In this article, we study the commutativity between the pull-back and the push-forward functors on constructible functions in Cluckers–Loeser motivic integration.

We lay the groundwork in this first installment of a series of papers aimed at developing a theory of Hrushovski–Kazhdan style motivic integration for certain types of nonarchimedean $o$-minimal fields, namely power-bounded $T$-convex valued fields, and closely related structures. The main result of the present paper is a canonical homomorphism between the Grothendieck semirings of certain categories of definable sets that are associated with the $\text{VF}$-sort and the $\text{RV}$-sort of the language ${\mathcal{L}}_{T\text{RV}}$. Many aspects of this homomorphism can be described explicitly. Since these categories do not carry volume forms, the formal groupification of the said homomorphism is understood as a universal additive invariant or a generalized Euler characteristic. It admits not just one, but two specializations to $\unicode[STIX]{x2124}$. The overall structure of the construction is modeled on that of the original Hrushovski–Kazhdan construction.

We define a notion of $p$-adic measure on Artin $n$-stacks that are of strongly finite type over the ring of $p$-adic integers. $p$-adic measure on schemes can be evaluated by counting points on the reduction of the scheme modulo ${{p}^{n}}$. We show that an analogous construction works in the case of Artin stacks as well if we count the points using the counting measure defined by Toën. As a consequence, we obtain the result that the Poincaré and Serre series of such stacks are rational functions, thus extending Denef's result for varieties. Finally, using motivic integration we show that as $p$ varies, the rationality of the Serre series of an Artin stack defined over the integers is uniform with respect to $p$.

We study the McKay correspondence for representations of the cyclic group of order $p$ in characteristic $p$. The main tool is the motivic integration generalized to quotient stacks associated to representations. Our version of the change of variables formula leads to an explicit computation of the stringy invariant of the quotient variety. A consequence is that a crepant resolution of the quotient variety (if any) has topological Euler characteristic $p$ as in the tame case. Also, we link a crepant resolution with a count of Artin–Schreier extensions of the power series field with respect to weights determined by ramification jumps and the representation.

For an arbitrary connected reductive group $G$, we consider the motivic integral over the arc space of an arbitrary $ \mathbb{Q} $-Gorenstein horospherical $G$-variety ${X}_{\Sigma } $ associated with a colored fan $\Sigma $ and prove a formula for the stringy $E$-function of ${X}_{\Sigma } $ which generalizes the one for toric varieties. We remark that, in contrast to toric varieties, the stringy $E$-function of a Gorenstein horospherical variety ${X}_{\Sigma } $ may be not a polynomial if some cones in $\Sigma $ have nonempty sets of colors. Using the stringy $E$-function, we can formulate and prove a new smoothness criterion for locally factorial horospherical varieties. We expect that this smoothness criterion holds for arbitrary spherical varieties.

We define a motivic analogue of the Haar measure for groups of the form $G(k((t)))$, where $k$ is an algebraically closed field of characteristic zero, and $G$ is a reductive algebraic group defined over $k$. A classical Haar measure on such groups does not exist since they are not locally compact. We use the theory of motivic integration introduced by M. Kontsevich to define an additive function on a certain natural Boolean algebra of subsets of $G(k((t)))$. This function takes values in the so-called dimensional completion of the Grothendieck ring of the category of varieties over the base field. It is invariant under translations by all elements of $G(k((t)))$, and therefore we call it a motivic analogue of Haar measure. We give an explicit construction of the motivic Haar measure, and then prove that the result is independent of all the choices that are made in the process.

We define invariants of the blow-Nash equivalence of Nash function germs, in a similar way to the motivic zeta functions of Denef and Loeser. As a key ingredient, we extend the virtual Betti numbers, which were known for real algebraic sets, to a generalized Euler characteristic for projective constructible arc-symmetric sets. Actually we prove more: the virtual Betti numbers are not only algebraic invariants, but also Nash invariants of arc-symmetric sets. Our zeta functions enable one to distinguish the blow-Nash equivalence classes of Brieskorn polynomials of two variables. We prove moreover that there are no moduli for the blow-Nash equivalence in the case of an algebraic family with isolated singularities.

The present work is devoted to the study of motivic integration on quotient singularities. We give a new proof of a form of the McKay correspondence previously proved by Batyrev. The paper contains also some general results on motivic integration on arbitrary singular spaces.