Affiliation:
1. Mathematisches Institut , Universität Bonn , Endenicher Allee 60, 53115 Bonn , Germany
Abstract
Abstract
We prove that the perfect loop functor LX of a quasi-projective scheme X over a local non-archimedean field k satisfies arc-descent, strengthening a result of Drinfeld. Then we prove that for an unramified reductive group G, the map
L
G
→
L
(
G
/
B
)
{LG\rightarrow L(G/B)}
is a v-surjection. This gives a mixed characteristic version (for v-topology) of an equal characteristic result (in étale topology) of Bouthier–Česnavičius.
In the second part of the article, we use the above results to introduce a well-behaved notion of Deligne–Lusztig spaces
X
w
(
b
)
{X_{w}(b)}
attached to unramified p-adic reductive groups. We show that in various cases these sheaves are ind-representable, thus partially solving a question of Boyarchenko. Finally, we show that the natural covering spaces
X
˙
w
˙
(
b
)
{\dot{X}_{\dot{w}}(b)}
are pro-étale torsors over clopen subsets of
X
w
(
b
)
{X_{w}(b)}
, and analyze some examples.
Subject
Applied Mathematics,General Mathematics
Reference55 articles.
1. J. Adams, X. He and S. Nie,
From conjugacy classes in the Weyl group to semisimple conjugacy classes,
Pure Appl. Math. Q. 17 (2021), no. 4, 1159–1189.
2. J. Anschütz,
Extending torsors on the punctured
Spec
(
A
inf
)
{{\rm Spec}(A_{\rm inf})}
,
preprint (2018), https://arxiv.org/abs/1804.06356;
to appear in J. reine angew. Math.
3. A. Beauville and Y. Laszlo,
Un lemme de descente,
C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 3, 335–340.
4. E. T. Beazley,
Codimensions of Newton strata for
SL
3
(
F
)
{\rm SL}_{3}(F)
in the Iwahori case,
Math. Z. 263 (2009), no. 3, 499–540.
5. B. Bhatt and A. Mathew,
The arc-topology,
Duke Math. J. 170 (2021), no. 9, 1899–1988.