Some representations of SL∗(2,A)

The study of groups has been of interest to mathematicians since the 19th century. Although much is known about the structure of groups, many group theoretic problems remain unsolved. Representation theory allows us to employ linear algebra to solve such problems. The representation theory of linear groups over finite fields has been a particularly interesting topic. Studying these representations is of interest to mathematicians and other scientists as it relates to physics and modern number theory.

In the 1960s Andre Weil introduced a method for finding a special unitary representation for symplectic groups over locally compact fields. This unitary representation is now referred to as the Weil representation. In 2010 Luis Gutiérrez, José Pantoja and Jorge Soto-Andrade were able to generalize Weil's method to a larger class of linear groups namely the ∗-analogue of Sl₂.

Originally, Weil constructed this unitary representation, decomposed it into irreducibles and, in this way, produced the irreducible complex representations of Sp(2n,k). Later, Shalika went in the other direction, first finding the irreducible representations and then computing their multiplicities in the Weil representation. We intend to follow Shalika's method.

In this thesis we look to explore the representation theory of Sl_∗(2, A) where A is the direct sum of the upper and lower n × n block matrices in M(2n,k), k a finite field. We use Wigner and Mackey's Method of Little Groups to construct these representations.