The arc space of horospherical varieties and motivic integration
Victor Batyrev, Anne Moreau

TL;DR
This paper extends motivic integration techniques to horospherical varieties, providing a formula for their stringy E-function and a new smoothness criterion, generalizing known results from toric varieties.
Contribution
It introduces a formula for the stringy E-function of horospherical varieties and proposes a new smoothness criterion, broadening the scope beyond toric varieties.
Findings
Stringy E-function formula for horospherical varieties
Non-polynomial nature of the E-function in certain cases
A new smoothness criterion for locally factorial horospherical varieties
Abstract
For arbitrary connected reductive group G we consider the motivic integral over the arc space of an arbitrary Q-Gorenstein horospherical G-variety associated with a colored fan and prove a formula for the stringy E-function of a horospherical variety X 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 may be not a polynomial if some cones in the fan of X 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.
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