Poles of triple product L-functions involving monomial representations

In this paper, we study the order of the pole of the triple tensor product [Formula: see text]-functions [Formula: see text] for cuspidal automorphic representations [Formula: see text] of [Formula: see text] in the setting where one of the [Formula: see text] is a monomial representation. In the view of Brauer theory, this is a natural setting to consider. The results provided in this paper give crucial examples that can be used as a point of reference for Langlands’ beyond endoscopy proposal.

The monomial representations of the Clifford group

We show that the Clifford group---the normaliser of the Weyl-Heisenberg group---can be represented by monomial phase-permutation matrices if and only if the dimension is a square number. This simplifies expressions for SIC vectors, and has other applications to SICs and to Mutually Unbiased Bases. Exact solutions for SICs in dimension 16 are presented for the first time.

Local Fourier transform and epsilon factors

Laumon introduced the local Fourier transform forℓ-adic Galois representations of local fields, of equal characteristicpdifferent fromℓ, as a powerful tool for studying the Fourier–Deligne transform ofℓ-adic sheaves over the affine line. In this article, we compute explicitly the local Fourier transform of monomial representations satisfying a certain ramification condition, and deduce Laumon’s formula relating the ε-factor to the determinant of the local Fourier transform under the same condition.