The value ring of geometric motivic integration and the Iwahori Hecke algebra of SL₂

In \cite{HK}, an integration theory for valued fields was developed with a Grothendieck group approach. It was shown that the semiring of semi-algebraic sets with measure preserving morphisms is isomorphic to a certain semiring formed out of twisted varieties over the residue field and rational polytopes over the value group. With a view to representation-theoretic applications, we require a simpler description of the possible values of the integration, and in particular natural homomorphisms into fields. In the present paper we obtain such results after tensoring with Q. Since this operation trivializes the full semiring, we restrict to bounded sets. We show that the resulting Q-algebra is generated by its one-dimensional part. In the "geometric" case, i.e. working over an elementary submodel as a base, we determine the structure precisely. As a corollary we obtain useful canonical homomorphisms in the general case. In the appendix we define the Iwahori Hecke algebra of $SL_2$ over an algebraically closed valued field using motivic integration (to replace the Haar measure) and study its structure.