Cycles in the chamber homology for SL(2,F)

Wemedh Aeal

Abstract

We emphasized finding the explicit cycles in the chamber homology groups and the K-theory groups in term of each representation for SL(2,F). This led to an explicit computing of chamber homology and the K-theory groups. We have identified the base change effect on each of these cycles. The base change map on the homology group level works by sending a generator of the homology group of SL(2,E) labeled by a character of E× to the generator of the homology group of SL(2,F) labeled by a character of F× multiplied by the residue field degree. Whilst, it works by sending the K-theory group generator of the reduceC∗-algebraofSL(2,E)labeledbythe1-cycle(resp. 0-cycle)tothemultiplicationoftheresiduefield degree with a generator of the K-theory group of SL(2,F) labeled by the base changed effect on 1-cycle (resp. 0-cycle). 

Keywords

Local Langlands, Base change, K-theory, Chamber Homology, Baum-Conns map, representation theory, Non-commutative Geometry, Number Theory.

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