Executive Summary : | Initiated by Frobenius and Schur more than a century ago, the representation theory of finite groups is currently a thriving field with many recent developments and discoveries, such as progress on representations of linear and arithmetic groups. Still, many of the problems of these two areas remain open. For example, a conjecture of Larsen and Lubotzky, which is still open for almost all cases, states that the abscissa of convergence of representation zeta functions of any two irreducible lattices in a higher rank semisimple group are equal. The above conjecture has been proved for groups of type A_2 by Avni, Klopsch, Voll and Onn and is open for A_k with k greater than two. The techniques involved use Margulis super-rigidity, the representation theory of simple algebraic groups, and Deligne-Lusztig theory for finite groups of Lie type. Further, it is clear that understanding of representations of groups of Lie type over finite principal ideal local rings is still missing from this picture, and research is very much ongoing on this subject. The first example of these groups is of GL_n(O_k), where O is the ring of integers of a non-archimedean local field with maximal ideal P and O_k= O/P^k. Our big goal is to determine all complex irreducible representations of GL_n(O_k) for k greater than one. The next step will be to extend this result to all other linear groups with entries from O_k. We point out that a construction of all irreducible representations of GL_n(O_k) for k = 1 was given by Green in 1955. However, for k greater than one, currently we only know the construction of all irreducible representations of GL_2(O_k), GL_n(O_2) and only a few partial results for GL_n(O_k) for general n and k. In this project, our goal is to decompose the tensor product of known irreducible representations of GL_n(O_k). Our objectives for this problem are twofold. On the one hand, this process leads to constructing and understanding new irreducible representations of a given group. On the other hand, because of its intricate nature, this problem gives rise to many new insights regarding the group and its known representations. We would like to harvest both of these benefits through this project. The decomposition of the tensor product of representations of finite groups is a hard problem in general. It is still not completely understood for the complex irreducible representations of the symmetric groups and the general linear groups over the finite field. The advantage that we have on hand is that in recent publications we have explored various results, including their construction regarding the "regular representations" of GL_n(O_k). We will utilise all the knowledge gained from those projects to tackle the current problem. We have obtained some preliminary results and have a few conjectures regarding the GL_2(O_k) case. Completing these results and extending those to GL_n(O_l) will be the goal of this project. |