Local Galois representations associated to additive polynomials (Q6564501)

Let \(p\) be a prime number and \(q\) a power of it. An additive polynomial \(R(x)\) over a finite field \(\mathbb F_{q}\) is a one-variable polynomial with coefficients in \(\mathbb F_{q}\) such that \(R(x+y)=R(x)+R(y)\). It is known that \(R(x)\) has the form\N\[\NR(x)=\sum_{i=0}^\nu a_{i}x^{p^{i}},a_\nu\neq 0,\text{ for some integer }\nu\geq 0\N\]\NLet \(F\) be be a non-archimedean local field with residue field \(\mathbb F_{q}\). Let \(W_{F}\) be the Weil group of \(\overline{F}/F\), where \(\overline{F}\) is a separable closure of \(F\). The author of the paper under review defines an irreducible smooth \(W_{F}\)-representation \(\tau_{\psi,R,m}\) over \(\mathbb Q_{l}\) of degree \(p^\nu\), where \(l\neq p\) is prime number, \(\psi:\mathbb F_{q}\to \mathbb Q_{l}^\times\) is a non-trivial character, \(R= R(x)\) a non-zero additive polynomial over \(\mathbb F_{q}\), and \(m\) is a positive integer. The integer \(m\) is related to the Swan conductor exponent of \(\tau_{\psi,R,m}\). An irreducible representation of a group is said to be primitive if it is not isomorphic to an induction of any representation of a proper subgroup. In the paper under review, the author finds a necessary and sufficient condition for \(\tau_{\psi,R,m}\) to be primitive. For this work he is using his previous works [\textit{N. Imai} and \textit{T. Tsushima}, Pac. J. Math. 326, No. 1, 37--83 (2023; Zbl 07781682); \textit{T. Tsushima}, J. Number Theory 214, 414--439 (2020; Zbl 1459.11139)] and some other authors works.