The Value Ring of Geometric Motivic Integration, and the Iwahori Hecke Algebra of Sl2 Ehud Hrushovski, David Kazhdan

In [1], an integration theory for valued fields was developed with a Grothendieck group approach. Two types of categories were studied. The first was of semi-algebraic sets over a valued field, with all semi-algebraic morphisms. The Grothendieck ring of this category was shown to admit two natural homomorphisms, esssentially into the Grothendieck ring of varieties over the residue field. These can be viewed as generalized Euler characteristics. The objects of the second category are semi-algebraic sets with volume forms; the morphisms are semialgebraic bijections preserving the absolute value of the volume form. (Some finer variants were also studied.) The Grothendieck ring of bounded objects in this category can be viewed as a universal integration theory. Even before the restriction to bounded sets, an isomorphism was shown between the semiring of semi-algebraic sets with measure preserving morphisms, and certain semirings formed out twisted varieties over the residue field, and rational polytopes over the value group. Though this description is very precise, the target remains complicated. With a view to representationtheoretic 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, in particular introducing additive inverses. 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. Let F be a valued field of residue characteristic 0. Let V be an F -variety. A semialgebraic subset of V is a Boolean combination of subvarieties and of sets defined by valuation inequalities {x ∈ U : valf(x) ≤ valg(x)}, where U is a relatively closed F -subvariety of V , and f, g are regular functions on U . (It is possible to think of the F -points defined by these equalities, but better to think of K-points where K is an undetermined valued field extension of F .) Let VolF be the category of semi-algebraic sets with bounded semi-algebraic volume forms; see 3.19 for a precise definition. The Jacobian of any semi-algebraic map between such objects can then be defined, outside a lower dimensional variety; morphisms are semi-algebraic bijections whose Jacobian has valuation zero (outside a lower dimensional variety.) The Grothendieck ring K(VolF ) of this category can be viewed as a universal integration theory for semialgebraic sets and volume forms over F . This ring is graded by dimension, but one can form out of it a ring K (VolF ) of “pure numbers”, ratios of integrals of equal dimension (see §1.1). We state there a version of Theorem 3.22 in the case of a higher dimensional local field. Let VarF be the category of algebraic varieties over the residue field of F . K df Q (VarF) is the dimension-free Grothendieck ring with rational coefficients this category. There exists a natural