Associated Representations of finite pattern groups

Abstract: In this paper, we consider the construction of irreducible representations of finite pattern groups in terms of Panov's associative polarization, which is a finite-field analogue of Kirillov's orbital method. Using this construction, first, we are able to classify the irreducible representations of the unipotent radical of the standard parabolic subgroups of $\mathrm{GL}_n$ with 4 parts; second, we can parameterize irreducible characters of degree $q$ in terms of coadjoint orbits of cardinality $q^2$, for any finite pattern groups $G$ over $\mathbb{F}_q,$ where $\mathbb{F}_q$ is a finite field with $q$ elements.