NEW DOCTORAL DEGREE Unitary duals of p-adic groups SO(5) and Mp(2)

The problem of the determination of the unitary dual of a reductive algebraic group is one of the most important problems in representation theory, with numerous applications in harmonic analysis and the theory of automorphic forms. In this thesis, we classify the unitary duals of some low-rank reductive $p$-adic groups. In the initial chapter, we systematically present the classification of the unitary dual of the group $SO(5, F)$ (modulo cuspidal representations), where $F$ is a local non-archimedean field. We use the method of Jacquet modules and intertwining operator methods. In the next chapter, we determine non-cuspidal part of the unitary dual of the non-linear metaplectic group $Mp(2)$, an unique nontrivial two-fold cover of the group $Sp(2)$ over a local non-archimedean field of characteristic different than two. We rely on the new results of authors Marcela Hanzer and Goran Mui\u27{c}, where they have related reducibilities of the representations of metaplectic groups with those of the special odd orthogonal groups, using theta correspondence. Besides that, those results allow us to use the method of Jacquet modules in investigating the composition series of the representations of the metaplectic group. In cases when this powerful methods happen to be insufficient tool, we use the theory of theta correspondence for dual pairs $Mp(n) times O(2r+1)$. Generalizing some of the results obtained while determining the unitary dual of the group $Mp(2)$, in the following chapter we prove irreducibility of the unitary principal series for $Mp(n)$. Let $Spin(2n+1)$ denote the simply-connected algebraic group of the type $B_{n}$, which can also be obtained as a two-fold cover of the classical group $SO(2n+1)$ if we are working over an algebraically closed field. In the representation theory, it is very important to know what the Levi subgroups in considered group look like, especially when investigating the parabolically induced representations. As a contribution to the study of this problem, in the final chapter we give a precise description of Levi subgroups of the group $Spin(2n+1)$ over algebraic closure of a $p$-adic field