up a pole at s = 1/2. Same Weil part, different monodromy, different verdict. [1:28]

The theorem and the local criterion

For split adjoint groups of type G2, F4, E6, E7 or E8 over a global function field, every local component of a globally generic cuspidal π is tempered. There is no restriction on the characteristic, the ramification depth or the local type. [1:51]

The key step is the pure completion criterion: if r admits one completion (r, N*) whose adjoint is pure of weight zero, then τ is tempered. [2:18]

Suppose τ isn't tempered. Its Langlands datum (σ, ν > 0) gives a second operator N_M over the same r.
Shahidi's local coefficient: the intertwiner acts on the one-dimensional Whittaker line by a nonzero scalar. That makes the adjoint L-factor of (r, N_M) regular at 1. [2:51]
(r, N*) is regular at 1 too. Regularity means the orbit under the centralizer of r is open, and only one orbit is open, so N_M ∼ N* and purity passes across.
On the nilradical n̂, purity gives |det| = 1, but the twist by ν gives a value strictly below 1. That's a contradiction. [3:39]

Where the purity comes from

Unramified Ramanujan makes Ad∘Σ pointwise pure. Deligne's theorem on local monodromy then gives a pure completion at every place, semisimple compatibility matches its Weil part with the parameter of π_v, and the criterion applies. [4:21]

Rule to use [4:50]: don't try to match monodromy operators by hand. Produce one pure weight-zero completion over the same r, and let adjoint regularity at 1 make the operators conjugate.

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