On tensor products of regular characters of the general linear and unitary groups of degree two over the principal ideal local rings of finite length
Let $R$ be a principal ideal local ring of finite length with a finite residue field of odd characteristic. Let $G(R)$ denote either the general linear group or the general unitary group of degree two over $R$. We study the decomposition of tensor products of irreducible representations of $G(R)$. It is known that the irreducible representations of $G(R)$ are built from regular representations, which are classified into three types: cuspidal, split semisimple, and split non-semisimple. We prove that the tensor product of any two regular irreducible representations of distinct types has irreducible constituents with multiplicity at most two. Moreover, we show that the regular part of the tensor product of a cuspidal representation with any other regular representation is multiplicity free. When both factors are of split semisimple type, we show that the multiplicity of any regular irreducible constituent is at most $\mathrm{length}(R) + 1$, and that this bound is achieved only when the constituent is also split semisimple. In contrast, we demonstrate that the multiplicity in the tensor product of two split non-semisimple representations can grow with the cardinality of the residue field when the length of the ring is at least two. In the case when $R$ is a finite field, all such tensor product multiplicities are uniformly bounded above by two. This highlights a significant difference between the behaviour of tensor products in the field case and in the more general finite local ring setting.