Let $ F_q $ be a finite field with $ q $ elements. Let $ g $ be a multiplicative generator of $ F_{q^2}^* $. It implies that $ <g^{q+1}> = F_q^* $. Let $ l $ be a prime greater than $ q^2-1 $ and dividing $ q^{2(q-1)}-1 $.
Identify $ F_l^{q+1} $ with $ R = F_l [x]/(x^{q+1}- q^{-2}) $. Define the polynomial $$ m(x) = \sum_{b \in g+F_q} x^{\log_g b} $$ in $R$. Define two $ F_l $-linear maps on $ R $. The first one is $ M $, which is the multiplication by $ m(x) $ in $ R $. The second one is $ T $, which sends $ x^k $ to $ x^k x^{\log_g (c)} $ where $ c = (g^{k(q-1)+1}-1)/(g^{q}-g) $ for $ 0\leq k < q+1 $.
It can be proved that $ 1 $ is an eigenvalue of $ M T $. We have verified that for many $ q $, the eigenvalue $ 1 $ has multiplicity $ 1 $. Is it true for all prime power $ q>3 $ ? Has the similar problem been studied before?
