Non-existence of non-trivial normal elements in the Iwasawa Algebra of Chevalley groups

TL;DR

This paper proves that in the Iwasawa algebra of certain Chevalley groups over zp, the only normal elements are units, removing previous restrictions on the prime p and confirming a conjecture.

Contribution

It provides a new proof of the non-existence of non-trivial normal elements in the Iwasawa algebra, eliminating the 'nice prime' condition and confirming a conjecture.

Findings

All non-zero normal elements are units in the Iwasawa algebra.

The proof does not require the prime p to be 'nice'.

The result confirms the conjecture by Ardakov, Wei, and Zhang.

Abstract

For a prime $p>2$, let $G$ be a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_p$, $G(1)$ be the first congruence kernel of $G$ and $\Omega_{G(1)}$ be the mod-$p$ Iwasawa algebra defined over the finite field $\mathbb{F}_p$. Ardakov, Wei, Zhang have shown that if $p$ is a "nice prime " ($p \geq 5$ and $p \nmid n+1$ if the Lie algebra of $G(1)$ is of type $A_n$), then every non-zero normal element in $\Omega_{G(1)}$ is a unit. Furthermore, they conjecture in their paper that their nice prime condition is superfluous. The main goal of this article is to provide an entirely new proof of Ardakov, Wei and Zhang's result using explicit presentation of Iwasawa algebra developed by the second author of this article and thus eliminating the nice prime condition, therefore proving their conjecture.

Equations

$${\Lambda }_G:=\underset{N⊴G}{\underleftarrow{\lim }}{\mathbb{Z}}_p[G/N],$$

$${\Omega }_G:=\underset{N⊴G}{\underleftarrow{\lim }}{\mathbb{F}}_p[G/N],$$

$$M⟶{\mathrm{Hom}}_{A^{\mathsf{op}}}({\mathrm{Hom}}_A(M,A),A)$$

$$\{A_n(n\ge 1),B_n(n\ge 2),C_n(n\ge 3),D_n(n\ge 4),E_6,E_7,E_8,F_4,G_2\}$$

$$G(1):=\mathrm{Ker}(G({\mathbb{Z}}_p)⟶G({\mathbb{Z}}_p/p{\mathbb{Z}}_p)).$$

$$G(k):=\ker (G({\mathbb{Z}}_p)⟶G({\mathbb{Z}}_p/p^k{\mathbb{Z}}_p)).$$

$$\{H_{{\delta }_1},\cdots ,H_{{\delta }_{\ell }},X_{\alpha },\alpha \in \Phi ,{\delta }_1,\cdots ,{\delta }_{\ell }\in \Pi \}$$

$$h_{\alpha }(t):={\omega }_{\alpha }(t){{\omega }_{\alpha }(1)}^{-1},$$

$$\omega :G(1)$$

$$x$$

$$\{x_{\beta }(p),h_{\delta }(1+p),x_{\alpha }(p),\beta \in {\Phi }^{-},\delta \in \Pi ,\alpha \in {\Phi }^{+}\}$$

$$c:{\mathbb{Z}}_p^d$$

$$(y_1,\cdots ,y_d)$$

$$c^{\ast }:C(G(1))\simeq C({\mathbb{Z}}_p^d).$$

$${\Lambda }_{G(1)}={\mathrm{Hom}}_{{\mathbb{Z}}_p}(C(G(1)),{\mathbb{Z}}_p)$$

$$x_{\alpha }(p)-1$$

$$h_{\delta }(1+p)-1$$

$$x_{{\alpha }_1}(p^{r+1})x_{{\alpha }_2}(p^{s+1})=x_{{\alpha }_2}(p^{s+1})x_{{\alpha }_1}(p^{r+1}).$$

$$[y_{{\alpha }_1}^{p^r},y_{{\alpha }_2}^{p^s}]=[y_{{\alpha }_1}^{p^r},y_{{\alpha }_2}^{p^s}{]}_{\circ }=0,$$

$$x_{{\alpha }_1}^{p^{r+1}}{x}_{{\alpha }_2}^{p^{s+1}}=\prod _{i,j>0}{x}_{i{\alpha }_1+j{\alpha }_2}(c_{ij}{p}^{i(r+1)+j(s+1)})\times x_{{\alpha }_2}(p^{s+1})x_{{\alpha }_1}(p^{r+1}).$$

$$(1+y_{{\alpha }_1}^{p^r})(1+y_{{\alpha }_2}^{p^s})=$$

$$\times (1+y_{{\alpha }_2}^{p^s})(1+y_{{\alpha }_1}^{p^r}).$$

$$[y_{\alpha }^{p^r},y_{\delta }^{p^s}]=$$

$$\times h_{\delta }((1+p{)}^{p^s}).$$

$$x_{\alpha }((1+p{)}^{p^s⟨\alpha ,\delta ⟩}{p}^{r+1})h_{\delta }((1+p{)}^{p^s})=h_{\delta }((1+p{)}^{p^s})x_{\alpha }(p^{r+1}).$$

$$x_{\alpha }((1+p{)}^{p^s⟨\alpha ,\delta ⟩}{p}^{r+1})=h_{\delta }((1+p{)}^{p^s})x_{\alpha }(p^{r+1})h_{\delta }((1+p{)}^{-p^s}).$$

$$x_{\alpha }(-p^{r+1})x_{\alpha }((1+p{)}^{p^s⟨\alpha ,\delta ⟩}{p}^{r+1})=x_{\alpha }(-p^{r+1})h_{\delta }((1+p{)}^{p^s})x_{\alpha }(p^{r+1})h_{\delta }((1+p{)}^{-p^s}).$$

$$x_{\alpha }(p^{r+1}(1+p{)}^{p^s⟨\alpha ,\delta ⟩}-p^{r+1})=x_{\alpha }(-p^{r+1})h_{\delta }((1+p{)}^{p^s})x_{\alpha }(p^{r+1})h_{\delta }((1+p{)}^{-p^s}).$$

$$[y_{\alpha }^{p^r},y_{\delta }^{p^s}]=(1+y_{\alpha }{)}^{p^r}[1-(1+y_{\alpha }{)}^{p^r(1+p{)}^{p^s⟨\alpha ,\delta ⟩}-p^r}](1+y_{\delta }{)}^{p^r}.$$

$$[y_{\alpha }^{p^r},y_{\delta }^{p^s}{]}_{\circ }=-⟨\alpha ,\delta ⟩y_{\alpha }^{p^{r+s+1}}.$$

$$\displaystyle\left[\begin{array}[]{cc}1&-p^{r+1}\\ 0&1\end{array}\right]\left[\begin{array}[]{cc}1&0\\ p^{s+1}&1\end{array}\right]\left[\begin{array}[]{cc}1&p^{r+1}\\ 0&1\end{array}\right]\left[\begin{array}[]{cc}1&0\\ -p^{s+1}&1\end{array}\right]$$

$$=$$

$$\displaystyle\times\left[\begin{array}[]{cc}1&0\\ -p^{r+2s+3}(1+p^{r+s+2})^{-1}&1\end{array}\right].$$

$${\mathrm{SL}}_2({\mathbb{Z}}_p)⟶⟨{\mathfrak{X}}_{\alpha },{\mathfrak{X}}_{-\alpha }⟩$$

Full text

The second author would like to thank PIMS-CNRS and the University of British Columbia for postdoctoral research grant. He is also thankful to Beijing Institute of Technology for its hospitality during a visit on August 2018 when this collaboration took place.

1. 1 Introduction

2. 2 Basic Setup on Lazard Ordered Basis

3. 3 Calculation of the Lowest Degree Terms of Commutators

4. 4 Partial Differential Equations

5. 5 Main Result and Its Proof

6. 6 Proof of Claim 5.3

7. 6.1 Applications to center

8. 6.2 Future questions

1. Introduction

Let $p$ be a prime integer, and let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers. A group $G$ is compact p-adic analytic if it is a topological group which has the structure of a $p$-adic analytic manifold - that is, it has an atlas of open subsets of $\mathbb{Z}^{n}_{p}$ , for some $n\geq 0$. There is a more intrinsic way to characterize such kinds of groups. A topological group $G$ is compact $p$-adic analytic if and only if $G$ is a closed subgroup of the general linear group ${\rm GL}_{n}(\mathbb{Z}_{p})$ for some $n\geq 1$. The research object of this article is the so-called Iwasawa algebras of $G$:

[TABLE]

where the inverse limit is taken over the open normal subgroups $N$ of $G$. Modulo $p$, the epimorphic image of $\Lambda_{G}$ is denoted by $\Omega_{G}=\Lambda_{G}\bigotimes_{\mathbb{Z}_{p}}\mathbb{F}_{p}$ and

[TABLE]

where $\mathbb{F}_{p}$ is the finite field of $p$ elements. For any odd prime $p$, Clozel in his paper [6] gave explicit presentations for the afore-mentioned two Iwasawa algebras over the first congruence subgroup of ${\rm SL}_{2}(\mathbb{Z}_{p})$, which is $\Gamma_{1}({\rm SL}_{2}(\mathbb{Z}_{p}))={\rm Ker}({\rm SL}_{2}(\mathbb{Z}_{p})\longrightarrow{\rm SL}_{2}(\mathbb{F}_{p}))$. Ray [20, 21, 22, 23] generalized Clozel’s work to the following three cases: the first congruence kernel of a semi-simple, simply connected Chevalley group over $\mathbb{Z}_{p}$, general uniform pro-$p$ groups, and pro-$p$ Iwahori subgroups of ${\rm GL}_{n}(\mathbb{Z}_{p})$.

In this article, we are going to look at the normal elements in $\Omega_{G}$ and the ideals generated by them. They are defined as $r\in\Omega_{G}$ such that $r\Omega_{G}=\Omega_{G}r$. Each normal element $r$ gives rise to a two-sided reflexive ideal $r\Omega_{G}$. Let us first recall the definition of reflexive ideals. Let $A$ be any algebra and $M$ be a left $A$-module. We call $M$ reflexive if the canonical mapping

[TABLE]

is an isomorphism. A reflexive right $A$-module is defined similarly. We will call a two-sided ideal $I$ of $A$ reflexive if it is reflexive as a right and as a left $A$-module.

Let $\Phi$ be a root system of $G$ having fixed a split maximal torus, so that the Dynkin diagram of any indecomposable component of $\Phi$ belongs to

[TABLE]

Let $\Phi(\mathbb{Z}_{p})$ denote the $\mathbb{Z}_{p}$-Lie algebra constructed by using a Chevalley basis associated to $\Phi$.

We say that $p$ is a nice prime for $\Phi$ if $p\geq 5$ and if $p\nmid n+1$ when $\Phi$ has an indecomposable component of type $A_{n}$. Ardakov, the third author of the current article and Zhang have showed that

Theorem 1.1**.**

[2, Theorem A and Theorem B],[3, Corollary 0.3]* Let $G$ be a torsionfree compact $p$-adic analytic group whose $\mathbb{Q}_{p}$-Lie algebra $\mathcal{L}(G)$ is split semisimple over $\mathbb{Q}_{p}$. Suppose that $p$ is a nice prime for the root system $\Phi$ of $\mathcal{L}(G)$. Then the mod-$p$ Iwasawa algebra $\Omega_{G}$ has no non-trivial two-sided reflexive ideals. In particular, every non-zero normal element of $\Omega_{G}$ is a unit.*

Ardakov, the third author of the current article and Zhang conjecture (see paragraph before section 0.4 of [3]) that the nice prime condition in their theorem above is superfluous. Now, for the rest of this paper, we assume that $G$ is a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_{p}$, and that $G(1)$ is the first congruence kernel defined by

[TABLE]

In this paper we prove that “nice prime” condition in indeed superfluous for $G(1)$ and thereby confirming to Ardakov, the third author of the current article and Zhang’s conjecture. Furthermore, our method of proof is completely different from that of Ardakov and in based on an explicit presentation of Iwasawa algebras and works for any prime $p>2$. The main theorem of this article, extending earlier works of [13, 14, 31, 32] for ${\rm SL}_{2}(\mathbb{Z}_{p})$, ${\rm SL}_{3}(\mathbb{Z}_{p})$ and ${\rm SL}_{n}(\mathbb{Z}_{p})$, is the following.

Theorem**.**

(Theorem 5.1) Let $p$ be a prime with $p\geq 3$ and $G$ be a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_{p}$. Suppose that $G$ is one of the following Chevalley groups of Lie type: $A_{\ell}\,(\ell\geq 1),B_{\ell}\,(\ell\geq 2),C_{\ell}\,(\ell\geq 2),D_{\ell}\,(\ell\geq 3),E_{6},E_{7},E_{8},F_{4},G_{2}$. Then for any nonzero element $W\in\Omega_{G(1)}$, $W$ is a normal element if and only if $W$, as a noncommuttaive formal power series, contains constant terms. In this case, $W$ is a unit.

A direct consequence of this result is reproving that the center of $\Omega_{G(1)}$ is trivial (again originally done by Ardakov in [1].) This is carried out in Proposition 6.1.

It should be remarked that we need $p>2$, otherwise $G(1)$ is not torsion free and $\Omega_{G(1)}$ is not an integral domain and our proof is heavily based on Lazard’s theory of $p$-valued group and Iwasawa algebras where we need our Iwasawa algebras to be an integral domain.

The roadmap of this article is as follows. After Introduction, we recall the basic notions of Lazard basis for $p$-adic analytic groups in Section 2. Section 3 is contributed to the computations of the lowest degree terms of the commutators between the generators of $\Omega_{G(1)}$ (see Proposition 3.1). In Section 4, we calculate the partial differential equations (cf. Lemma 4.1). The proof of our main result (Theorem 5.1) is given in Section 5. While proving our main theorem, we need an important claim (see Claim 5.3) which we prove later in Section 6 using Dynkin diagrams and the partial differential equations. Section 6.1 contains applications to center reproving Ardakov’s result. Future questions are discussed in Section 6.2.

2. Basic Setup on Lazard Ordered Basis

Let $G$ be a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_{p}$, $\Phi$ be the root system with respect to a maximal torus, $\Pi$ be a set of simple roots. We can view $G$ as a group scheme over $\mathbb{Z}$ (cf. [12, XXV]) and the congruence kernels can be defined as

[TABLE]

Let

[TABLE]

be a Chevalley basis in the Lie algebra Lie$(G)$ of $G$ (cf. [27, Page 6]). Let $x_{\alpha}(t)=\exp\,tX_{\alpha}$, which is the one dimensional unipotent subgroup in the $\mathbb{Q}_{p}$ points of $G$ if $t\in\mathbb{Q}_{p}$. For $t\in\mathbb{Q}_{p}^{*}$, we define

[TABLE]

where $\omega_{\alpha}(t):=x_{\alpha}(t)x_{-\alpha}(-t^{-1})x_{\alpha}(t)$ (cf. [27, Lemma 19 of Section 3]). By invoking [24, Page 171], we know that the function

[TABLE]

is a $p$-valuation on $G(1)$ in the sense of Lazard [19, III, 2.1.2]. Here, by convention, we put $\omega(1)=\infty$. We recall, from [20, Theorem 2.2], that the elements

[TABLE]

form a Lazard ordered basis (cf. [19, III, 2.2.4]) for ($G(1),\omega$). The ordering in (2.1) is given by a fixed order on the roots such that the height function on the roots increases. Also, in (2.1), $\Phi^{-}$ (resp. $\Phi^{+}$) denotes the negative (resp. positive) roots in $\Phi$. Moreover, since $\omega(x_{\beta}(p))=\omega(x_{\alpha}(p))=\omega(h_{\delta}(1+p))$ and $p$ is assumed to be $>2$, $G(1)$ is also $p$-saturated with respect to its $p$-valuation  (cf. [19, III, 2.2.7.1]). Furthermore, we would like to point out that $G(1)$ is a uniform pro-$p$ group in the sense of [11]. A nice exposition can be found in Sections 2, 3 and 4 of the Arxiv version ([20, Arxiv]) of [20].

Now, let $g_{1},\cdots,g_{d}\,(d=|\Phi|+|\Pi|)$ be the ordered basis as in (2.1). Taking into account the definition of Lazard ordered basis, we have a homomorphism

[TABLE]

Let $C(G(1))$ be the set of continuous functions from $G(1)$ to $\mathbb{Z}_{p}$. The mapping $c$ induces, by pulling back functions, an isomorphism of $\mathbb{Z}_{p}$-modules

[TABLE]

Dualizing this isomorphism, we obtain

[TABLE]

Identifying $y_{\alpha}$ with $x_{\alpha}(p)-1$ and $y_{\delta}$ with $h_{\delta}(1+p)-1$, we observe that any element in $\Lambda_{G(1)}$ (and $\Omega_{G(1)}$) can be written as a uniquely determined power series in $y_{\alpha}$’s and $y_{\delta}$’s with coefficients in $\mathbb{Z}_{p}$ (resp. $\mathbb{F}_{p}$). Note that the ordering of $y_{\alpha}$ and $y_{\delta}$ are according to increasing height function on the roots as in the case of Lazard ordered basis (cf. $y_{\alpha}=x_{\alpha}(p)-1$, $y_{\delta}=h_{\delta}(1+p)-1$, see equation (2.1) and the discussion following it). We refer the reader to [24] for a nice introduction to Iwasawa algebras and Lazard ordered basis.

3. Calculation of the Lowest Degree Terms of Commutators

Let $y_{\alpha}=x_{\alpha}(p)-1$ and $y_{\delta}=h_{\delta}(1+p)-1$. In this section we are going to find out the lowest degree terms of all commutator relation between $y_{\alpha}$ and $y_{\delta}$ in $\Omega_{G(1)}$. The strategy is to use firstly Steinberg’s Chevalley relation in $G(1)$ following [27] (which gave us the explicit relations for the Iwasawa algebra in [20, Lemma 3.1]) and then determine the commutators of elements $y_{\alpha}$’s and $y_{\delta}$’s. Furthermore we determine the lowest degree terms of the commutators (cf. Proposition 3.1).

Throughout this section $\alpha\in\Phi$, $\delta\in\Pi$, $\Pi=\{\delta_{1},\cdots,\delta_{\ell}\}$, $\ell=|\Pi|$, $y_{\alpha}=x_{\alpha}(p)-1$ and $y_{\delta}=h_{\delta}(1+p)-1$. We use $[-,-]$ to denote the commutators of elements $y_{\alpha}$’s and $y_{\delta}$’s and use $[-,-]_{\circ}$ to denote the lowest degree terms of the commutators $[-,-]$. We recall that we are working in the mod-$p$ Iwasawa algebra $\Omega_{G(1)}$.

Proposition 3.1**.**

Let $r,s$ be non-negative integers and $p$ be a prime integer with $p>5$. Then we have

1. (1)

$[y^{p^{r}}_{\alpha_{1}},y^{p^{s}}_{\alpha_{2}}]_{\circ}=0$, whenever $\alpha_{1},\alpha_{2}\in\Phi$, $\alpha_{1}\neq-\alpha_{2}$, $\alpha_{1}+\alpha_{2}\notin\Phi$, 2. (2)

$[y^{p^{r}}_{\alpha_{1}},y^{p^{s}}_{\alpha_{2}}]_{\circ}=c_{11}y^{p^{r+s+1}}_{\alpha_{1}+\alpha_{2}}$, whenever $\alpha_{1}+\alpha_{2}\in\Phi$, $\alpha_{1}\neq-\alpha_{2}$, 3. (3)

$[y^{p^{r}}_{\alpha},y^{p^{s}}_{\delta}]_{\circ}=-\langle\alpha,\delta\rangle y^{p^{r+s+1}}_{\alpha}$, 4. (4)

$[y^{p^{r}}_{\alpha},y^{p^{s}}_{-\alpha}]_{\circ}=-\sum_{i=1}^{\ell}n_{i}y^{r+s+1}_{\delta_{i}}$,

where $c_{11}\in\{\pm 1,\pm 2,\pm 3\}$, $\langle\alpha,\delta\rangle=\frac{2(\alpha\,|\,\delta)}{(\delta\,|\,\delta)}\in\mathbb{Z}$ is the so-called Cartan integer (cf. [27, Page 2]). The $n_{i}$’s in $(4)$ are such that $\alpha=\sum_{i=1}^{\ell}n_{i}\delta_{i}$, $\delta_{i}\in\Pi$. By the discussion in [27, Page 5], we know that $\langle\alpha,\delta\rangle\in$ $\{\pm 1,0,\pm 1,\pm 2\}$.

Proof.

If $\alpha_{1},\alpha_{2}\in\Phi$, $\alpha_{1}\neq-\alpha_{2}$, $\alpha_{1}+\alpha_{2}\notin\Phi$, then Example (a) in [27, Page 24] gives

[TABLE]

Note that $y^{p^{r}}_{\alpha_{1}}=x_{\alpha_{1}}(p^{r+1})-1$  (we are working in $\Omega_{G(1)}$, which is defined over $\mathbb{F}_{p}$), we therefore get

[TABLE]

which is the desired assertion (1).

If $\alpha_{1}+\alpha_{2}\in\Phi$ and $\alpha_{1}\neq-\alpha_{2}$, then applying the relation R2 of [27, Page 30] yields

[TABLE]

By Page 64 of [12, Exposé XXIII], we see that $c_{ij}\in\{\pm 1,\pm 2,\pm 3\}$. Thus we arrive at

[TABLE]

And hence $[y^{p^{r}}_{\alpha_{1}},y^{p^{s}}_{\alpha_{2}}]_{\circ}=c_{11}y^{p^{r+s+1}}_{\alpha_{1}+\alpha_{2}}$, $c_{11}\in\{\pm 1,\pm 2,\pm 3\}$. This is the assertion (2). Note that assertion (2) also holds for $p=5$. The same proof works.

Now let $\alpha\in\Phi$, $\delta\in\Pi$ as before. Then

[TABLE]

By the relation R8 [27, Page 30], we get

[TABLE]

And hence

[TABLE]

So

[TABLE]

By invoking the relation R1 in [27, Page 30], we see that

[TABLE]

This implies that

[TABLE]

We therefore have

[TABLE]

This proves the assertion (3).

For the Chevalley group ${\rm SL}_{2}(\mathbb{Z}_{p})$, we by [31, Page 1025] know that

[TABLE]

Recall that we have a homomorphism (cf. [27, Corollary 6 of Page 46])

[TABLE]

[TABLE]

So we conclude that

[TABLE]

One should note that

[TABLE]

It follows from the properties of $p$-adic integers that there exists one element $\beta$ such that

[TABLE]

where $\beta=\beta_{0}+\beta_{1}p+\beta_{2}p^{2}+\cdots+\beta_{r+s}p^{r+s}+\beta_{r+s+1}p^{r+s+1}+\cdots$, $\beta_{k}\in\mathbb{Z}$ and $0\leq\beta_{k}\leq(p-1)$. According to the expansion formula of $(1+p^{r+s+2})^{-1}$, we can compute all $\beta_{k}$. For instance, $\beta_{0}=\beta_{1}=\cdots=\beta_{r+s}=0$, $\beta_{r+s+1}=p-1$, $\beta_{r+s+2}=(\frac{p^{r+s+1}-1}{2}-1)\mod p$. Now,

[TABLE]

Since $G$ is simply connected, by the proof of [27, Corollary 5 in Page 44]  (see also [27, Corollary of Lemma 28 in Page 44]), we know that there exist integers $n_{i}$ such that

[TABLE]

uniquely. Thus we get

[TABLE]

In order to determine the lowest degree term of $[y^{p^{r}}_{\alpha},y^{p^{s}}_{-\alpha}]$, our next step is to find out how large the $n_{i}$’s can be. It should be remarked that the highest root  (the root in $\Phi$ with maximum height) denoted by $\alpha_{\max}$ has the maximum $n_{i}$’s  (cf. [4, Proposition 25 in Page 178]). That is, let us set $\alpha_{\max}=\underset{\delta_{i}\in\Pi}{\sum}n_{i}^{\max}\delta_{i}$ and let $\alpha$ be any root with $\alpha=\sum_{\delta_{i}\in\Pi}n_{i}\delta_{i}$. Then $n_{i}\leq n_{i}^{\max}$, $i\in[1,\ell]$, $\ell=|\Pi|$. Now, one can check, case by case, for type $A_{\ell}$, $B_{\ell}$, $C_{\ell}$, $D_{\ell}$ and exceptional Lie type algebras these $n_{i}^{\max}$ from Bourbaki table (cf. [4, Chapter VI, Section 4]).

1. (1)

Type $A_{\ell}$: $\underset{i}{\rm Max}\{n_{i}^{\max}\}=1$, 2. (2)

Type $B_{\ell}\,(\ell\geq 2)$: $\underset{i}{\rm Max}\{n_{i}^{\max}\}=2$  (Page 214 of [4, Chapter IV, Section 5]), 3. (3)

Type $C_{\ell}\,(\ell\geq 2)$: $\underset{i}{\rm Max}\{n_{i}^{\max}\}=2$  (Page 216 of [4]), 4. (4)

Type $D_{\ell}\,(\ell\geq 3)$: $\underset{i}{\rm Max}\{n_{i}^{\max}\}=2$ (Page 220 of [4]), 5. (5)

Type $E_{6}$: $\underset{i}{\rm Max}\{n_{i}^{\max}\}=3$  (Page 229 of [4]), 6. (6)

Type $E_{7}$: $\underset{i}{\rm Max}\{n_{i}^{\max}\}=4$  (Page 227 of [4]), 7. (7)

Type $E_{8}$: $\underset{i}{\rm Max}\{n_{i}^{\max}\}=6$  (Page 226 of [4]), 8. (8)

Type $F_{4}$: $\underset{i}{\rm Max}\{n_{i}^{\max}\}=4$  (Page 223 of [4]), 9. (9)

Type $G_{2}$: $\underset{i}{\rm Max}\{n_{i}^{\max}\}=3$  (Page 232 of [4]).

Let $p$ satisfies the following hypothesis.

[TABLE]

Note that $HYP_{\Phi}$ depends on the type of the Lie algebra. Also note that if $p>5$ (which was the hypothesis assumed in proposition 3.1) then $p$ satisfies $HYP_{\Phi}$ for all root types $\Phi$.

For primes $p$ satisfying $HYP_{\Phi}$, the lowest degree terms of $[y^{p^{r}}_{\alpha},y^{p^{s}}_{-\alpha}]$ will come from the part $\prod_{\delta_{i}\in\Pi}(1+y_{\delta_{i}})^{n_{i}\beta}$ because $\beta_{0}=\beta_{1}=\cdots=\beta_{r+s}=0$, $\beta_{r+s+1}=p-1$ and $p^{2r+s+2}$ or $p^{r+2s+2}$ is strictly larger than $n_{i}p^{r+s+1}$, $r,s\geq 0$, because $p$ satisfies $HYP_{\Phi}$

Now

[TABLE]

Therefore we obtain

[TABLE]

This completes the proof. ∎

Let us extend Proposition 3.1 to the cases of $p=3$ and $p=5$.

We are going to deduce the lowest degree terms of the commutators whenever $p=5$ and $p=3$.

For the prime integer $p=5$, note that the relations (1), (2), (3) of Proposition 3.1 remain true because $c_{11}\in\{\pm 1,\pm 2,\pm 3\}$, $\langle\alpha,\delta\rangle\in\{0,\pm 1,\pm 2\}$. We only need to modify (4) of Proposition 3.1 for $p=5$. Notice that (4) of Proposition 3.1 is the lowest degree term of the commutator relation $[y_{\alpha}^{p^{r}},y_{-\alpha}^{p^{s}}]_{\circ}$. By 3.1 we know that

[TABLE]

For the prime integer $p=5$, the argument given before to deduce the lowest degree term $[y_{\alpha}^{p^{r}},y_{-\alpha}^{p^{s}}]_{\circ}$ of $[y_{\alpha}^{p^{r}},y_{-\alpha}^{p^{s}}]$ only fails for type $E_{8}$ where $\underset{i}{\rm max}\,\{n_{i}^{\rm max}\}=6$ (That is $p=5$ does not satisfy $HYP_{\Phi}$). From [4, Page 225], we see that if $\alpha$ is a $+ve$ root and $\alpha=\sum_{i=1}^{8}n_{i}\delta_{i}$ is the root system of type $E_{8}$, then there must exist some $n_{i}$’s such that $n_{i}$’s $<5$. In other words, all of $n_{i}$’s can not be $\geq 5$ for all $i\in[1,8]$. Then, for type $E_{8}$ and prime integer $p=5$, the lowest degree term

[TABLE]

We get the above lowest degree term by using the similar argument as (4) of Proposition 3.1 using equation 3.1 and noticing that $p^{2r+s+2}$ or $p^{r+2s+2}$ are less than or equal to $n_{i}p^{r+s+1}$ for $r=s=0$ and $n_{i}\geq 5$. The point is $[y_{\alpha}^{5^{r}},y_{-\alpha}^{5^{s}}]_{\circ}$ contains only powers $y_{-}^{5^{r+s+1}}$ and this is what we will use in the future. This completes the proof of case $p=5$.

Now let us consider the case of $p=3$. It should be remarked that (1) and (3) of Proposition 3.1 remain unchanged as $\langle\alpha,\delta\rangle\in\{0,\pm 1,\pm 2\}$. Let us look at $[y_{\alpha_{1}}^{p^{r}},y_{\alpha_{2}}^{p^{s}}]_{\circ}$ if $\alpha_{1}+\alpha_{2}\in\Phi$, $\alpha_{1}\neq-\alpha_{2}$ (i.e. part (2) of Proposition 3.1) for $p=3$. Note that we have

[TABLE]

Here $c_{ij}\in\{\pm 1,\pm 2,\pm 3\}$. Suppose that $c_{11}\neq\pm 3$. Then

[TABLE]

Suppose that $c_{11}=\pm 3$. Since $(r+1)+(s+1)=r+s+2\leq i(r+1)+j(s+1)-1$ for all $(i,j)$ with $i,j>0$ and $(i,j)\neq(1,1)$, we have

[TABLE]

where $s_{11}\neq 0$ and $s_{ij}$ for $i,j>0$, $(i,j)\neq(1,1)$ might be zero. Here also, the main point is not the exact values of the $s_{i,j}$’s but the fact that $[y^{3^{r}}_{\alpha_{1}},y^{3^{s}}_{\alpha_{2}}]_{\circ}$ contains only powers $y_{-}^{3^{r+s+2}}$. This will later be used in the proof of our main theorem.

Let us next look at (4) of Proposition 3.1 for $p=3$ and $p$ does not satify $HYP_{\Phi}$. We just need to find $[y_{\alpha}^{3^{r}},y_{-\alpha}^{3^{s}}]_{\circ}$ for $p=3$. Again from Bourbaki classification of root systems and corresponding expressions for $\alpha_{\rm max}$ for type $F_{4},E_{6},E_{7},E_{8}$ and $G_{2}$ [4, Page 223–Page 232], we see that if $\alpha\in\Phi$, $\alpha$ $+ve$ root, $\alpha=\sum_{i=1}^{\ell}n_{i}\delta_{i}$, then there must exist some $n_{i}$ such that $n_{i}\leq 2$. So, by the same argument as for the case $p=5$ and type $E_{8}$, we get

[TABLE]

This completes the computation of the lowest degree terms for the prime integer $p=3$.

4. Partial Differential Equations

Let

[TABLE]

where the sum is finite, $n_{\underline{{\xi}}}=((n_{\xi}),\xi\in\Phi^{-},\Pi,\Phi^{+})$, $n_{\xi}\in\mathbb{N}\cup\{0\}$, $a_{\underline{\xi}}\in\mathbb{F}_{p}$, $\underline{\xi}=(\xi\in\Phi^{-},\Pi,\Phi^{+})$, i.e., $\underline{\xi}$ is just the collection of the $-ve$, simple and $+ve$ roots ordered according to the order of Lazard basis, i.e. according to an order compatible with increasing height function on the roots.

Explanation of multi-index notation in (4.1): Before proceeding, for the convenience of the reader, let us explain our multi-index notation (4.1) in a little more details because we will be using it throughout the text. For example, let us take the particular case when $G=SL_{2}(\mathbb{Z}_{p})$. Then the only negative root is $\xi=(2,1)\in\Phi^{-}$ and the corresponding element $y_{\xi}=\left[\begin{matrix}0&0\\ p&0\end{matrix}\right]$. Therefore working over $\mathbb{F}_{p}$, $y_{\xi}^{p^{s}}=\left[\begin{matrix}0&0\\ p^{s+1}&0\end{matrix}\right]$.

For the simple root $\xi=(1,2)\in\Pi$, the corresponding element $y_{\xi}=\left[\begin{matrix}(1+p)-1&0\\ 0&(1+p)^{-1}-1\end{matrix}\right]$. Hence $y_{\xi}^{p^{s}}=\left[\begin{matrix}(1+p)^{s}-1&0\\ 0&(1+p)^{-s}-1\end{matrix}\right]$.

For the positive root $\xi=(1,2)\in\Phi^{+}$, the corresponding element $y_{\xi}=\left[\begin{matrix}0&p\\ 0&0\end{matrix}\right]$. Therefore, $y_{\xi}^{p^{s}}=\left[\begin{matrix}0&p^{s+1}\\ 0&0\end{matrix}\right]$.

So if $G$ is $SL_{2}$, then $\omega$ denotes just a finite sum of monomials of the form $\prod_{\xi\in\Phi^{-},\Pi,\Phi^{+}}(y_{\xi}^{p^{s}})^{n_{\xi}}$ (where $n_{\xi}$ are non-negative integers) multiplied with scalars $a_{\underline{\xi}}$ depending on each monomial. This is certainly an element in the mod-$p$ Iwasawa algebra of $G(1)$.

Let $\gamma\in\Phi$ or $\Pi$. Our goal in this section is to show the following lemma which will be crucial to show our main theorem (ref. Theorem 5.1).

Lemma 4.1**.**

[TABLE]

where the sum is over all roots $\eta$ such that $[y^{p^{r}}_{\gamma},y^{p^{s}}_{\eta}]_{\circ}\neq 0$.

Proof.

[TABLE]

Suppose $\eta$ is the first root in the ordering such that $[y^{p^{r}}_{\gamma},y^{p^{s}}_{\eta}]_{\circ}\neq 0$. In the following, we are going to change $y^{p^{r}}_{\gamma}y^{p^{s}}_{\eta}\rightarrow y^{p^{s}}_{\eta}y^{p^{r}}_{\gamma}$ popping out a $[y^{p^{r}}_{\gamma},y^{p^{s}}_{\eta}]$ term. We obtain

[TABLE]

(Here $\xi<\eta$ means all the roots $\xi$ which come before the root $\eta$ in the total order on the roots compatible with the ordering in Lazard’s order basis). Iterating the above step until $y^{p^{r}}_{\gamma}$ passes through all $y^{p^{s}}_{\eta}$ in (4.1) and pops out $[y^{p^{r}}_{\gamma},y^{p^{s}}_{\eta}]_{\circ}$ term in (4.3). Then we get

[TABLE]

Applying Proposition 3.1, we have

[TABLE]

But degree of $[y^{p^{r}}_{\gamma},y^{p^{s}}_{\eta}]_{\circ}y^{p^{s}}_{\xi}$ is $p^{r+2s+1}$. Therefore,

[TABLE]

We therefore get

[TABLE]

Note that here we are working under the background of the lowest-degree terms. So the “=” sign does make sense against the backdrop of $[-,-]_{\circ}$.

Iterating this method until $y^{p^{s}}_{\gamma}$ passes through all $y^{p^{s}}_{\xi}$ for all $\xi>\eta$, we finally arrive at

[TABLE]

This completes proof of this lemma.

∎

5. Main Result and Its Proof

In this section we will state and prove our main result assuming Claim 5.3, which we will prove in the next section. Let $G$ be a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_{p}$, $G(1)$ be the first congruence kernel of $G$ and $\Omega_{G(1)}$ be the mod-$p$ Iwasawa algebra of $G(1)$ over $\mathbb{F}_{p}$. Let us recall that an element $r\in\Omega_{G(1)}$ is said to be normal if $r\Omega_{G(1)}=\Omega_{G(1)}r$. Clearly, $\mathbb{F}_{p}$ and those elements of $\Omega_{G(1)}$ which contain constant terms are normal elements. It is a natural question whether the converse statement of this result holds true. Our main objective is to determine which elements in the $\Omega_{G(1)}$ are eligible for normal elements. Our main theorem is the following

Theorem 5.1**.**

Let $p$ be a prime with $p\geq 3$ and $G$ be a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_{p}$. Suppose that $G$ is one of the following Chevalley groups of Lie type: $A_{\ell}\,(\ell\geq 1),B_{\ell}\,(\ell\geq 2),C_{\ell}\,(\ell\geq 2),D_{\ell}\,(\ell\geq 3),E_{6},E_{7},E_{8},F_{4},G_{2}$. Then for any nonzero element $W\in\Omega_{G(1)}$, $W$ is a normal element if and only if $W$, as a noncommuttaive formal power series, contains constant terms. In this case, $W$ is a unit.

Proof.

Let $W$ be a nonzero element of $W\in\Omega_{G(1)}$. Suppose that $W$, as a noncommutative formal power series, contains constant terms. Then it is straightforward to check that $W$ is invertible. And hence $W$ is a normal element of $\Omega_{G(1)}$. In this case, $W$ is a unit.

Let $W$ be a nonzero normal element of $\Omega_{G(1)}$ and $W$ is of the form

[TABLE]

where $w_{d}$ ($d=m,m+1,m+2,\cdots,m\geq 1$) are homogeneous polynomials with respect to $y_{\xi}$ ($\xi$ varying through $\Phi$ and $\Pi$) of degree $d$.

[TABLE]

(the multi-index notation has same meaning as in (4.1) with explanations given after (4.1)).

Moreover, we put

[TABLE]

which will be frequently invoked in the sequel.

Since $W$ is a normal element, there exists an element $D_{\gamma}(r)\in\Omega_{G(1)}$ such that

[TABLE]

for each $\gamma\in\Phi$ or $\gamma\in\Pi$. We define

[TABLE]

We divide the proof of this theorem into two cases: $s=s_{m}$ and $s<s_{m}$.

Case 1: $s=s_{m}$.

In this case, we by (5.2) get

[TABLE]

for each $y_{\gamma}$ ($\gamma\in\Phi$ or $\gamma\in\Pi$). Recall that $[y^{p^{r}}_{\gamma},w_{m}]_{\circ}$ and $(D_{\gamma}(r))_{\circ}$ stand for the lowest degree terms in $[y^{p^{r}}_{\gamma},w_{m}]$ and $D_{\gamma}(r)$, respectively. It should be pointed out that $[y^{p^{r}}_{\gamma},w_{m}]_{\circ}$ is a homogenous polynomial of degree $m-p^{s}+p^{r+s+1}$.

We can assume that the lowest-degree homogenous polynomial $w_{m}$ of $W$ is of the form

[TABLE]

(as in the beginning of Section 4) such that

[TABLE]

where $\mathbb{F}_{p}[y_{\xi}^{p^{s}},\ \xi\in\Phi^{-},\Pi,\Phi^{+}]$ denotes the polynomial ring generated by $y_{\xi}^{p^{s}}$ for $\xi\in\Phi^{-},\Pi,\Phi^{+}$ over the field $\mathbb{F}_{p}$. It follows from Lemma 4.1 that

[TABLE]

Claim 5.2**.**

$\frac{\partial w_{m}}{\partial y^{p^{s}}_{\gamma}}$ are not exactly zeroes for all $\gamma\in\Phi$ or $\Pi$.

Proof.

In view of (5.3), we can write

[TABLE]

where $v_{n_{\gamma}}(\underline{y}^{\gamma})$ is a polynomial over $\mathbb{F}_{p}$ in $y_{\xi}^{p^{s}}$ for $\xi\in\Phi\cup\Pi\setminus\{\gamma\}$. Suppose on the contrary that

[TABLE]

Considering the polynomial above related to $y^{p^{s}}_{\gamma}$, we see that $n_{\gamma}v_{n_{\gamma}}(\underline{y}^{\gamma})=0$, and we therefore get $p\,|\,n_{\gamma}$, $\gamma\in\Phi$ or $\Pi$. But then $w_{m}\in\mathbb{F}_{p}[y_{\xi}^{p^{s+1}},\ \xi\in\Phi^{-},\Pi,\Phi^{+}]$, which is a contradiction. This proves this claim.

∎

Now we complete the proof of Theorem 5.1 for the case $s=s_{m}$. By Claim 5.2, there exists $\gamma_{1}\in\Phi$ or $\gamma_{1}\in\Pi$ such that $\frac{\partial w_{m}}{\partial y^{p^{s}}_{\gamma_{1}}}\neq 0$. Certainly there exists $\alpha\in\Phi$ or $\Pi$ such that $[y^{p^{r}}_{\alpha},y^{p^{s}}_{\gamma_{1}}]_{\circ}\neq 0$ (if $\gamma_{1}\in\Phi$, one can choose $\alpha\in\Phi$ such that $\alpha+\gamma_{1}\in\Phi$; and if $\gamma_{1}\in\Pi$, we can choose any $\alpha\in\Phi$ and use Proportion 3.1). Then

[TABLE]

By Proposition 3.1, for $p>5$, we know that

[TABLE]

Suppose we have the first one, i.e. $[y^{p^{r}}_{\alpha},y^{p^{s}}_{\gamma_{1}}]_{\circ}=c_{11}y_{\alpha+\gamma_{1}}^{p^{r+s+1}}$. Then

[TABLE]

Comparing the coefficient of $y_{\alpha+\gamma_{1}}^{p^{r+s+1}}$ of the above relation on both sides we see that

[TABLE]

where $U(\underline{y})$ is the polynomial form coming from $(D_{\alpha}(r))_{\circ}$. Now comparing the degree of $y_{\gamma_{1}}^{p^{s}}$ on both sides of relation (5.6) we immediately arrive at a contradiction (Note that in the above argument we can also choose $\alpha$ to be $-\gamma_{1}$ and similar contradiction will arise). Now if

[TABLE]

by the same argument as before (replacing $c_{11}y_{\alpha+\gamma_{1}}^{p^{r+s+1}}$ in the argument above by $-\langle\alpha,\gamma_{1}\rangle y_{\alpha}^{p^{r+s+1}}$) we arrive at a contradiction by comparing degrees of $y^{p^{s}}_{\gamma_{1}}$. The argument for $p=3,5$ extending Proposition 3.1 is the same. This implies that $W$, as a non-zero normal element in $\Omega_{G(1)}$, must contain constant terms and completes the proof of Theorem 5.1 for the case $s=s_{m}$.

Case 2. $s<s_{m}$.

Now there exists some fixed $d$ with $d>m$ such that $s=s_{d}<s_{m}$, and it follows from (5.2) that

[TABLE]

for each $y_{\gamma}$ ($\gamma\in\Phi$ or $\gamma\in\Pi$) provided $r\gg 0$.

To proceed our discussion, we assume that $w_{d}$ is of the form

[TABLE]

where the sum is over all the component $n_{\xi}$ of $n_{\underline{\xi}}$ varying from [math] to $p-1$. Here

[TABLE]

where $\mathbb{F}_{p}[y_{\xi}^{p^{s}},\ \xi\in\Phi^{-},\Pi,\Phi^{+}]$ denotes the polynomial ring generated by $y_{\xi}^{p^{s}}$ for $\xi\in\Phi^{-},\Pi,\Phi^{+}$ over the field $\mathbb{F}_{p}$.

Claim 5.3**.**

$w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\gamma}}$ for all $\gamma\in\Phi$ or $\gamma\in\Pi$.

Assume that Claim 5.3 holds true now. We are going to prove it later in Section 6 after finishing the proof of our main theorem. Let us simplify our notations and denote the polynomial ring $\mathbb{F}_{p}[y_{\xi}^{p^{s}},\ \xi\in\Phi^{-},\Pi,\Phi^{+}]$ by $\mathbb{F}_{p}[\underline{y}^{p^{s}}]$.

Claim 5.4**.**

For $w_{m},w_{d}$, there exist $u\in\mathbb{F}_{p}[\underline{y}^{p^{s}}]$ and $v\in\mathbb{F}_{p}[\underline{y}^{p^{s+1}}]$ such that $w_{d}=w_{m}u+v$

Proof.

According to Claim 5.3, there exists $\,u_{\gamma}\in\mathbb{F}_{p}[\underline{y}^{p^{s}}]$ such that

[TABLE]

Suppose that $u_{\gamma}$ is of the following form:

[TABLE]

Using (5.7) we get

[TABLE]

Here we need to keep tract of the components $n_{\xi}$ of $n_{\underline{\xi}}$ for the roots $\xi$ coming before $\gamma$, that is $\xi<\gamma$ (in the total order on the roots according to Lazard’s ordered basis), $\xi=\gamma$ and $\xi>\gamma$. We also include this in the subindices of the polynomial $h_{n_{\underline{\xi}}}(\underline{y}^{p^{s+1}})$. In the following we write $h_{n_{\underline{\xi}}}(\underline{y}^{p^{s+1}})$ as $h_{((n_{\xi\,(\xi<\gamma)},n_{\gamma},n_{\xi\,(\xi>\gamma)}))}(\underline{y}^{p^{s+1}})$. So

[TABLE]

Taking (5.9) and (5.10) into (5.8) yields

[TABLE]

This shows that

[TABLE]

where the components of $n_{\underline{\xi}}=(n_{\xi\,(\xi<\gamma)},n_{\gamma},n_{\xi\,(\xi>\gamma)})$ are not all zeros, i.e. $n_{\underline{\xi}}\neq\underline{0}$ ($\underline{0}$ is the zero vector). That is, for each $h_{n_{\underline{\xi}}}(\underline{y}^{p^{s+1}})$, there exists a corresponding $h^{*}_{n_{\underline{\xi}}}(\underline{y}^{p^{s+1}})$ such that

[TABLE]

Taking this back into (5.7), we arrive at

[TABLE]

where the summation is taken over $n_{\underline{\xi}}$ such that the components $n_{\xi}$ varies from [math] to $p-1$, but all the components $n_{\xi}$ can not be simultaneously zero. Let us put

[TABLE]

This proves that

[TABLE]

where $u\in\mathbb{F}_{p}[\underline{y}^{p^{s}}]$ and $v\in\mathbb{F}_{p}[\underline{y}^{p^{s+1}}]$ and Claim 5.4 follows from Claim 5.3. ∎

Now we continue the proof for the Case $w<s_{m}$ of Theorem 5.1. Let us consider the following subset of $\Omega_{G(1)}$:

[TABLE]

where $s(W)$ is the $s$ corresponding to $W$. For any $W\in N(w_{m})$, we assume that $s(W)=s_{d}$ for some $d>m$. Thus one can write $W$ as

[TABLE]

Then $w_{d}=w_{m}u+v$, where $u\in\mathbb{F}_{p}[\underline{y}^{p^{s_{m}-1}}]$ and $v\in\mathbb{F}_{p}[\underline{y}^{p^{s_{m}}}]$. For convenience, we denote the index of $w_{d}$ by $d(W)$. Let us write $W=W_{0}$ and $W_{1}=W(1-u)$. Then

[TABLE]

It is easy to verfiy that $W_{1}\in N(w_{m})$ and $d(W_{0})<d(W_{1})$. Likewise, for $W_{1}$, there exist $u^{\prime}$ and $v^{\prime}$ such that $w_{1d}=w_{m}u^{\prime}+v^{\prime}$, where $w_{1d}$ is the first homogeneous polynomial satisfying the condition $s(W_{1})=s_{m}-1$ in $W_{1}$. Let us set $W_{2}=W_{1}(1-u^{\prime})$. It is also easy to check that $W_{2}\in N(w_{m})$ and $d(W_{1})<d(W_{2})$. Repeating this process continuously, we finally construct an infinite sequence of normal elements

[TABLE]

Let us set $\underset{n\rightarrow\infty}{\rm lim}\,W_{n}=V$. Then $V$ is a normal element with the form

[TABLE]

where $v_{m}=w_{m}$. It follows that $s(V)>s_{m-1}$, a contradiction. This shows that $W$, as a non-zero normal element in $\Omega_{G(1)}$, must contain constant terms in the case that $s<s_{m}$. This completes the proof of our main theorem (Theorem 5.1) provided the Claim 5.3 is true.

∎

6. Proof of Claim 5.3

We will give a detailed proof for Claim 5.3 in this section. Let us restate Claim 5.3: $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\gamma}}$, for all $\,\gamma\in\Phi$ or $\gamma\in\Pi$.

Proof.

Let $\gamma\in\Phi$. We by the partial differential equations in Lemma 4.1 get

[TABLE]

Note that $[y^{p^{r}}_{-\gamma},y^{p^{s}}_{\gamma}]_{\circ}\neq 0$ by Proposition 3.1. Also note that

[TABLE]

So

[TABLE]

Let us write

[TABLE]

Combining (6.2) with (6.3) gives

[TABLE]

Now we use again the partial differential equations in Lemma 4.1. Pick any $\delta\in\Pi$,

[TABLE]

We also have

[TABLE]

Similarly writing $(D_{\delta}(r))_{\circ}=U^{\gamma}(\underline{y})y^{p^{r}}_{\gamma}+\text{other terms}$ and using (6.5) and (6.6) we deduce that

[TABLE]

Notice that from (6.4) we have $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\gamma}}y_{\delta_{i}}^{p^{r+s+1}-p^{s}}$. So, $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\gamma}}$. This completes the case if $\gamma\in\Phi$.

The case when $\gamma\in\Pi$ is much more tricky and its proof will use diagram chasing using Dynkin diagram of the Lie type of the group $G(1)$. Given $\Phi$ and $\Pi$, we will denote $\alpha_{\max}$ to be the highest root. The philosophy of the proof is to write the partial differential equations in Lemma 4.1 for all $y_{\alpha_{i}}$, where $\alpha_{i}$ is a simple root equal to $\delta_{i}$ ($\alpha_{i}=\delta_{i}$). Note that $y_{\alpha_{i}}$ and $y_{\delta_{i}}$ are different elements. When we write $y_{\alpha_{i}}$, we mean the element $y_{\alpha_{i}}=x_{\alpha_{i}}(p)-1$ but $y_{\delta_{i}}=h_{\delta_{i}}(1+p)-1$. So we will write $[y_{\alpha_{i}},w_{d}]_{\circ}$, $[y_{-\alpha_{i}},w_{d}]_{\circ}$, $[y_{-\alpha_{\max}},w_{d}]_{\circ}$ and $[y_{\alpha_{\max}},w_{d}]_{\circ}$. This will give us divisibility relations of $w_{m}$ which we will then solve to show that $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{i}}}$ for all $i$. Let us start with the case $B_{\ell}$, which suit us best to explain the philosophy of our diagram chasing argument.

Dynkin diagram for ${\ell}\geq 3$ (cf. [4, Page 214]):

\alpha_{\max}$$\delta_{1}$$\delta_{2}$$\delta_{3}$$\delta_{4}$$\delta_{\ell-1}$$\delta_{\ell}

Note that, by [4, Page 207] and the discussion on Dynkin graphs, the fact that $\alpha_{\max}$ is linked to $\delta_{2}$ only means that $\langle\alpha_{\max},\delta_{2}\rangle\neq 0$ and $\langle\alpha_{\max},\delta_{i}\rangle=0$ for all $i=1,3,4,\cdots,\ell$. Let us first write the partial differential equation for $\alpha_{\max}$, so

[TABLE]

By expanding $(D_{\alpha_{\max}}(r))_{\circ}$ and by the same method as that of (6.1)-(6.7) (noticing that $[y^{p^{r}}_{\alpha_{\max}}$, y^{p^{s}}_{\delta_{2}}]_{\circ}$$=-\langle\alpha_{\max},\delta_{2}\rangle y_{\alpha_{\max}}^{p^{r+s+1}}), we obtain $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{2}}}y_{\alpha_{\max}}^{p^{r+s+1}-p^{r}}$. Replacing $y_{\alpha_{max}}$ in (6.8) by $y_{-\alpha_{max}}$, we get $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{2}}}y_{-\alpha_{\max}}^{p^{r+s+1}-p^{r}}$. This implies that $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{2}}}$.

Recall the Dynkin diagram above

\alpha_{\max}$$\delta_{1}$$\delta_{2}$$\delta_{3}$$\delta_{4}$$\delta_{\ell-1}$$\delta_{\ell}

Notice that writing the partial differential equations for $[y^{p^{r}}_{\alpha_{\max}},w_{d}]_{\circ}$ and $[y^{p^{r}}_{-\alpha_{\max}},w_{d}]_{\circ}$ gives us $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{2}}}$ (the main fact is that $\alpha_{\max}$ is only linked with $\delta_{2}$).

Now write the partial differential equations $[y^{p^{r}}_{\alpha_{1}},w_{d}]_{\circ}$ and $[y^{p^{r}}_{-\alpha_{1}},w_{d}]_{\circ}$, where $\alpha_{1}=\delta_{1}$, $y_{\alpha_{1}}=x_{\alpha_{1}}(p)-1$ is not the same as $y_{\delta_{1}}=h_{\delta_{1}}(1+p)-1$. Since $\delta_{1}$ is only connected with $\delta_{2}$, we will arrive at $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{1}}}$. We clarify this below for the convenience of the reader.

[TABLE]

Thus we get

[TABLE]

Replacing $\alpha_{1}$ by $-\alpha_{1}$ yields

[TABLE]

As $\gcd(y_{\alpha_{1}},y_{-\alpha_{1}})=1$, we have

[TABLE]

Now we have already shown that $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{2}}}$. This gives that $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{1}}}$ as $\langle\alpha_{1},\delta_{1}\rangle=\langle\delta_{1},\delta_{1}\rangle\neq 0$ (by [4]). We proceed this diagram chasing for other simple roots in the Dynkin diagram of $B_{\ell}$.

Now writing $[y^{p^{r}}_{\alpha_{2}},w_{d}]_{\circ}$ and $[y^{p^{r}}_{-\alpha_{2}},w_{d}]_{\circ}$ for $\alpha_{2}=\delta_{2}$, we get

[TABLE]

which is due to the fact that $\delta_{2}$ is only connected with $\delta_{1}$ and $\delta_{3}$. Since we have already shown that $w_{m}|\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{1}}}$ and $w_{m}|\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{2}}}$, we will obtain $w_{m}|\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{3}}}$. We proceed this to complete and finally we conclude that $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{i}}}$, $i\in[1,\ell]$. The above argument also works for types

\alpha_{\max}$$C_{\ell}$$\delta_{1}$$\delta_{2}$$\delta_{3}$$\delta_{4}$$\delta_{\ell-1}$$\delta_{\ell}

\alpha_{\max}$$F_{4}$$\delta_{1}$$\delta_{2}$$\delta_{3}$$\delta_{4}

D_{\ell}$$\alpha_{\max}$$\delta_{1}$$\delta_{2}$$\delta_{3}$$\delta_{4}$$\delta_{\ell-3}$$\delta_{\ell-2}$$\delta_{\ell}$$\delta_{\ell-1}

For $D_{\ell}$, just as in the case for $B_{\ell}$, we start from $\alpha_{\max}$ and do diagram chasing until we get $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{\ell-2}}}$ from the equations

[TABLE]

and

[TABLE]

Finally, by invoking the relations

[TABLE]

we arrive at $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{\ell}}}$ . Similarly, using the relations

[TABLE]

we obtain $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{\ell-1}}}$.

Here is the Dynkin diagram for $E_{8}$.

\delta_{1}$$E_{8}$$\delta_{3}$$\delta_{4}$$\delta_{2}$$\delta_{5}$$\delta_{6}$$\delta_{7}$$\delta_{8}$$\alpha_{\max}

Case $E_{8}$ is also easy to see, we adopt the same trick as we did for the case of $D_{\ell}$.

Here is the Dynkin diagram for $E_{7}$.

\alpha_{\max}$$E_{7}$$\delta_{1}$$\delta_{3}$$\delta_{2}$$\delta_{4}$$\delta_{5}$$\delta_{6}$$\delta_{7}

Case $E_{7}$ is also similar to $E_{8}$ and $D_{\ell}$ and we do the same diagram chasing by using the partial differential equations as we did earlier.

Here is the Dynkin diagram for $E_{6}$.

E_{6}$$\delta_{1}$$\delta_{3}$$\delta_{2}$$\alpha_{\max}$$\delta_{4}$$\delta_{5}$$\delta_{6}

This $E_{6}$ case is a little tricky and we will demonstrate it in a detailed way. In view of the facts

[TABLE]

and

[TABLE]

we observe that $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{2}}}$. Taking into account the relations

[TABLE]

and

[TABLE]

we obtain $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{4}}}$ (because $\delta_{2}$ is linked with $\delta_{4}$). Now we write, for $\alpha_{3}=\delta_{3}$,

[TABLE]

and these will give us

[TABLE]

Applying the fact $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{4}}}$ yields that

[TABLE]

It follows from the relations

[TABLE]

that

[TABLE]

This implies that

[TABLE]

Now by [27, Page 5], we know that

[TABLE]

So $\langle\delta_{1},\delta_{3}\rangle=\langle\delta_{3},\delta_{1}\rangle$. We therefore assume that $\langle\delta_{1},\delta_{3}\rangle=+1$. By the relation (6.9) we see that

[TABLE]

Let us set $x=\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{3}}}$ and $y=\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{1}}}$. Thus $w_{m}\,|\,(2x+y)$ and $w_{m}\,|\,(2y+x)$. And hence $w_{m}\,|\,[2(2x+y)-(2y+x)=3x]$. Therefore, $w_{m}\,|\,x$ and then from (6.10) we get $w_{m}\,|\,y$. So $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{1}}}$ and $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{3}}}$. Similarly, using the equations

[TABLE]

and solving them we get $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{5}}}$ and $w_{m}\,|\,\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{6}}}$. This completes the proof of case $E_{6}$.

Here is the Dynkin diagram for $G_{2}$.

G_{2}$$\delta_{1}$$\delta_{2}$$\alpha_{\max}

This case is easy and similar to the cases of $B_{\ell}$ and $D_{\ell}$.

Let us next deal with the case of $A_{\ell}$. The Dynkin diagram for $A_{\ell}$ is the following

A_{\ell}$$\alpha_{\max}$$\delta_{1}$$\delta_{2}$$\delta_{3}$$\delta_{\ell-1}$$\delta_{\ell}

Let us write $x_{i}=\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{i}}}$, $\alpha_{i}=\delta_{i}$. Then the following two relations

[TABLE]

give

[TABLE]

which is $w_{m}\,|\,(-2\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{1}}}+\frac{\partial w_{d}}{\partial y^{p^{s}}_{\delta_{2}}})$ due to [4, Page 217]. Thus we have

[TABLE]

Similarly, writing

[TABLE]

we get

[TABLE]

ie.

[TABLE]

Writing the other partial differential equations, we finally arrive at

[TABLE]

Now note that there exist constants $d_{1},d_{2},\cdots,d_{\ell-1}$ such that $\sum_{i=2}^{\ell-1}d_{i}C_{i}+d_{1}C_{1}$ is of the form $ax_{1}+x_{2}$, where $d_{i}$ for $i\in[1,\ell-1]$ and $a$ are positive constants. We therefore conclude that there exists a positive constant $a$ such that $w_{m}\,|\,(ax_{1}+x_{2})$. Now, $w_{m}\,|\,(-2x_{1}+x_{2}))$, and so $w_{m}\,|\,[(a+2)x_{1}=(ax_{1}+x_{2})-(-2x_{1}+x_{2})]$. This shows that $w_{m}\,|\,x_{1}$. Now recursively, using the relations after (6.12), we eventually arrive at $w_{m}\,|\,x_{i}$ for all $i$. This completes the proof of Claim 5.3. ∎

6.1. Applications to center

Recall that $G$ is a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_{p}$, $G(1)$ is the first congruence kernel of $G$ and $\Omega_{G(1)}$ is the mod-$p$ Iwasawa algebra of $G(1)$ over $\mathbb{F}_{p}$.

As a direct consequence of Theorem 5.1, we have

Proposition 6.1**.**

Let $p$ be a prime with $p\geq 3$ and $G$ be a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_{p}$. Suppose that $G$ is one of the following Chevalley groups of Lie type: $A_{\ell}\,(\ell\geq 1),B_{\ell}\,(\ell\geq 2),C_{\ell}\,(\ell\geq 2),D_{\ell}\,(\ell\geq 3),E_{6},E_{7},E_{8},F_{4},G_{2}$. Then the center of $\Omega_{G(1)}$ is trivial, i.e. if $r$ is central element of $\Omega_{G(1)}$, then $r\in\mathbb{F}_{p}$.

Proof.

Suppose that $r$ is a central element of $\Omega_{G(1)}$. Then by Theorem 5.1, we can write

[TABLE]

where $r_{d}\,(d=0,1,2,\cdots)$ are homogeneous polynomials with respect to $y_{12}$, $y_{13}$, $\cdots$, $y_{1122}$, $\cdots$, $y_{n(n-1)}$ of degree $d$.

Since $r$ is a central element, we have $[x,r]=0$ for all $x\in\Omega_{G(1)}$. This implies

[TABLE]

for all $x\in\Omega_{G(1)}$. By Theorem 5.1 again, we assert that

[TABLE]

and the result follows. ∎

We therefore say that the center of $\Omega_{G(1)}$ is exactly the finite field $\mathbb{F}_{p}$. This accounts to reproving Ardakov’s result [1, Corollary A].

6.2. Future questions

Clozel [7] considered the Iwasawa algebra of the pro-$p$ Iwahori subgroup of ${\rm GL}_{2}(L)$ for an unramified extension $L$ of degree $r$ of $\mathbb{Q}_{p}$ and gave a presentation of it by generators and relations. Inspired by Clozel’s systematic works, Ray [22] extend his result to determine the explicit ring-theoretic presentation, in the form of generators and relations, of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of ${\rm GL}_{n}(\mathbb{Z}_{p})$.

Clozel’s and Ray’s works show that there is considerable interest towards understanding the structure of Iwasawa algebras over the pro-$p$ Iwahori subgroup $G$ of ${\rm GL}_{n}(\mathbb{Z}_{p})$. Moreover, Bushnell and Henniart [5, Chapter 4, Section 17] together with Herzig [17, Lemma 10] have pointed out that for the pro-$p$ Iwahori subgroup $G$ of ${\rm GL}_{2}(\mathbb{Q}_{p})$ and any nonzero irreducible (resp. smooth) representation $V$, the class of irreducible (resp. smooth) representation $(\pi,V)$ of ${\rm GL}_{2}(\mathbb{Q}_{p})$ for which the set of $G$-fixed vectors $V^{G}\neq 0$ is particularly subtle and useful in mod-$p$ representation theory of $p$-adic groups. Furthermore, the pro-$p$ Iwahori and its associated Hecke algebra have several applications in the emerging Langlands program (see the works of M. F. Vigneras). Therefore a natural question is to understand the pro-$p$ Iwahori subgroups and their associated Iwasawa algebra.

Let $L/\mathbb{Q}_{p}$ be a finite extension and $e$ its ramification index. Suppose that $L$ is mildly ramified, i.e.

[TABLE]

see [19, Chapter III, 3.2]. Let $\mathfrak{p}$, $\mathcal{O}=\mathcal{O}_{L}$ denote the prime ideal, the integers of $L$, respectively.

We denote by $G$ the pro-$p$ Iwahori subgroup of ${\rm GL}_{2}(\mathbb{Z}_{p})$, i.e.

[TABLE]

Proposition 6.2**.**

[19, Chapter III (3.2.7)]* Let $p>3$, then the pro-$p$ Iwahori subgroup $G$ of ${\rm GL}_{2}(\mathbb{Z}_{p})$ is a $p$-valued $p$-saturated Sylow subgroup in the sense of Lazard.*

Applying the arguments of [7, Section 2] to ${\rm GL}_{2}(\mathbb{Z}_{p})$, we see that

[TABLE]

form a topological generating set for the pro-$p$ Iwahori subgroup $H$ of ${\rm SL}_{2}(\mathbb{Z}_{p})$ and that

[TABLE]

construct a topological generating set for the pro-$p$ Iwahori subgroup $G$ of ${\rm GL}_{2}(\mathbb{Z}_{p})$. When certain complicated computations are needed, the type and the number of topological generators will be useful. Let us set

[TABLE]

Then ${\bf x},\,{\bf h},\,{\bf y}\in{\mathbb{F}}_{p}[H]\subseteq\Omega_{H}$ and ${\bf x},\,{\bf h},\,{\bf z},\,{\bf y}\in{\mathbb{F}}_{p}[G]\subseteq\Omega_{G}$. Thus we can produce various monomials in the ${\bf x},\,{\bf h},\,{\bf z},\,{\bf y}$: if $\alpha=(i,j,k)$ is a 3-tuple of nonnegative integers, we define

[TABLE]

If $\beta=(i,j,k,l)$ is a 4-tuple of nonnegative integers, one can define

[TABLE]

It should be remarked that the expressions of these monomials depend on our choice of ordering of the ${\bf x}$’s, ${\bf h}$’s, ${\bf z}$’s, ${\bf y}$’s, because $\Omega_{H}$ and $\Omega_{G}$ are noncommutative unless $H$ and $G$ are abelian. The following result shows that $\Omega_{H}$ and $\Omega_{G}$ are both “noncommutative formal power series rings”.

Proposition 6.3**.**

(cf. [24, Chapter VI, Section 28])* Every element $a$ of the mod-$p$ Iwasawa algebra $\Omega_{H}$ can be written as the sum of a uniquely determined convergent series*

[TABLE]

where $a_{\alpha}\in{\mathbb{F}}_{p}$ for all $\alpha\in{\mathbb{N}}^{3}$; each element $b$ of the mod-$p$ Iwasawa algebra $\Omega_{G}$ is equal to the sum of a uniquely determined convergent series

[TABLE]

where $b_{\beta}\in{\mathbb{F}}_{p}$ for all $\beta\in{\mathbb{N}}^{4}$.

One can analogously define the pro-$p$ Iwahori of subgroup of ${\rm SL}_{n}(\mathbb{Z}_{p})$ and ${\rm GL}_{n}(\mathbb{Z}_{p})$ which are $p$ saturated if $p>n+1$.

We should distinguish the pro-$p$ Iwahori subgroups of ${\rm GL}_{n}(\mathbb{Z}_{p})$ and ${\rm SL}_{n}(\mathbb{Z}_{p})$ from their uniform pro-$p$ subgroups, such as their first congruence subgroups. Note that the first congruence subgroup $\Gamma_{1}({\rm GL}_{n}(\mathbb{Z}_{p}))$ (resp. $\Gamma_{1}({\rm SL}_{n}(\mathbb{Z}_{p}))$) of ${\rm GL}_{n}(\mathbb{Z}_{p})$ (resp. ${\rm SL}_{n}(\mathbb{Z}_{p})$) is a uniform pro-$p$ subgroup. We should be aware of the fact that an arbitrary uniform pro-$p$ group is $p$-valued and $p$-saturated. Although the pro-$p$ Iwahori subgroup of ${\rm GL}_{n}(\mathbb{Z}_{p})$ (resp. the pro-$p$ Iwahori subgroup of ${\rm SL}_{n}(\mathbb{Z}_{p})$) is $p$-valued and $p$-saturated as well, it is not in general uniform and does contain the first congruence subgroup $\Gamma_{1}({\rm GL}_{n}(\mathbb{Z}_{p}))$ (resp. $\Gamma_{1}({\rm SL}_{n}(\mathbb{Z}_{p}))$) properly. It is easily seen that uniform pro-$p$ groups form a subclass of the class of $p$-saturated groups. (cf. Remark after [25, Lemma 4.3]). Klopsch [18] and Schneider [24] illustrated by examples that the standard notion of uniform pro-$p$ groups is more restrictive and less flexible than Lazard’s concept of $p$-saturated groups. Klopsch [18, Proposition 2.4] also pointed out that the Sylow pro-$p$ subgroups of many classical groups are $p$-saturated, but typically fail to be uniform powerful. Consequently, the Iwasawa algebras of the pro-$p$ Iwahori subgroups are much more larger than those of the first congruence subgroups. We therefore say that the Iwasawa algebras over the first congruence subgroups can be looked on as subalgebras of the Iwasawa algebras over the pro-$p$ Iwahori subgroups.

In our future work we hope to generalize our methods to more general $p$-saturated groups like the pro-$p$ Iwahori (which is a non-uniform group) of ${\rm GL}_{n}(\mathbb{Z}_{p})$ and ${\rm SL}_{n}(\mathbb{Z}_{p})$ in order to determine the normal elements in their mod-$p$ Iwasawa algebras. We now need to look at the lowest degree commutators from Ray’s article [22] and construct the partial differential equations similar to section 4.

Acknowledgements We have accumulated quite a debt of gratitude in writing this paper. Most of all to Professor Konstantin Ardakov and Professor Simon Wadsley for many invaluable suggestions and discussions and for constant inspiration. The second author would like to thank PIMS-CNRS and the University of British Columbia for postdoctoral research grant. He is also thankful to Beijing Institute of Technology for its gracious hospitality during a visit on August 2018 when this collaboration took place.