Representation theory of rank two Kac-Moody groups

• Main Author: Rose, Andrew James
• Published: University of Warwick 2011
• Subjects: 510; QA Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.560184

An example of an affine Kac-Moody group of rank two can be found in a central extension of the group of unimodular, two by two matrices over the non-archimedean local field Fq((z)). This thesis studies the representation theory of rank two Kac-Moody groups, paying particular attention to this example. We begin by determining an explicit formula for the cocycle associated to the central extension mentioned above. An exposition of the irreducible principle series representations of SL2(Fq((z))) follows along with a study of the irreducible finite dimensional representations of split and non-split tori in the central extension of SL2(Fq((z))). The geometry of affine Deligne-Lusztig varieties will be used in an attempt to generalise the theory of Deligne-Lusztig representations of finite groups of Lie type and to create representations of the locally compact group we are studying. The theory is developed along with some basic examples, although more complex results are yet to be completed. Finally the Hecke algebras of general rank two Kac-Moody groups are determined and their finite dimensional irreducible representations are classified. The local Shimura correspondence between representations of a group and representations of its Hecke algebra is established in this context.