| Topological central extensions play a role in several areas of mathematics, e.g., in consideration of the congruence subgroup problem and of automorphic forms of fractional weight. The D in the title is a division algebra over a non-archimedean local field k. This thesis examines the topological central extensions of any parahoric sub-group P of SL{dollar}sb{lcub}n{rcub}(D),{dollar} where SL{dollar}sb{lcub}n{rcub}(D){dollar} is the set of n x n matrices with entries in D and of reduced norm one. Topological central extensions of P are determined by the second continuous cohomology group H{dollar}sp2(P,IR/doubz).{dollar} The main theorem of this thesis shows the p-primary component of H{dollar}sp2(P,IR/doubz){dollar} is a finite, cyclic p-group. Using the main theorem and Theorem 1.3 due to G. Prasad, I show the p-primary part of H{dollar}sp2(P,IR/doubz){dollar} is isomorphic to the p-primary component of H{dollar}sp2{dollar}(SL{dollar}sb{lcub}n{rcub}(D),IR/doubz).{dollar} I also compute the prime to p component of H{dollar}sp2(P,IR/doubz){dollar}. |
