Hecke algebras for the basic characters of the unitriangular group

Abstract:

Let $U_{n}(q)$ denote the unitriangular group of degree $n$ over the finite field with $q$ elements. In a previous paper we obtained a decomposition of the regular character of $U_{n}(q)$ as an orthogonal sum of basic characters. In this paper, we study the irreducible constituents of an arbitrary basic character $\xi _{{\mathcal {D}}}(\varphi )$ of $U_{n}(q)$. We prove that $\xi _{ {\mathcal {D}}}(\varphi )$ is induced from a linear character of an algebra subgroup of $U_{n}(q)$, and we use the Hecke algebra associated with this linear character to describe the irreducible constituents of $\xi _{{\mathcal {D}}}(\varphi )$ as characters induced from an algebra subgroup of $U_{n}(q)$. Finally, we identify a special irreducible constituent of $\xi _{{\mathcal {D}}}(\varphi )$, which is also induced from a linear character of an algebra subgroup. In particular, we extend a previous result (proved under the assumption $p \geq n$ where $p$ is the characteristic of the field) that gives a necessary and sufficient condition for $\xi _{{\mathcal {D}}}(\varphi )$ to have a unique irreducible constituent.