# 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
*shuffle operators*.

Our study led us to consider the general situation of a nilpotent operator *T* on a vector space *V*. We consider the Grassmannian
*T*-*shuffle operators* generalizing the shuffle operators (cf. Section 2.1).

#### 1.1.1 An alternate definition of
𝒮
T

Let
*T*-invariant *n*-planes *U* such that

More can be said in the special case where
*T* has Jordan type given by the partition

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
*V* be a finite-dimensional vector space over
*n* be a non-negative integer. Write
*n*-planes in *V*. Let
*V*, and let
*n* vector subbundle of

Further, let us assume that
*T*-invariant subspaces (see e.g. [10, Section 5.4]). When *n* and *V* are clear from context, we will simply write

For any *T*-invariant *n*-plane *U*, we consider the characteristic polynomial

We define for each
*shuffle operators*, which are linear operators

To define these, consider the
*n*-th wedge power

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
*shuffle ideal*. In our previous work, we proved the following.

### 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*
*n* linear transformations. We have a natural map
*X* inside the Stiefel variety. For example, we define the scheme

### 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
*R*, we have

The map φ sends a pair

Because of the characteristic polynomial equation defining

Let
*V* be a vector space of dimension *de* and suppose *T* is the nilpotent endomorphism of *V* that is given by a standard Jordan form corresponding to the partition

### 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
*R*, consider the ring
*T* defines an *S*-module structure on
*S*-module structure on

It is clear that

We also have a map
*T* this map will not be well behaved. However, our assumptions on the Jordan type of *T* imply that
*free*
*S*-module of rank *d*. Noting that there is an isomorphism

given by
*nd*. Therefore, the map

### 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
*t*. We therefore obtain a closed embedding

Explicitly, we identify

This realizes

We may also consider

#### 2.4.1 Big cells

Consider the decomposition

and the induced projection map
*big cell* of
*L* such that the restricted projection map
*R*-points are given by

We can intersect the big cell of

Let
*R*-points are given by

#### 2.4.2 Schubert varieties

Recall that the

The Schubert varieties relevant to us are
*n* times the first and last fundamental coweight, which are the minimal multiples that lie in the cocharacter (equivalently, coroot) lattice for

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
*Sato Grassmannian*

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
*j*, and where the superscript

#### Remark 2.6.

Consider the group functor
*R*-points are

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
*p*. In [7, Corollary 2] (see also the proof of [1, Proposition 6.1]), we proved that the intersection

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
*L* may be uniquely represented by a matrix polynomial
*R*-basis of
*t* is the companion matrix of the matrix polynomial

In particular, the characteristic polynomial of this matrix is
*p*. This in particular gives another proof of [10, Theorem 4.6].

## 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
*c* parts of size *e* and one part of size *f*. Then the reduced scheme of
*t* acts trivially on

## 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
*R* to:

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
*X* and
*X* is equal as a scheme to the intersection
*X* is reduced. More directly, the map
*X* and the nilpotent cone

Now consider the scheme theoretic intersection
*X* and

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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**Received:**2020-03-12

**Revised:**2020-08-06

**Published Online:**2020-10-08

**Published in Print:**2021-03-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.