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The arc space of horospherical varieties and motivic integration
- Part of
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- Journal:
- Compositio Mathematica / Volume 149 / Issue 8 / August 2013
- Published online by Cambridge University Press:
- 19 June 2013, pp. 1327-1352
- Print publication:
- August 2013
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- Article
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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.
