Abstract
In their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of
1 Introduction
For integers
of
Soon after the Pappas–Rapoport conjecture was announced in 2000, Weyman ([15]) proved two cases: (i) when the base field is characteristic zero, and (ii) when the base field has arbitrary characteristic and
Pappas and Rapoport also proved interesting results about nilpotent orbits and affine Grassmannians conditional upon the reducedness of
Pappas and Rapoport’s conjecture arose in their investigation of local models of Shimura varieties. Shimura varieties serve as a bridge between arithmetic geometry and automorphic forms, and as such they play an important role in the Langlands program. Local models of Shimura varieties, which capture the behavior that occurs when considering reduction mod p, often allow one to reduce arithmetic problems to questions about affine Grassmannians and flag varieties ([13]). These problems tend to be difficult but often tractible (e.g. [6, 16, 5]).
1.1 Our approach
Our solution follows naturally by building on the ideas of our previous work [10] and earlier work by the first two authors and Kamnitzer [7]. One of our stated goals in [10, Section 1.3.2] is to use the framework developed there to understand the scheme structure of nilpotent orbit closures in positive characteristic. The present work is our first success in this program.
In [10], we proved a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman about the equations defining affine Grassmannians for
Our study led us to consider the general situation of a nilpotent operator T on a vector space V. We consider the Grassmannian
1.1.1 An alternate definition of
𝒮
T
Let
More can be said in the special case where
See Section 2.1 and Section 2.3 for explanation. The key fact is the following.
Theorem 1.1.
The morphism
This is a corollary of one of Pappas and Rapoport’s main theorems [11, Theorem 4.1]. For the benefit of the reader we explain in Section 2.3 how their proof works in our simpler setting.
Pappas and Rapoport consider a version of (1.1) over a discrete valuation ring that base changes to (1.1) over the residue field. In particular, the object that base changes to
1.1.2 Applying our previous results and Frobenius splitting
In [7], we developed some techniques for attacking a conjecture about the reduced scheme structure on Schubert varieties in the affine Grassmannian. We use this to prove that a certain family of
We note that this strategy is similar to the proof of [4, Theorem 4.5.1], where the special fiber
Finally, we show that for each n and e there is some
After analyzing our argument, we found a simplified but less conceptual version that is almost completely elementary. The only non-elementary ingredient is the Frobenius splitting of Schubert varieties. We present this simplified argument in Section 3.1.
2 Preliminaries
2.1 Grassmannians and associated schemes
Let
Further, let us assume that
For any T-invariant n-plane U, we consider the characteristic polynomial
We define for each
To define these, consider the
These operators were introduced in [10, Section 6.1], where also an explicit formula appears.
For each
and we refer to the homogeneous ideal defining
Theorem 2.1 ([10, Theorem 6.4 and Proposition 6.6]).
The subscheme
Remark 2.2.
In fact, we conjecture ([10, Conjecture 7.3]) that
The Stiefel variety
2.2 Alternative description of
𝒮
T
Theorem 2.3.
We have an equality
Proof.
Recall that
One can alternately define
factor through maps
Write
Because
2.3 Smoothness of the map
φ
:
𝒮
T
′
~
→
𝒩
n
,
e
We have a map
The map φ sends a pair
Because of the characteristic polynomial equation defining
Let
Theorem 2.4 ([11, Theorem 4.1]).
The morphism
The proof of [11, Theorem 4.1] holds in a more complicated arithmetic situation, and it simplifies greatly in the present one. For the benefit of the reader, we will briefly explain how their proof works in our situation.
Define a closed subscheme
It is clear that
We also have a map
given by
Remark 2.5.
For a general T with
2.4 Type A affine Grassmannians
Let
This condition means that we are only considering the connected component of the full affine Grassmannian of
For each pair of integers
Explicitly, we identify
This realizes
We may also consider
2.4.1 Big cells
Consider the decomposition
and the induced projection map
We can intersect the big cell of
Let
2.4.2 Schubert varieties
Recall that the
The Schubert varieties relevant to us are
We have
which is an explicit set-theoretic description of these Schubert varieties in terms of lattices. Because of Theorem 2.1, one immediately obtains closed embeddings
that are bijective on points. Below (Proposition 3.1) we will show that these are actually isomorphisms of schemes. In particular, the naïve lattice description of these Schubert varieties is scheme-theoretically correct once we add a further condition about the characteristic polynomial of t (as in the definition of
2.4.3 Diagram automorphism
Presentation (2.4) gives rise to a closed embedding of
Consider the involution of the ring of symmetric functions given by swapping elementary and homogeneous symmetric functions (denoted ω in [8, Section I.2]). This induces an involution ω of
where J is the
Remark 2.6.
Consider the group functor
Note that there are subtle technicalities involved in precisely constructing the quotient
3 The main proof
Proposition 3.1.
For all
from (2.5) are isomorphisms. In particular,
Proof.
First consider the embedding
such that
The second isomorphism follows by applying the diagram automorphism (Section 2.4.3) and observing that the shuffle ideal considered in [10] is invariant under it (see [10, Corollary 4.3, Theorem 4.6]). ∎
Remark 3.2.
Let L be an R-point of
In particular, the characteristic polynomial of this matrix is
Proposition 3.3.
Let
Proof.
It is clear that
that is bijective on points. Because the target of this map is reduced, so is its source. ∎
Remark 3.4.
The intersection
Recall by Theorem 2.3 and Theorem 2.4, we have a smooth map
Proposition 3.5.
Let e be an integer with
Proof.
As surjectivity is a set-theoretic statement, it is sufficient to work with reduced schemes.
Dividing with remainder, write
Theorem 3.6.
For every n, e with
Proof.
The scheme
is a torsor for
is smooth and surjective. In particular, it is faithfully flat, and the property of being reduced descends along faithfully flat morphisms. ∎
3.1 Another proof
After closely inspecting the above argument, we found the following direct proof of Theorem 3.6. The idea is essentially the same as the more conceptual proof above, but gets to the punchline quickly and by more elementary means. The only non-elementary ingredient is the Frobenius splitting of Schubert varieties.
Another proof of Theorem 3.6.
For any
We see that
Let X be the closed subscheme of
Observe that the condition
and
So X is equal to a closed subscheme of
The involution (2.6) of
Now consider the scheme theoretic intersection
On the other hand, we directly compute
such that
Recall that all schemes of the form
Corollary 3.7.
For all
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: JP19K14495
Funding source: Australian Research Council
Award Identifier / Grant number: DP180102563
Funding statement: Dinakar Muthiah was supported by JSPS KAKENHI Grant Number JP19K14495. Alex Weekes was supported in part by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. Oded Yacobi is supported by the ARC grant DP180102563.
Acknowledgements
We would like to thank an anonymous referee for their helpful report, and also the referee of [10] for suggesting the connection between our work and that of Pappas and Rapoport. We thank Geordie Williamson and Xinwen Zhu for their comments on an earlier version of this manuscript. We are also grateful to the Centre de Recherches Mathématiques and to Joel Kamnitzer and Hugh Thomas for organizing the program “Quiver Varieties and Representation Theory” where this project was started.
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