Hodge-Newton indecomposability and a combinatorial identity

TL;DR

This paper offers a new perspective on Hodge-Newton indecomposability and provides a unified proof of a combinatorial identity related to affine Deligne-Lusztig varieties with finite Coxeter elements.

Contribution

It introduces an alternative viewpoint on Hodge-Newton indecomposability and simplifies the proof of a key combinatorial identity in the context of affine Deligne-Lusztig varieties.

Findings

Unified proof of a combinatorial identity

New interpretation of Hodge-Newton indecomposability

Enhanced understanding of affine Deligne-Lusztig varieties

Abstract

We present a simple alternative viewpoint on Hodge-Newton indecomposability, illustrating its explanatory value through a uniform proof of a combinatorial identity arising from affine Deligne-Lusztig varieties with finite Coxeter part.

Equations

$$\sum _{[b]\in B(G,\mu {)}_{\text{indec}}}(\mathbf{q}-1{)}^{?}{\mathbf{q}}^{-??}=1.$$

$$B(G,\mu {)}_{indec}:=\{v\in (V^{+}{)}^{\sigma }:v\le \mu \text{ and }⟨\mu -v,{\varpi }_{{\mathcal{O}}_i}⟩\in {\mathbb{Z}}_{\ge 0}\text{leavevmode}\forall i\in \mathbb{S}\setminus I(v)\}.$$

$$\sum _{v\in B(G,\mu {)}_{\text{indec}}}(1-\mathbf{p}{)}^{{\sum }_{i\in \mathbb{S}/⟨\sigma ⟩}\lceil ⟨\mu -v,{\varpi }_{{\mathcal{O}}_i}⟩\rceil -\#(\mathbb{S}/⟨\sigma ⟩)}{\mathbf{p}}^{\#(\mathbb{S}/⟨\sigma ⟩)-\#(I(v)/⟨\sigma ⟩)}=1.$$

$$\left\{([i],⟨\mu ,{\varpi }_{{\mathcal{O}}_i}⟩-m):i\in \mathbb{S}\text{ and }m\in {\mathbb{Z}}_{\ge 0}\text{ such that }m<⟨\mu ,{\varpi }_{{\mathcal{O}}_i}⟩\right\}.$$

$$R-env(C):=\{v\in (V^{+}{)}^{\sigma }:v\le \mu \text{ and }⟨v,{\varpi }_{{\mathcal{O}}_i}⟩\ge C([i])\text{leavevmode}\forall i\in \mathbb{S}\}.$$

$$\mathrm{prob}(v)=(1-\mathbf{p}{)}^{{\sum }_{i\in \mathbb{S}/⟨\sigma ⟩}\lceil ⟨\mu -v,{\varpi }_{{\mathcal{O}}_i}⟩\rceil -\#(\mathbb{S}/⟨\sigma ⟩)}{\mathbf{p}}^{\#(\mathbb{S}/⟨\sigma ⟩)-\#(I(v)/⟨\sigma ⟩)}.$$

$$\sum _{[b]\in B(G,\mu {)}_{\mathrm{indec}}}(\mathbf{q}-1{)}^{\#(S/⟨\sigma ⟩)-\#(I({\nu }_b)/⟨\sigma ⟩)}{\mathbf{q}}^{-{\sum }_{i\in \mathbb{S}/⟨\sigma ⟩}\lceil ⟨\mu -{\nu }_b,{\varpi }_{{\mathcal{O}}_i}⟩\rceil +\#(I({\nu }_b)/⟨\sigma ⟩)}=1,$$

$${\varpi }_j^{\vee }=\sum _{i\in J}{c}_i{\alpha }_i^{\vee }+\sum _{k\in \partial J}{d}_k{\varpi }_k^{\vee }.$$

$$\#\{i\in \mathbb{S}:⟨v^{\prime },{\varpi }_{{\mathcal{O}}_i}⟩=C([i])\}>\#\{i\in \mathbb{S}:⟨v,{\varpi }_{{\mathcal{O}}_i}⟩=C([i])\}$$

$$\#\{i\in \mathbb{S}\setminus I(v^{\prime }):⟨v^{\prime },{\varpi }_{{\mathcal{O}}_i}⟩\ne C([i])\}<\#\{i\in \mathbb{S}\setminus I(v):⟨v,{\varpi }_{{\mathcal{O}}_i}⟩\ne C([i])\}$$

$$\{i\in \mathbb{S}:⟨v,{\varpi }_{{\mathcal{O}}_i}⟩=C([i])\}$$

$$=\{i\in \mathbb{S}\setminus J:⟨v^{\prime },{\varpi }_{{\mathcal{O}}_i}⟩=C([i])\}.$$

$$\sum _{\begin{array}{c}k\ge 1,1>\frac{a_1}{b_1}>\cdots >\frac{a_k}{b_k}>0\\ a_1+\cdots +a_k=i,b_1+\cdots +b_k=n\end{array}}(\mathbf{q}-1{)}^{k-1}{\mathbf{q}}^{1-k+\frac{{\sum }_{1\le l_1<l_2\le k}(a_{l_1}{b}_{l_2}-a_{l_2}{b}_{l_1})+{\sum }_{1\le l\le k}\gcd (a_l,b_l)}{2}}$$

$$={\mathbf{q}}^{\frac{i(n-i)-n}{2}+1}.$$

$$\sum _{P\in C}(1-\mathbf{p}{)}^{u(P)}{\mathbf{p}}^{b(P)}=1.$$

Full text

1. 1 Introduction
2. 2 Proof of the proposition
3. 3 An explicit identity as a motivation

1. Introduction

1.1. Background

In [HNY22], He-Nie-Yu studies the affine Deligne-Lusztig varieties with finite Coxeter parts. They study such types of varieties using the Deligne-Lusztig reduction method from [DL76] and carefully investigating the reduction path. In the approach, they establish the “multiplicity one” result which is, roughly speaking, for any $\sigma$-conjugacy class $[b]\in B(G)$, there is at most one path in the reduction tree that corresponds to $[b]$. The proof of the “multiplicity one” result is obtained by showing that a certain combinatorial identity (of two $\mathbf{q}$-polynomials, or more precisely, of the class polynomials) of the following form holds:

[TABLE]

They first reduce this to the case when $G$ is split and simply-laced and $\mu$ is a fundamental coweight ([HNY22, 6.5 and 6.6]). Then, for type $A$, they check the identity using some geometric properties of affine Deligne-Lusztig varieties, such as dimension formulae and injectiveness of the projection map from the affine flag variety to the affine Grassmannian ([HNY22, 5.4]). Also, for example, the type $E$ case is proven by computer. In our paper, we prove it all at once in the quasi-split case via a combinatorial proof.

1.2. The general identity and the sketch of proof

As in [HNY22, 2.1], let $G$ be a quasi-split reductive group over a local field $F$ and $T$ be a maximal torus constructed in loc.cit.. Denote by $W$ the relative Weyl group and by $\mathbb{S}$ the set of simple reflections, equipped with a Frobenius action $\sigma$.

Moreover, denote by $\Gamma_{0}$ the inertia group of $F$ and define $V\colonequals X_{*}(T)_{\Gamma_{0},\mathbb{Q}}$. The subset of dominant vectors will be denoted by $V^{+}$. Now, for each $i\in\mathbb{S}$, we denote the corresponding root, coroot, fundamental weight, and fundamental coweight by $\alpha_{i}$, $\alpha_{i}^{\vee}$, $\varpi_{i}$, and $\varpi_{i}^{\vee}$. Moreover, we denote by ${\mathcal{O}}_{i}$ the $\sigma$-orbit of $i$ and define $\alpha^{\vee}_{{\mathcal{O}}_{i}}\colonequals\frac{1}{\#{\mathcal{O}}_{i}}\sum_{j\in{\mathcal{O}}_{i}}\alpha_{j}^{\vee}$ and $\varpi_{{\mathcal{O}}_{i}}\colonequals\sum_{j\in{\mathcal{O}}_{i}}\varpi_{j}$.

Definition 1.1**.**

Given $v\in V$, define $I(v)$ as $\{i\in\mathbb{S}:\langle v,\alpha_{i}\rangle=0\}$ and define

[TABLE]

Note that $I(v)$ is $\sigma$-stable for any $v\in(V^{+})^{\sigma}$. We also note that our $B(G,\mu)_{\operatorname{indec}}$ is defined in this way for the sake of simplicity of the proof, but it is the (bijective) image of the original one via the dominant Newton map (see [HNY22, 6.2]).

Our main theorem is the following.

Theorem 1.2**.**

Let $\mu\in V^{+}$. Then, as polynomials of a variable $\mathbf{p}$,

[TABLE]

1.2.1. The setup and the proof

We explain the proof straight, but, in order to see the main idea more clearly, we suggest seeing Section 3.

In $(\mathbb{S}/\langle\sigma\rangle)\times\mathbb{R}_{>0}$, consider the following set $T$ of points

[TABLE]

Definition 1.3**.**

Fix a subset $C\subset T$. For each $i\in\mathbb{S}$, let us define $C([i])$ as $\max\{r\in\mathbb{R}_{>0}:([i],r)\in C\}$ with the convention that $\max\emptyset=0$.

We also define the rough envelope of $C$ by

[TABLE]

We will denote by $\operatorname{env}(C)$, the envelope of $C$, the set $\operatorname{R-env}(C)\cap B(G,\mu)_{indec}$.111In particular, we are not assuming that $\langle\mu-v,\varpi_{{\mathcal{O}}_{i}}\rangle\in\mathbb{Z}_{\geq 0}$ for all $i\in\mathbb{S}\setminus\!I(v)$ when considering $\operatorname{R-env}(C)$.

The main proposition of our paper is as follows.

Proposition 1.4**.**

Let $v\in\operatorname{env}(C)$. TFAE:

1. (1)

$v$* is the smallest among the ones in $\operatorname{env}(C)$.* 2. (2)

$v$* is minimal among the ones in $\operatorname{env}(C)$.* 3. (3)

For all $i\in\mathbb{S}\setminus\!I(v)$, we have $\langle v,\varpi_{{\mathcal{O}}_{i}}\rangle=C([i])$.

By taking any minimal element of $\operatorname{env}(C)$, we get the following corollary.

Corollary 1.5**.**

For any $C\subset T$, the smallest element of $\operatorname{env}(C)$ exists.

Now, we can prove the main theorem as follows.

Proof of Theorem 1.2.

We may assume that $0<\mathbf{p}<1$. Now, consider the following probabilistic process: For each $t\in T$, we select it with the probability $\mathbf{p}$. Then, for the set of the selected points, say $C$, we take the smallest element $v$ in $\operatorname{env}(C)$ whose existence is guaranteed by Corollary 1.5.

Given $v\in B(G,\mu)_{\operatorname{indec}}$, let $\operatorname{prob}(v)$ be the probability that this process results in $v$. Then, it is obvious that $\sum_{v\in B(G,\mu)_{\operatorname{indec}}}\operatorname{prob}(v)=1$ by the definition of $\operatorname{env}(C)$. Now, it is enough to show that

[TABLE]

By Proposition 1.4, the element $v$ is the smallest in $\operatorname{env}(C)$ if and only if $C([i])\leq\langle v,\varpi_{{\mathcal{O}}_{i}}\rangle$ for all $i\in\mathbb{S}$ with the equality holding for each $i\in\mathbb{S}\setminus\!I(v)$. The inequality part is equivalent to that the points above $([i],\langle v,\varpi_{{\mathcal{O}}_{i}}\rangle)$ are not selected. There are $\lceil\langle\mu-v,\varpi_{{\mathcal{O}}_{i}}\rangle\rceil-1$ number of such points so that we get the $(1-\mathbf{p})$-part. The equality part means that the point $([i],\langle v,\varpi_{{\mathcal{O}}_{i}}\rangle)$ must be selected for all $i\in\mathbb{S}\setminus\!I(v)$, giving $\mathbf{p}^{\#(\mathbb{S}\setminus\!I(v))/\langle\sigma\rangle}$ which is $\mathbf{p}^{\#(\mathbb{S}/\langle\sigma\rangle)-\#(I(v)/\langle\sigma\rangle)}$ as $\mathbb{S}$ and $I(v)$ are $\sigma$-stable. ∎

1.2.2. The identity $\spadesuit^{\prime}$ of [HNY22, 6.1]

By applying $\mathbf{p}=\frac{\mathbf{q}-1}{\mathbf{q}}$ to Theorem 1.2, we get

[TABLE]

where $B(G,\mu)_{\operatorname{indec}}$ here is the original one defined in [HNY22, 2.3].

2. Proof of the proposition

2.1. Two lemmas

Given a subset $J\subset\mathbb{S}$, we say that $J$ is connected if the corresponding graph in the Dynkin diagram is connected. We denote by $\partial J$ the set of vertices of $\mathbb{S}$ that is adjacent to $J$ but not in $J$. For example, when $J=\mathbb{S}$, we have $\partial J=\emptyset$. Note that if $J$ is $\sigma$-stable, then so is $\partial J$.

The following is crucial in the proofs of Proposition 1.4 and Corollary 1.5.

Lemma 2.1**.**

Let $J\subset\mathbb{S}$ be connected and $j$ be a vertex of $J$. Then, there exist $c_{i}\in\mathbb{Q}_{>0}$ for each $i\in J$ and $d_{k}\in\mathbb{Q}_{>0}$ for each $k\in\partial J$ such that

[TABLE]

The dual version also holds (with possibly different $c_{i}$’s and $d_{k}$’s).

Proof.

The main idea is that $J$ is a Dynkin diagram again. We now set $c_{i}$’s to be the coefficients, when expressing fundamental coweights of $J$, of the linear combination by coroots of $J$. Then, they are all positive (cf. [Lim23, Lemma 2.18]). Now, we need to compute what $\varpi_{j}^{\vee}-\sum_{i\in J}c_{i}\alpha_{i}^{\vee}$ looks like. We do this by computing $\langle-,\alpha_{l}\rangle$ for each $\alpha_{l}$. By construction, it is nonzero if and only if $l\in\partial J$. In that case, it is not just nonzero, but it is always negative as $\langle\alpha_{i}^{\vee},\alpha_{l}\rangle<0$ when $i$ and $l$ are adjacent, and $c_{i}$ is positive. ∎

Lemma 2.2**.**

For a subset $C\subset T$ and $v\in\operatorname{R-env}(C)$, suppose that there exists $i_{0}\in\mathbb{S}\setminus\!I(v)$ such that $\langle v,\varpi_{{\mathcal{O}}_{i_{0}}}\rangle\neq C([i_{0}])$. Then, one can find $v^{\prime}\in\operatorname{R-env}(C)$ such that either (S) or (Ic) holds.

[TABLE]

[TABLE]

Proof.

Definition of $u^{\diamond}$: Define $A(v)\colonequals\{i\in\mathbb{S}:\langle v,\varpi_{{\mathcal{O}}_{i}}\rangle\not=C([i])\}$ and $J$ be its connected component containing $i_{0}$. It is $\sigma$-stable. Applying lemma 2.1 to those $J$ and $i_{0}$, we denote by $u$ the resulting $\sum_{i\in J}c_{i}\alpha_{i}^{\vee}$ part and by $u^{\diamond}$ its $\sigma$-average. We want to set $v^{\prime}=v-\epsilon u^{\diamond}$ for some $\epsilon\in\mathbb{Q}_{>0}$.

Construction of $v^{\prime}$: Note that $\langle v,\varpi_{{\mathcal{O}}_{i}}\rangle-C([i])>0$ for all $i\in J$ by definition of $A(v)$. As $\langle u^{\diamond},\varpi_{{\mathcal{O}}_{i}}\rangle=\sum_{j\in{\mathcal{O}}_{i}}c_{j}>0$ for all $i\in J$ by lemma 2.1, we can find the maximal one $\epsilon_{1}$ such that $\langle v-\epsilon_{1}u^{\diamond},\varpi_{{\mathcal{O}}_{i}}\rangle-C([i])\geq 0$ for all $i\in J$. We take $\epsilon=\max\{\epsilon_{1},\#{\mathcal{O}}_{i_{0}}\langle v,\alpha_{i_{0}}\rangle\}$ and let $v^{\prime}\colonequals v-\epsilon u^{\diamond}$.

Verification of $v^{\prime}\in\operatorname{R-env}(C)$: Observe that $\langle u^{\diamond},\alpha_{i}\rangle>0$ only for $i\in{\mathcal{O}}_{i_{0}}$ and $\langle u^{\diamond},\alpha_{i}\rangle<0$ only for $i\in\partial J$. Other cases give $\langle u^{\diamond},\alpha_{i}\rangle=0$. Hence, we only need to consider $i\in{\mathcal{O}}_{i_{0}}$. However, $\langle v-\epsilon u^{\diamond},\alpha_{i}\rangle=\langle v,\alpha_{i_{0}}\rangle-\frac{1}{\#{\mathcal{O}}_{i_{0}}}\epsilon$ as $v$ is $\sigma$-invariant. It is $\geq 0$ since $\epsilon\geq$. As $u^{\diamond}$ is $\sigma$-invariant, so is $v^{\prime}$.

Verification of “(S) or (Ic)”: As $J\subset A(v)$ and $\langle u^{\diamond},\varpi_{{\mathcal{O}}_{i}}\rangle=0\leavevmode\nobreak\ \forall i\in\mathbb{S}\setminus\!J$,

[TABLE]

So, (S) is equivalent to $\{i\in J:\langle v-\epsilon u^{\diamond},\varpi_{{\mathcal{O}}_{i}}\rangle=C([i])\}\neq\emptyset$ which is equivalent to that $\epsilon=\epsilon_{1}$. Next, from the previous paragraph, we know that $\mathbb{S}\setminus\!I(v^{\prime})=(\mathbb{S}\setminus\!I(v)\setminus\!{\mathcal{O}}_{i_{0}})\cup\partial J$ if and only if $\epsilon=\#{\mathcal{O}}_{i_{0}}\langle v,\alpha_{i_{0}}\rangle$.222If $\epsilon_{1}=\epsilon<\#{\mathcal{O}}_{i_{0}}\langle v,\alpha_{i_{0}}\rangle$, then $\mathbb{S}\setminus\!I(v^{\prime})=(\mathbb{S}\setminus\!I(v))\cup\partial J$. However, for $i\in\partial J$, we have $\langle v,\varpi_{{\mathcal{O}}_{i}}\rangle=\langle v^{\prime},\varpi_{{\mathcal{O}}_{i}}\rangle=C([i])$ but $i_{0}$ belongs to only the right-hand side set in (Ic). This proves the claim. ∎

2.2. The proposition

Proof of Proposition 1.4.

(1)$\Rightarrow$(2): Trivial.

(2)$\Rightarrow$(3): Assume that (3) does not hold. As $v\in\operatorname{env}(C)\subset\operatorname{R-env}(C)$, we can apply lemma 2.2. If the new $v^{\prime}$ satisfies the assumption again, we keep repeating this process. As the number in (S) is $\leq\#S$ and that in (Ic) is $\geq 0$, the process terminates in finite steps. The final $v_{\operatorname{fin}}$ then satisfies that $\langle v_{\operatorname{fin}},\varpi_{{\mathcal{O}}_{i}}\rangle=C([i])$ for all $i\in\mathbb{S}\setminus\!I(v_{\operatorname{fin}})$. Noting that $C([i])=\langle\mu,\varpi_{{\mathcal{O}}_{i}}\rangle-m$ for some $m\in\mathbb{Z}_{\geq 0}$, we get $\langle\mu-v_{\operatorname{fin}},\varpi_{{\mathcal{O}}_{i}}\rangle\in\mathbb{Z}_{\geq 0}$ for all $i\in\mathbb{S}\setminus\!I(v_{\operatorname{fin}})$ meaning that $v_{\operatorname{fin}}\in\operatorname{env}(C)$. It contradicts to (2).

(3)$\Rightarrow$(1): Suppose that $w\in\operatorname{env}(C)$ and let us show that $w\geq v$. This is equivalent to that $\langle w-v,\varpi_{i}\rangle\geq 0$ for all $i\in\mathbb{S}$. Now recall that $\langle w,\varpi_{{\mathcal{O}}_{i}}\rangle\geq C([i])$ for all $i\in\mathbb{S}$ as $w\in\operatorname{env}(C)$. Moreover, we have $\langle v,\varpi_{{\mathcal{O}}_{i}}\rangle=C([i])$ for all $i\in\mathbb{S}\setminus\!I(v)$ by assumption. Hence, $\langle w-v,\varpi_{{\mathcal{O}}_{i}}\rangle\geq 0$ for all $i\in\mathbb{S}\setminus\!I(v)$. We get $\langle w-v,\varpi_{i}\rangle\geq 0$ for all $i\in\mathbb{S}\setminus\!I(v)$ as $w$ and $v$ are $\sigma$-invariant.

Let $j\in I(v)$. The dual version of lemma 2.1 applied to the connected component of $I(v)$ containing $j$ tells us that $\varpi_{j}$ is a nonnegative linear combination of $\alpha_{i}$ ($i\in I(v)$) and $\varpi_{k}$ ($k\in\mathbb{S}\setminus\!I(v)$). However, $\langle w,\alpha_{i}\rangle\geq 0$ as $w\in V^{+}$ and $\langle v,\alpha_{i}\rangle=0$ for all $i\in I(v)$, so we have $\langle w-v,\alpha_{i}\rangle\geq 0$ for all $i\in I(v)$. Therefore, $\langle w-v,\varpi_{j}\rangle\geq 0$ for all $j\in I(v)$. ∎

3. An explicit identity as a motivation

The main (probabilistic) idea of our paper arose proving this theorem:

Theorem 3.1** ([HNY22, 1.3]).**

For natural numbers $i<n$, the following holds:

[TABLE]

Now, we consider $\mathbb{R}^{2}$ and let $O=(0,0)$, $A=(i,n-i)$, and $B=(i,0)$. After modifications, Theorem 3.1 is equivalent to the following, for $\mathbf{p}=\frac{\mathbf{q}-1}{\mathbf{q}}$. Here, we only consider lattice points interior to $\triangle OAB$.

Proposition 3.2**.**

Let $C$ be the set of all convex piecewise linear curves which has only lattice points as non-linear points and lies in $\triangle OAB$ not touching $\overline{OB}$ and $\overline{AB}$. Then, the following identity holds. Here, $u(P)$ and $b(P)$ are the number of lattice points under $P$ and of non-linear points of $P$.

[TABLE]

Proof.

We may assume $0<\mathbf{p}<1$ and consider the following process. For each lattice point, choose (or abandon) it with the probability $\mathbf{p}$ (or $1-\mathbf{p}$). Then, take the minimal length curve from $O$ to $A$ under the chosen points.

Given $P\in C$, the process ends in $P$ exactly when

1. the lattice points under $P$ are abandoned ($=(1-\mathbf{p})^{u(P)}$) and

2. the non-linear points of $P$ are chosen ($=\mathbf{p}^{b(P)}$).

So, the probability of ‘the process ends in $P$’ is $(1-\mathbf{p})^{u(P)}\mathbf{p}^{b(P)}$ and, obviously, their sum must be $1$.

∎