Zur Konstruktion einfacher Charaktere und der Fortsetzungen ihrer Heisenbergdarstellungen für lokale zentral-einfache Algebren

by Grabitz, Martin

In this thesis, we try to explain how simple characters for arbitrary central simple algebras over a non-archimedian local field $F$ can be constructed. Moreover, we introduce a kind of matching of simple characters between different algebras of fixed reduced degree. If the index of the algebra $A$ is odd or $A=M_l(D)$, where $l$ is an arbitrary prime number and $D$ a central division algebra over $F$, we can extend the Heisenberg representations associated to the simple characters to level-0 and obtain a hypothetical list of simple types. For $A=M_l(D)$ and if the residual field of $F$ is not the field with two elements, we can proof that all so-called maximal simple types in our list are simple types in the sense of \cite{BK4} and their extensions to their stabelizers induce supercupidal representations of $G_l(D)$. Using the the heuristical relation via the abstract matching theorem of \cite{BDKV} to the cases of a division algebra due to \cite{Z5} and to the split case due to \cite{BK1}, we conjecture that all supercuspidal representations of $Gl_l(D)$ can be obtained by this way.