Back to Search
Start Over
Intersection complexes and unramified đż-factors
- Source :
- Journal of the American Mathematical Society. 35:799-910
- Publication Year :
- 2021
- Publisher :
- American Mathematical Society (AMS), 2021.
-
Abstract
- Let X X be an affine spherical variety, possibly singular, and L + X \mathsf L^+X its arc space. The intersection complex of L + X \mathsf L^+X , or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified L L -functions. Such relationships were previously established in BravermanâFinkelbergâGaitsgoryâMirkoviÄ for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in BouthierâNgôâSakellaridis for toric varieties and L L -monoids. In this paper, we compute this intersection complex for the large class of those spherical G G -varieties whose dual group is equal to G Ë \check G , and the stalks of its nearby cycles on the horospherical degeneration of X X . We formulate the answer in terms of a Kashiwara crystal, which conjecturally corresponds to a finite-dimensional G Ë \check G -representation determined by the set of B B -invariant valuations on X X . We prove the latter conjecture in many cases. Under the sheafâfunction dictionary, our calculations give a formula for the Plancherel density of the IC function of L + X \mathsf L^+X as a ratio of local L L -values for a large class of spherical varieties.
- Subjects :
- Combinatorics
Intersection
Applied Mathematics
General Mathematics
Mathematics
Subjects
Details
- ISSN :
- 10886834 and 08940347
- Volume :
- 35
- Database :
- OpenAIRE
- Journal :
- Journal of the American Mathematical Society
- Accession number :
- edsair.doi...........ab96f2e64ee2c4701403da42c2a32ab3