Abstract
Let \(k'/k\) be a finite purely inseparable field extension and let \(G'\) be a reductive \(k'\)-group. We denote by \(G=\mathrm {R}_{k'/k}(G')\), the Weil restriction of \(G'\) across \(k'/k\), a pseudo-reductive group. This article gives bounds for the exponent of the geometric unipotent radical \({\mathscr {R}}_{u}(G_{\bar{k}})\) in terms of invariants of the extension \(k'/k\), starting with the case \(G'={{\,\mathrm{GL}\,}}_n\) and applying these results to the case where \(G'\) is a simple group.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Let k be a field, which we assume is imperfect of characteristic p. In this case, there exist pseudo-reductive k-groups which are not reductive. Recall that a smooth, connected, affine algebraic k-group G is pseudo-reductive if the largest smooth connected unipotent normal k-subgroup \({\mathscr {R}}_{u,k}(G)\) is trivial.
We are interested in the structure of the geometric unipotent radical \(R:={\mathscr {R}}_u(G_{{\bar{k}}})={\mathscr {R}}_{u,{\bar{k}}}(G_{{\bar{k}}})\) of G. Since R is a p-group, it makes sense to study its exponent: the minimal integer s such that the \(p^s\)-power map on the geometric unipotent radical factors through the trivial group. We denote the exponent of R by e(R).
M. Bate, B. Martin, G. Röhrle, and the third author previously gave some bounds for e(R) in [1]. For example, [1, Lem. 4.1] implies that if \(k'/k\) is a simple purely inseparable field extension with \((k')^{p^e}\subseteq k\), then \(e(R)\le e\).
The monographs [4] and [5] contain a classification of pseudo-reductive groups. They are all related in some way or other to the Weil restrictions of reductive groups, which are themselves pseudo-reductive. We focus on groups which are Weil restrictions, since general pseudo-reductive groups contain central pseudo-reductive subgroups whose classification is thought to be intractable. Thus we assume unless stated otherwise that \(G=\mathrm {R}_{k'/k}(G')\) for some reductive \(k'\)-group \(G'\) where \(k'\) is a finite non-zero reduced k-algebra. Since we are interested in e(R), by the remarks in [1, §4], we may as well consider the case that k is separably closed and \(k'\) is a purely inseparable field extension of k. In this case, \(G'\) is a split reductive group and as such is classified by its root datum.
Recall that the exponent of the extension \(k'/k\) is the smallest integer e such that \((k')^{p^e}\subseteq k\). To describe our results, we need more sensitive data about the structure of \(k'/k\). Since \(k'/k\) is purely inseparable, we can appeal to the results in [10] and [11], which guarantee a normal generating sequence \(\alpha _1,\ldots ,\alpha _l\) of \(k'\) over k with certain properties—see Definition 2.6 below for more details. In particular, these elements come with certain exponents which are invariants of \(k'/k\). These are the integers \(e_1\ge \cdots \ge e_l\) such that \(e_i\) is minimal subject to
It is immediate that \(e_1=e\) and it follows from the tower law that \([k':k]=\prod p^{e_i}\). Let \(r\in {\mathbb {N}}\). We define integers
In other words, \({{\,\mathrm{em}\,}}(k'/k)\) is the minimal s such that \(p^s\ge {{\,\mathrm{m}\,}}(k'/k)\), and similarly for \({{\,\mathrm{emr}\,}}\).
In answering a question of [1] on the exponent of the intersection of R with a pseudo-Borel subgroup, we discovered that these numbers give lower bounds on e(R) and for large enough rank determine the exponent of R exactly. We first state our results in the case that \(G'\) is a general linear group.
Theorem 1.1
Let \(k'/k\) be a finite purely inseparable field extension of a separably closed field k of characteristic p. Suppose \(k'/k\) has exponent e and a normal generating sequence \(\alpha _1,\ldots ,\alpha _l\) with sequence of exponents \(e_1,\ldots ,e_l\). For \(r\in {\mathbb {N}}\), let \(G':={{\,\mathrm{GL}\,}}_r\), \(G:=\mathrm {R}_{k'/k}(G')\), and \(R:={\mathscr {R}}_u(G_{{\bar{k}}})\). Furthermore let \(B'\) denote a Borel subgroup of \(G'\) and \(B:=\mathrm {R}_{k'/k}(B')\subseteq G\) the corresponding pseudo-Borel subgroup of G. The following hold:
-
(i)
if s satisfies \(p^s\ge p^e r\), then e(R) is at most s; i.e. \(e(R)\le \lceil e+\log _p(r)\rceil \);
-
(ii)
we have \(e(R\cap B_{{\bar{k}}})={{\,\mathrm{E}\,}}(k'/k,r)\);
-
(iii)
in particular, \(e(R)\ge {{\,\mathrm{E}\,}}(k'/k,r)\);
-
(iv)
if \(\sum _{i=r+1}^l(p^{e_i}-1)<r-1\), then \(e(R)={{\,\mathrm{E}\,}}(k'/k,r)={{\,\mathrm{em}\,}}(k'/k)\).
Remark 1.2
-
(i)
The bounds in Theorem 1.1 are both arithmetic and Lie-theoretic in character: the integers \({{\,\mathrm{m}\,}}(k'/k)\) and \({{\,\mathrm{em}\,}}(k'/k)\) depend only on the arithmetic structure of \(k'/k\) but \({{\,\mathrm{mr}\,}}(k'/k,r)\), \({{\,\mathrm{emr}\,}}(k'/k,r)\), and \({{\,\mathrm{E}\,}}(k'/k,r)\) depend on both \(k'/k\) and the rank r of \(G'\).
-
(ii)
Part (i) of Theorem 1.1 improves [1, Lem. 4.4], which required \(p^s\ge r^2(p^e-1)\). As a corollary of (i) and (iv), if \(G=\mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_r)\) for \(p\ge r\ge 2\), then the exponent of \({\mathscr {R}}_u(G_{\bar{k}})\) is e if \(k'/k\) is a simple extension and is \(e+1\) otherwise. If \(r=1\), then \(e(R)=e\) as \(\mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_1)\) is sent into the canonical \({{\,\mathrm{GL}\,}}_1\) k-subgroup by the \(p^e\)-power map, since it also sends \(k'\) into k.
-
(iii)
In [1, Rem. 4.5(iv)], it was asked if the exponent e(R) always coincides with the exponent \(e(R\cap B_{{\bar{k}}})\). Clearly Theorem 1.1(ii)-(iv) answers this question in the affirmative when \(G'={{\,\mathrm{GL}\,}}_r\) of large enough rank. We give an example to show that in general the answer is ‘no’ (see Proposition 4.3), coming from \({{\,\mathrm{SL}\,}}_2\) in characteristic 2. In fact, in Section 4, we give a complete description of e(R) when \(G'\) is \({{\,\mathrm{SL}\,}}_2\), \({{\,\mathrm{PGL}\,}}_2\), or \({{\,\mathrm{GL}\,}}_2\).
Together with the rank 1 results in Section 4, the following (proved in Section 5) confirms the large rank behaviour in all simple groups, with mild conditions on the characteristic.
Corollary 1.3
Let \(k'/k\) be a finite, purely inseparable field extension of a separably closed field k of characteristic p. Suppose \(k'/k\) has exponent e and a normal generating sequence \(\alpha _1,\ldots ,\alpha _l\) with sequence of exponents \(e_1,\ldots ,e_l\). Let \(G'\) be a (split) simple algebraic k-group of rank \(r\ge 2\), \(G:=R_{k'/k}(G')\), and let \(R:={\mathscr {R}}_{u}(G_{{\bar{k}}})\). If \(G'\) is type B, C, D, F, then assume \(p\ne 2\) or \(G'\) is \({{\,\mathrm{SO}\,}}_{n}\) or \({{\,\mathrm{Sp}\,}}_{2n}\), and if G is of type \(E_6\), then assume \(p\ne 3\). Then the exponent \(e(R)\ge {{\,\mathrm{E}\,}}(k'/k,r)\); if \(\sum _{i=r+1}^l(p^{e_i}-1)<r-1\), then in fact, \(e(R)={{\,\mathrm{E}\,}}(k'/k,r)={{\,\mathrm{em}\,}}(k'/k)\).
2 Preliminaries and notation
Let k be a field of characteristic p and G be a k-group, by which we mean an affine algebraic group scheme of finite type over k. The k-unipotent radical is the maximal smooth connected normal unipotent k-subgroup of G and we denote this by \({\mathscr {R}}_{u,k}(G)\). The geometric unipotent radical is the \(\bar{k}\)-unipotent radical of the base change \(G_{\bar{k}}\) of G to \({\bar{k}}\). This is denoted by \({\mathscr {R}}_{u}(G_{\bar{k}})\). We call a smooth connected k-group G reductive if \({\mathscr {R}}_{u}(G_{\bar{k}})=1\) and pseudo-reductive if \({\mathscr {R}}_{u,k}(G)=1\). From now on, G will always be pseudo-reductive. We recall the definition and key properties of Weil restriction from [4, A.5].
Definition 2.1
Let \(B\rightarrow B'\) be a finite flat map of Noetherian rings, and \(X'\) a quasi-projective \(B'\)-scheme. The Weil restriction is a finite type B-scheme satisfying the universal property
for all B-algebras A.
The following proposition makes use of the natural map
which is induced by \(k'\otimes _k A \rightarrow A, a\otimes b \mapsto ab\) for any \(k'\)-algebra A.
Proposition 2.3
([4, Prop. A.5.11, Thm. 1.6.2], [1, §2], [1, Lem. 3.4]). Let \(k'/k\) be a finite and purely inseparable field extension, \(G'\) a non-trivial smooth connected \(k'\)-group, and \(G=\mathrm {R}_{k'/k}(G')\).
-
(i)
The kernel of \(q_{G'}\) is a smooth connected unipotent \(k'\)-subgroup and thus is contained in \({\mathscr {R}}_{u,k'}(G_{k'})\).
-
(ii)
If \(G'\) is reductive over \(k'\), then the kernel of \(q_{G'}\) has field of definition over k equal to \(k'\subseteq \bar{k}\). Thus the kernel \(\ker (q_{G'})\) coincides with \({\mathscr {R}}_{u,k'}(G_{k'})\), which is a \(k'\)-descent of \({\mathscr {R}}_u(G_{\bar{k}})\) (i.e. \({\mathscr {R}}_{u,k'}(G_{k'})_{{\bar{k}}}\cong {\mathscr {R}}_u(G_{\bar{k}})\)).
-
(iii)
If \(H'\) is a reductive subgroup of \(G'\), then the geometric unipotent radical of \(\mathrm {R}_{k'/k}(H')\) is a subgroup of the geometric unipotent radical of \(\mathrm {R}_{k'/k}(G')\).
-
(iv)
If \(f:G_1\rightarrow G_2\) is an étale isogeny of reductive \(k'\)-groups, then \(\mathrm {R}_{k'/k}(f)_{{\bar{k}}}\) induces an isomorphism of geometric unipotent radicals of the respective Weil restrictions.
We are interested in the following invariant of a unipotent k-group:
Definition 2.4
Let k be a field of characteristic p and U a unipotent k-group. The exponent \(\exp (U)\) of U is the minimal s such that the \(p^s\)-power map on U factors through the trivial group.
Clearly, this definition is insensitive to base change. As in [1, §4], we may therefore assume k is separably closed for the remainder of the article.
When calculating with matrices, we make use of the fact that if \(U(k)\subset U\) is dense (e.g. if k is perfect [6, Thm. 34.4]), then a map factors through the trivial group if and only if it maps to the identity on U(k). Since \(R={\mathscr {R}}_u(G_{{\bar{k}}})\) descends to \(R':={\mathscr {R}}_{u,k'}(G_{k'})\) whose \(k'\)-points are dense (by the fact that \(k'\) is separably closed), we have that e(R) is the smallest s such that \(p^s\) kills \(R'(k')\).
Following [9], let \({\mathfrak {B}}:=k'\otimes _k k'\). Then \({\mathfrak {B}}\) is a local ring with maximal ideal
where \(\phi :k'\otimes _k k' \rightarrow k', a\otimes b \mapsto ab\). Moreover, \(\mathrm {R}_{k'/k}(G')_{k'}\) can be identified with \(\mathrm {R}_{{\mathfrak {B}}/k'}(G'_{\mathfrak {B}})\) and the \(k'\)-points of the \(k'\)-unipotent radical of the former is the kernel \(G'({\mathfrak {m}})\) of the map \(G'({\mathfrak {B}})\rightarrow G'(k)\) induced by \(\phi \). Suppose now \(G'={{\,\mathrm{GL}\,}}_r\) and \(G=\mathrm {R}_{k'/k}(G')\). As the entries of any \(M\in {{\,\mathrm{Mat}\,}}(r,{\mathfrak {m}})\) are nilpotent elements of \({\mathfrak {B}}\), it follows that \(I_r+M\) has invertible determinant, hence is an element of \(G'({\mathfrak {m}})\). Therefore
As \(I_r\) and M commute, we have \(A^{p^s}=I_r+M^{p^s}=I_r\) is equivalent to \(M^{p^s}=0\). Hence:
Lemma 2.5
The exponent of the geometric unipotent radical of \(\mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_r)\) is the smallest s such that \(M^{p^s}=0\) for all \(M\in {{\,\mathrm{Mat}\,}}(r,{\mathfrak {m}})\).
It was observed in [1, Lem. 4.1] that as \({\mathfrak {m}}^n=0\) for some \(n\in {\mathbb {N}}\), one knows that s in the lemma above is bounded above by the minimal t such that \(p^t\ge n\): the entries of \(M^{p^t}\) are homogeneous polynomials of degree \(p^t\) in elements of \({\mathfrak {m}}\) and so must vanish.
Definition 2.6
Let \(k'/k''/k\) be a tower of finite purely inseparable field extensions.
-
(i)
If \(x\in k'\), then the minimal \(s\in {\mathbb {N}}\) such \(x^{p^s}\in k''\) is called the exponent of x with respect to \(k''\) and denoted by \(e_{k''}(x)\).
-
(ii)
We say that \(x\in k'\) is normal in \(k'/k''\) if \(e_{k''}(x)\ge e_{k''}(y)\) for all \(y\in k'\).
-
(iii)
A sequence \(\alpha _1,\ldots , \alpha _l\) in \(k'\) is called a normal sequence if for every \(1\le i \le l\) and \(k_i=k[\alpha _1,\ldots , \alpha _i]\), we have that \(\alpha _i\) is normal in \(k'/k_{i-1}\), and \(\alpha _i \notin k_{i-1}\).
-
(iv)
A normal sequence \(\alpha _1,\ldots , \alpha _l\) is called a normal generating sequence of \(k'/k\) if \(k[\alpha _1,\ldots , \alpha _l]=k'\).
-
(v)
For a normal generating sequence \(\alpha _1,\ldots , \alpha _l\) of \(k'/k\), we set \(e_i=e_{k_{i-1}}(\alpha _i)\). We call \(e_1,\ldots ,e_l\) a sequence of exponents for \(k'/k\)
Note that if \(e_1,\ldots ,e_l\) is a sequence of exponents for \(k'/k\), then we have \(e_1\ge e_2 \ge \cdots \ge e_l\) due to [11, Prop. 5]. Crucially, by loc. cit., we also have
In particular, as the \(e_i\) are decreasing,
Combining this with the main result of [10], we get that the sequence of exponents is an invariant of the field extension \(k'/k\). More precisely:
Lemma 2.9
Let \(k'/k\) be a finite purely inseparable field extension. Suppose that \(\alpha _{1},\ldots ,\alpha _l\) and \(\beta _1,\ldots ,\beta _{l'}\) are normal generating sequences of \(k'/k\) with sequences of exponents \(e_1,\ldots , e_l\) and \(e'_1,\ldots , e'_{l'}\). Then \(l=l'\) and \(e_i=e'_i\) for \(1\le i< l\).
By the lemma, the numbers \({{\,\mathrm{m}\,}}(k'/k)\), \({{\,\mathrm{mr}\,}}(k'/k,r)\) defined in the introduction are both invariants of \(k'/k\) and \((k'/k,r)\) respectively. To relate these with \({\mathfrak {m}}\), we first prove:
Lemma 2.10
For a normal generating sequence \(\alpha _1,\ldots , \alpha _l\) of \(k'/k\), define \(a_i:=1\otimes \alpha _i-\alpha _i\otimes 1\in {\mathfrak {m}}\). Then:
-
(i)
the ideal \({\mathfrak {m}}\) is generated by the \(a_i\) over \(k'\);
-
(ii)
the \(p^{e_i}\)-power map takes \({\mathfrak {m}}\) into the \(k'\)-subspace \(k'[a_1^{p^{e_i}},\ldots ,a_{i-1}^{p^{e_i}}]\cap {\mathfrak {m}}\).
Proof
(i) Since \({\mathfrak {m}}\) is a codimension 1 (left) \(k'\)-subspace of \({\mathfrak {B}}\), it is a codimension \([k':k]\) k-subspace of \({\mathfrak {B}}\). Define \(J=\{ (b_1,b_2,\ldots ,b_l)\in {\mathbb {Z}}^l \mid 0\le b_i\le p^{e_i}-1 \}\). For \(b=(b_1,...,b_l)\in J\), we set \(x^b=\prod _{i=1}^{l}\alpha _{i}^{b_i}\). Then \(\{x^b\}_{b\in J}\) is a k-basis of \(k'\) and \(\{1\otimes x^b\}\) is a \(k'\)-basis of \({\mathfrak {B}}\). For \(b\ne (0,\ldots , 0)\), the elements \(a_b:=1\otimes x^b-x^b\otimes 1=1\otimes x^b-x^b (1\otimes 1)\) are therefore linearly independent, generating a codimension 1 subspace of \({\mathfrak {B}}\), which is contained in the codimension 1 kernel \({\mathfrak {m}}\) of the multiplication map \(x\otimes y\mapsto xy\); hence they form a \(k'\)-basis of \({\mathfrak {m}}\). We wish to show \(a_b\in {\mathfrak {m}}':={\mathfrak {m}}\cap k'[a_1,\ldots ,a_l]\) by induction on \(\sum b_i\), the case \(\sum b_i=1\) being trivial. So let \(\sum b_i>1\). Without loss of generality, suppose \(b_1>0\) and let \(b'=b-(1,0,\ldots ,0)\). Then inductively \(a_{b'}=1\otimes x^{b'}-x^{b'}\otimes 1\in {\mathfrak {m}}'\). Now one checks directly that \(a_b=x^{b'}a_1+\alpha _1x^{b'}-a_{b'}\cdot a_1\in {\mathfrak {m}}'\) as required. (And it follows that \({\mathfrak {m}}={\mathfrak {m}}'\).)
(ii) Define \({\mathfrak {m}}':=k'[a_1^{p^{e_i}},\ldots ,a_{i-1}^{p^{e_i}}]\cap {\mathfrak {m}}\). From (i), we can express any element of \({\mathfrak {m}}\) as a polynomial in \(a_1,\ldots , a_l\) over \(k'\). Since \(k'\) has characteristic p, the \(p^{e_i}\)-power map distributes over addition and so it suffices to check that \((a_1^{r_1}\cdots a_l^{r_l})^{p^{e_i}}\in {\mathfrak {m}}'\) for \(r_1,\ldots , r_l\in {\mathbb {Z}}_{\ge 0}\). For this, it suffices to check that \(a_j^{p^{e_i}}\in {\mathfrak {m}}'\) for \(j\ge i\). Now \(a_j^{p^{e_i}}=(1\otimes \alpha _j -\alpha _j\otimes 1)^{p^{e_i}}=1\otimes \alpha _j^{p^{e_i}}-\alpha _j^{p^{e_i}}\otimes 1\) and by (2.8), we can write \(\alpha _j^{p^{e_i}}\) as a k-polynomial P in \(k[\alpha _1^{p^{e_i}},\ldots ,\alpha _{i-1}^{p^{e_i}}]\) with no constant term. But applying part (i) with \(l=i-1\), \(a_j^{p^{e_i}}\) in place of \(a_j\) for \(1\le j\le i-1\), and \(k[a_1^{p^{e_i}},\dots , a_{i-1}^{p^{e_i}}]\) in place of \(k'\), we deduce that \(1\otimes P-P\otimes 1\) is indeed a polynomial in the \(a_j^{p^{e_i}}\) with \(1\le j\le i-1\) as required.\(\square \)
We may now identify the minimal integer n such that \({\mathfrak {m}}^n=0\) with the integer \({{\,\mathrm{m}\,}}(k'/k)\). In fact, we also prove something stronger:
Lemma 2.11
With notation as above, let \(d\le l\) and suppose \(m_1,\ldots ,m_d\in {\mathfrak {m}}\) and \(f_1,\ldots , f_d\in {\mathbb {Z}}_{\ge 0}\).
If \(\sum f_i\ge {{\,\mathrm{mr}\,}}(k'/k,d)-d+1=\sum p^{e^i}-d+1\), then \(m_1^{f_1}\cdots m_d^{f_d}=0\).
In particular, the minimum n such that \({\mathfrak {m}}^{n}=0\) is \({{\,\mathrm{m}\,}}(k'/k)\).
Proof
Assume for a contradiction that \(m:=m_1^{f_1}\cdots m_d^{f_d}\ne 0\). We proceed by induction on d, observing that the product of any factors of m is also non-zero. It does no harm to assume that \(f_1\ge \cdots \ge f_d\). If \(d=1\), then \({{\,\mathrm{mr}\,}}(k'/k,1)=p^{e_1}=p^e\) and indeed \(m=0\), a contradiction. Otherwise, let c be maximal such that \(f_c\ge p^{e_c}\). This implies \(\sum _{i=1}^c f_i\ge {{\,\mathrm{mr}\,}}(k'/k,c)-c+1\) so if \(c<d\), then \(m_1^{f_1}\cdots m_c^{f_c}\) is zero by induction, a contradiction. Therefore \(c=d\). Let \((q_i,r_i)\) be the quotient and remainder when \(p^{e_d}\) is divided into \(f_i\). As \(f_i\ge f_d\ge p^{e_d}\), we have each \(q_i\ge 1\). Now
and so
Since the second expression above is a multiple of \(p^{e_d}\), we have
Now as \(q_ip^{e_d}\le f_i\) for each i, we have \(m'=m_1^{q_1p^{e_d}}\cdots m_d^{q_dp^{e_d}}\ne 0\). By Lemma 2.10(ii), we may write each \(m_i^{p^{e_d}}\) as a polynomial in \(k'[a_1^{p^{e_d}},\ldots ,a_{d-1}^{p^{e_d}}]\). After expanding \(m'\), any constituent monomial will be of the form \(a_1^{\nu _1}\cdots a_{d-1}^{\nu _{d-1}}\) where \(\sum _{i=1}^{d-1}\nu _i=p^{e_d}\sum _{i=1}^d q_i\ge p^{e_1}+\cdots +p^{e_{d-1}}-p^{e_d}(d-2)\) and each \(\nu _i\) is divisible by \(p^{e_d}\). Furthermore, as \(a_{d-1}^{p^{e_{d-1}}}\) can be re-expressed in terms of the \(a_i\) with \(i<d-1\), we may assume \(\nu _{d-1}\le p^{e_{d-1}}-p^{e_d}\). It follows that
Continuing inductively, \(\nu _1\ge p^{e_1}-p^{e_d}+1\). Thus \(\nu _1\ge p^{e_1}\) and hence all monomials in \(m'\) are zero, a contradiction.
We now tackle the second statement. To see that \(n\ge {{\,\mathrm{m}\,}}(k'/k)\), observe that \(a:=\prod _{i=1}^{l}a_i^{p^{e_i}-1}\) is a product of \({{\,\mathrm{m}\,}}(k'/k)-1\) elements from \(\{a_1,\ldots , a_l\}\). Expanding this product in terms of the k-basis \(\{x^b\otimes x^c\mid b,c\in J\}\) of \({\mathfrak {B}}\), the coefficient of \(1\otimes x^b\) where \(b=(p^{e_1}-1,\ldots ,p^{e_l}-1)\) is easily seen to be 1. It follows that \(a\ne 0\).
For the upper bound, take \(d=l\) in the first part of the lemma, noting that \({{\,\mathrm{m}\,}}(k'/k)={{\,\mathrm{mr}\,}}(k'/k,l)-l+1\). \(\square \)
3 Bounds in the case of the general linear group
In this section, \(G'={{\,\mathrm{GL}\,}}_r\) and \(G=\mathrm {R}_{k'/k}(G')\), where \(k'/k\) is purely inseparable of exponent e. In [1, Lem. 4.4], it was shown that the exponent e(R) of \(R:={\mathscr {R}}_u(G_{{\bar{k}}})\) is at most s, where s is chosen such that \(p^s\ge r^2(p^e-1)\). Using the Cayley–Hamilton theorem in its full generality (see [2, III.8.11]) yields an improvement.
Proof of Theorem 1.1(i)
Let \(M\in {{\,\mathrm{Mat}\,}}(r,{\mathfrak {m}})\), as in Lemma 2.5. By the Cayley–Hamilton theorem, we have \(\chi _M(M)=0\), where \(\chi _M(\lambda )=\lambda ^r+\sum _{t=0}^{r-1}f_t \lambda ^t\) is the characteristic polynomial of M with degree r. Observe that the coefficient \(f_t\) is either zero or a homogeneous polynomial in the entries of \({\mathfrak {m}}\) with degree \(r-t\). Since the matrices \(M^{r-1},\ldots ,M,I_r\) all commute and \(k'\otimes _{k} k'\) has characteristic p, we get
since \(f_t^{p^e}=0\) for all \(f_t\in {\mathfrak {m}}\). Choosing s such that \(p^s\ge p^er\), we get \(M^{p^s}=M^{p^er}M^{p^s-p^er}=0\). \(\square \)
Let \(B'\subseteq G'\) be the upper Borel subgroup, whose points are upper triangular matrices, and \(B=\mathrm {R}_{k'/k}(B')\subseteq G\) its Weil restriction—a pseudo-Borel subgroup in the terminology of [5, C.2]. We now specify certain elements \(M\in {{\,\mathrm{Mat}\,}}(r,{\mathfrak {m}})\) with \(M\in B(k')=B'({\mathfrak {B}})\) to construct lower bounds on e(R) which depend on r and the exponents \(e_i\) of a normal generating sequence. Moreover, we calculate exactly the exponent of the intersection \(R\cap B_{{\bar{k}}}\).
Before we give the proof, we make some elementary remarks about powers of matrices. If A is a commutative ring and \(E_{i,j}\in {{\,\mathrm{Mat}\,}}(r,A)\) is the \(r\times r\) elementry matrix with \(e_{i,j}=1\) and zeros everywhere else, then we have \(E_{i,j}E_{l,s}=E_{i,s}\) if and only if \(j=l\) and 0 else. Write \(M=\sum a_{ij}E_{i,j}\) for a general element of \({{\,\mathrm{Mat}\,}}(r,A)\) . Then after expansion, any non-zero term in the power \(M^n\) is of the form \(a_w\cdot E_{i_1,i_{n+1}}\) where \(a_w=a_{i_1,i_2}\cdot a_{i_2,i_3}\cdots a_{i_l,i_{n+1}}\), and \(w\in (i_1,\ldots ,i_{n+1})\in \{1,\ldots ,r\}^{n+1}\). Furthermore, the (i, j)-th entry of \(M^n\) is the sum over all \(a_w\) such that \(i_1=i\) and \(i_{n+1}=j\). For \(w\in \{1,\ldots ,r\}^{n+1}\), say n is the length of w. Now, if M is an upper triangular matrix, then \(a_w\) will be zero unless \(i_j\le i_{j+1}\) for each \(1\le j\le n+1\). Hence the only w yielding non-zero \(a_w\) are non-decreasing sequences of integers from \(\{1,\ldots ,r\}\), whose length is bounded above by \({{\,\mathrm{m}\,}}(k'/k)\) (Lemma 2.11).
Proof of Theorem 1.1
(ii)-(iv). We work inside \(B'({\mathfrak {m}})\). There are two cases. Suppose we have \(\sum _{i=r+1}^l(p^{e_i}-1)<r-1\). Then there is an integer \(q<r\) with q maximal such that \((p^{e_{q+1}}-1)+\cdots +(p^{e_l}-1)\ge q-1\). Define the matrix \(M\in B'({\mathfrak {m}})\) as follows.
-
(i)
Start with \(M=0\);
-
(ii)
fill the first q spaces on the diagonal with \(a_1,\ldots , a_q\) respectively;
-
(iii)
Fill up the first \(q-1\) elements of the superdiagonal with \(p^{e_l}-1\) entries of \(a_l\), then \(p^{e_{l-1}}-1\) entries of \(a_{l-1}\) and so on.
-
(iv)
If one has now written down \(p^{e_{q+1}}-1\) entries \(a_q\), then stop. Otherwise put \(a_q\) in the q-th position of the superdiagonal. If, after this, there are only \(p^{e_{q+1}}-1-\tau \) entries \(a_q\) with \(\tau \ge 1\) on the superdiagonal, we set also the \((q+1)\)st diagonal element equal to \(a_q\).
We claim \(M^{{{\,\mathrm{m}\,}}(k'/k)-1}\ne 0\). Take the sequence
and observe that \(a_w\ne 0\). A little thought shows that this is the unique sequence w of this length with \(a_w\ne 0\) starting at 1 and ending at \(q+1\). This proves the claim. Hence we conclude \(e(R\cap B_{{\bar{k}}})\ge {{\,\mathrm{em}\,}}(k'/k)={{\,\mathrm{E}\,}}(k'/k,r)\). Since \(M^{{{\,\mathrm{m}\,}}(k'/k)}=0\) by Lemmas 2.11 and 2.5, we have \(e(R\cap B_{{\bar{k}}})\le {{\,\mathrm{em}\,}}(k'/k)\) and so this case is done.
Now suppose \(\sum _{i=r+1}^l(p^{e_i}-1)\ge r-1\). One constructs a similar M. Start as before with \(M=0\). One fills the diagonal with \(a_1,\ldots , a_r\) and the superdiagonal with any \(r-1\) elements from the list \((\underbrace{a_{r+1},\ldots ,a_{r+1}}_{p^{e_{r+1}}-1\text { times}},\ldots ,\underbrace{a_l,\ldots ,a_l}_{p^{e_l}-1 \text { times}})\). This gives a matrix M such that \(M^m\ne 0\) where \(m=\sum _{i=1}^r(p^{e_i}-1)+r-1={{\,\mathrm{mr}\,}}(k'/k,r)-1\). Hence \(e(R\cap B_{{\bar{k}}})\ge {{\,\mathrm{emr}\,}}(k'/k,r)\).
We need to demonstrate the upper bound for this case. Take a matrix \(M\in B'({\mathfrak {m}})\) and any sequence w of length \({{\,\mathrm{mr}\,}}(k'/k)\) such that \(a_{w}\ne 0\) in a product of \({{\,\mathrm{mr}\,}}(k'/k,r)\) copies of M. Suppose the entries on the diagonal of M are \(m_1,\ldots ,m_r\in {\mathfrak {m}}\). Then \(a_w=m_1^{f_1}\cdots m_r^{f_r}\cdot \mu \) for non-negative integers \(f_1,\ldots ,f_r\) and \(\mu \) a product of s entries from above the diagonal; we have \(s\le r-1\). Since \({{\,\mathrm{mr}\,}}(k'/k)=\sum f_i+s\), we have \(\sum f_i\ge {{\,\mathrm{mr}\,}}(k'/k,r)-r+1\). We are now done by Lemma 2.11.
For part (iii) of the theorem, we note that any lower bound for \(e(R\cap B_{{\bar{k}}})\) is automatically one for e(R). For part (iv), we just observe that \({{\,\mathrm{E}\,}}(k'/k,r)={{\,\mathrm{em}\,}}(k'/k)\) is also an upper bound for e(R) by Lemmas 2.11 and 2.5.\(\square \)
4 Extended example of
\({{\,\mathrm{SL}\,}}_2\) and \({{\,\mathrm{PGL}\,}}_2\). In this section, we give a complete description of e(R) in case \(G'={{\,\mathrm{SL}\,}}_2,{{\,\mathrm{PGL}\,}}_2\), or \({{\,\mathrm{GL}\,}}_2\).
Proposition 4.1
Let \(k'/k\) be purely inseparable of characteristic p and exponent e. Let G be \(\mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_2)\), \(\mathrm {R}_{k'/k}({{\,\mathrm{PGL}\,}}_2)\), or \(\mathrm {R}_{k'/k}({{\,\mathrm{SL}\,}}_2)\), the last of these only if \(p\ne 2\). Then e(R) is e if \(k'/k\) is simple and \(e+1\) otherwise.
Proof
If \(G'\) is \({{\,\mathrm{GL}\,}}_2\), then Theorem 1.1(iv) gives the exponent \(e(R)=e\) for \(k'/k\) simple of exponent e, and at least \(e+1\) when \(k'/k\) is non-simple. However, by Theorem 1.1(i), we have e(R) is at most \(e+1\). As the map \({{\,\mathrm{GL}\,}}_2\rightarrow {{\,\mathrm{PGL}\,}}_2\) is smooth, and \({{\,\mathrm{SL}\,}}_2\subset {{\,\mathrm{GL}\,}}_2\), we immediately obtain e or \(e+1\) respectively as upper bounds for both the cases \(G'={{\,\mathrm{SL}\,}}_2\) and \(G'={{\,\mathrm{PGL}\,}}_2\). In fact, the matrices constructed in the proof of Theorem 1.1(ii)-(iv) and their relevant powers are clearly non-zero in their image in \({{\,\mathrm{PGL}\,}}_2({\mathfrak {B}})\) and so the exponents of the radicals in case \(G'={{\,\mathrm{GL}\,}}_2\) and \(G'={{\,\mathrm{PGL}\,}}_2\) coincide. Now by Proposition 2.3(iv), these also coincide for \({{\,\mathrm{SL}\,}}_2\) outside of characteristic 2, since the isogeny \(f:{{\,\mathrm{SL}\,}}_2\rightarrow {{\,\mathrm{PGL}\,}}_2\) is smooth. \(\square \)
We see some different behaviour for \({{\,\mathrm{SL}\,}}_2\) in characteristic 2; in particular, the exponents of \(R_u(G_{\bar{k}})\cap R_u(B_{\bar{k}})\) and \(R_u(G_{\bar{k}})\) do not coincide.
Recall that a unital ring R has characteristic 2 if the equation \(1+1=0\) holds in R. We prove a general formula for \(2^s\)-th powers of matrices over rings of characteristic 2.
Lemma 4.2
Let R be a commutative ring of characteristic 2 and \(M=\begin{pmatrix} a &{}\quad b \\ c &{}\quad d \end{pmatrix}\) with \(a,b,c,d\in R\). Then for all \(s\in {\mathbb {N}}_0\), we have
where \(t=a+d\).
Proof
We argue by induction on s. For \(s=0\), we just get M on both sides. So assume the result for s and set \(l_s=\sum _{i=0}^{s-1} b^{2^i}c^{2^i}t^{2^s-2^{i+1}}\). We note that
holds. And so we get
\(\square \)
Proposition 4.3
Let \(G'={{\,\mathrm{SL}\,}}_2\), \(p=2\), and suppose \(k'/k\) has normal generating sequence \(\alpha _1,\ldots ,\alpha _l\) with exponents \(e_1\ge e_2\ge \cdots \ge e_l\). Let \(B'\) be an upper Borel subgroup of \(G'\) and \(B=\mathrm {R}_{k'/k}(B')\) a pseudo-Borel subgroup.
-
(i)
We have \(e(R\cap B_{{\bar{k}}})=e\).
-
(ii)
We have \(e(R)={\left\{ \begin{array}{ll}e+1\text { if }e_1=e_2,\\ e\text { otherwise.}\end{array}\right. }\)
Proof
We first calculate the exponent of \(B'({\mathfrak {m}})=\ker B'({\mathfrak {B}})\rightarrow B'(k')\). A general element is \(I+M\) with \(M=\begin{pmatrix} a &{}\quad b\\ 0 &{}\quad d\\ \end{pmatrix}\), where \(a,b,d\in {\mathfrak {m}}=\langle a_1,\ldots ,a_l\rangle \), satisfying \(a+d+ad=0\) from the condition \(\det (I+M)=1\). As \(a\in {\mathfrak {m}}\), \(1+a\) is invertible in \({\mathfrak {B}}\), and so it follows that \(d=a(1+a)^{-1}\) and \(a+d=ad=a^2(1+a)^{-1}\). By Lemma 4.2, the \(2^s\)-th power of this matrix is
Choosing for example \(a=a_1=1\otimes \alpha _1-\alpha _1\otimes 1\), one sees the minimal s for which \(M^{2^s}=0\) is at least \(e_1=e\) and also that if \(s\ge e\), then the diagonal entries vanish. For \(s=e\), the off-diagonal entry is a multiple of \((a^2)^{2^e-1}\), which has at least \(2^e\) factors of a, hence is also 0. So \(e(R\cap B_{{\bar{k}}})=e\), proving (i).
For (ii), a general element of \(G'({\mathfrak {m}})\) is a matrix \(I+M\) where \(M=\begin{pmatrix} a &{}\quad b\\ c &{}\quad d\\ \end{pmatrix}\), satisfying \(a+d+ad+bc=0\), since \(\det (I+M)=1\). If \(l=1\) (i.e. \(k'/k\) is simple of exponent \(e=e_1\)), then it is clear that \(e(R)=e\). Suppose \(e_2\le e_1-1=e-1\). We may write \(d=(1+a)^{-1}(a+bc)\), hence \(t=a+d=(1+a)^{-1}(a^2+bc)\). Therefore \(t^{2^{e-1}}=(1+a)^{-2^{e-1}}(a^{2^e}+b^{2^{e-1}}c^{2^{e-1}})=(1+a)^{-2^{e-1}}(b^{2^{e-1}}c^{2^{e-1}})\). Now by Lemma 2.10, we have \(m^{2^{e-1}}\in \langle a_1^{2^{e-1}}\rangle \subseteq {\mathfrak {m}}\) for any \(m\in {\mathfrak {m}}\). It follows that \(b^{2^{e-1}}c^{2^{e-1}}\) has a factor of \(a_1^{2^{e-1}}\cdot a_1^{2^{e-1}}=a_1^{2^e}=0\). By the same token, \(t^{2^{e-1}}=0\) and hence, using the lemma, \(M^{2^e}=0\). Therefore \(e(R)=e\) as required.
Now suppose \(e_1=e_2\). Since \(G\subseteq \mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_2)\), we have \(e(R)\le e+1\). Set \(b=a_1\), \(c=a_2\), \(d=0\), and \(a=a_1a_2\). Then
The off-diagonal entries are zero, but \(\sum _{i=1}^{e-1} (a_1a_2)^{2^i+2^e-2^{i+1}}=\sum _{i=1}^e (a_1a_2)^{2^e-2^{i}}\) and the \(i=e-1\) entry is \((a_1a_2)^{2^{e-1}}\). Provided \(e_2=e_1\), this element is non-zero by the same argument as in the proof of Lemma 2.11—in a k-basis of \({\mathfrak {B}}\), the coefficient of \(1\otimes \alpha _1^{2^{e-1}}\alpha _2^{2^{e-1}}\) in \((a_1a_2)^{2^{e-1}}\) is evidently 1. It follows that \(M^{2^e}\ne 0\) and \(e(R)=e+1\). \(\square \)
5 Simple groups of rank at least 2
For a positive integer r, recall that the algebraic k-group \(\mu _r\) is the functor returning the r-th roots of unity in any k-algebra A; it is the kernel of the r-power map \(\phi _r:{{\,\mathrm{{\mathbb {G}}_m}\,}}\rightarrow {{\,\mathrm{{\mathbb {G}}_m}\,}},x\mapsto x^r\). (We have \(\mu _r\) étale if and only if \(p\not \mid r\).) Say a \(k'\)-group L is a skewed \({{\,\mathrm{GL}\,}}_n\) if \(L\cong {{\,\mathrm{GL}\,}}_n/K\), where \(K\cong \mu _r\), with \(r\not \mid n\); the quotient by \(\mu _r\) induces the r-power map on the central torus of \({{\,\mathrm{GL}\,}}_n\). Since \(K\cap {{\,\mathrm{SL}\,}}_n=1\), the derived subgroup \({\mathscr {D}}(L)\) of L is still isomorphic to \({{\,\mathrm{SL}\,}}_n\) and notably there is still a central torus \(C\subset L\) which contains \(Z({{\,\mathrm{SL}\,}}_n)\). In [12, Ex. 5.17], it is explained that if \(\mu _n\subset T\) is a finite subgroup of a split k-torus of rank at least 2, then any automorphism of \(\mu _n\) (which is some r-power map on \(\mu _n\) with \((r,n)=1\)) lifts to an automorphism of T. In our case, the quotient map by \(K\times 1\) on \({{\,\mathrm{GL}\,}}_n\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\) lifts to an automorphism of \({{\,\mathrm{GL}\,}}_n\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\) defined on the central torus \({{\,\mathrm{{\mathbb {G}}_m}\,}}\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\) by \((x,y)\mapsto (x^ry^{-n},x^by^a)\) where a and b are integers such that \(ar+bn=1\). It follows that if L is a skewed \({{\,\mathrm{GL}\,}}_n\)-group, then \(L\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\cong {{\,\mathrm{GL}\,}}_n\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\).
The group \({{\,\mathrm{SL}\,}}_{n+1}\) contains a (Levi) subgroup \(L:={{\,\mathrm{GL}\,}}_n\) embedded block diagonally via \(g\mapsto {{\,\mathrm{diag}\,}}(g,\det {g}^{-1})\). If \(G'\) is a split simple k-group of type \(A_n\), then \(G'\cong {{\,\mathrm{SL}\,}}_{n+1}/\mu _r\) with \(\mu _r\subseteq Z({{\,\mathrm{SL}\,}}_{n+1})\cong \mu _{n+1}\), so \(r\mid n+1\). But then \(r\not \mid n\), hence L maps under the quotient by \(\mu _r\) to a skewed \({{\,\mathrm{GL}\,}}_n\)-subgroup of \(G'\). This observation establishes the type A case of the following lemma.
Lemma 5.1
Let \(G'\) be a split simple \(k'\)-group of rank r. Then one of the following holds:
-
(i)
\(G'\) contains a skewed \({{\,\mathrm{GL}\,}}_r\) subgroup L.
-
(ii)
There are étale isogenies \(\pi :L\rightarrow L/Z\), and \(\phi :{{\,\mathrm{GL}\,}}_r\rightarrow L/Z\) or \(\phi :L\rightarrow {{\,\mathrm{GL}\,}}_r\).
-
(iii)
\(p=2\) and either \(G'\) is of type \(F_4\) or \(G'\) is of type B, C, or D, but not isomorphic to \({{\,\mathrm{SO}\,}}_{2r}\), \({{\,\mathrm{SO}\,}}_{2r+1}\), or \({{\,\mathrm{Sp}\,}}_{2r}\).
-
(iv)
\(p=3\) and G is of type \(E_6\).
In cases (i) and (ii), if we let \(R={\mathscr {R}}_u(\mathrm {R}_{k'/k}(L)_{{\bar{k}}})\), then we have \(e(R)=e({\mathscr {R}}_u(\mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_r)_{{\bar{k}}}))\).
Proof
If \(G'={{\,\mathrm{SO}\,}}_{2r}\), \({{\,\mathrm{SO}\,}}_{2r+1}\), or \({{\,\mathrm{Sp}\,}}_{2r}\), then [7, 4.5] promises a Levi subgroup \(L\cong {{\,\mathrm{GL}\,}}_r\) as required so (i) holds in all these cases. The central isogenies \({{\,\mathrm{Sp}\,}}_{2r}\rightarrow {{\,\mathrm{PSp}\,}}_{2r}\), \({{\,\mathrm{Spin}\,}}_n\rightarrow {{\,\mathrm{SO}\,}}_n\), \({{\,\mathrm{SO}\,}}_n\rightarrow {{\,\mathrm{PSO}\,}}_n\), \({{\,\mathrm{Spin}\,}}_{2r}\rightarrow {{\,\mathrm{HSpin}\,}}_{2r}\), or \({{\,\mathrm{HSpin}\,}}_{2r}\rightarrow {{\,\mathrm{PSO}\,}}_{2r}\) (where \({{\,\mathrm{HSpin}\,}}_{2r}\) occurs only if n even) are all quotients by central subgroups isomorphic to \(\mu _2\), hence are étale maps when \(p>2\) and hence (ii) holds for all classical groups when \(p>2\). Otherwise (iii) holds.
Let now \(G'\) be exceptional. Note that groups of type \(E_7\) and \(E_8\) contain simple \(A_7\) and \(A_8\) maximal rank subgroups, evident from the Borel–de-Siebenthal algorithm. Hence each contains a skewed \({{\,\mathrm{GL}\,}}_7\) and \({{\,\mathrm{GL}\,}}_8\) subgroup L respectively, satisfying (i) as required. By [8, Thm. A], any Levi subgroup L of \(G_2\) is isomorphic to \({{\,\mathrm{GL}\,}}_2\) (rather than \({{\,\mathrm{SL}\,}}_2\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\) for example), hence satisfies (i). If \(G'\) is of type \(F_4\), then we take L to be an \(A_3\) Levi subgroup of the \({{\,\mathrm{Spin}\,}}_9\) subgroup of \(G'\). Now L admits an étale isogeny with the \({{\,\mathrm{GL}\,}}_4\) Levi subgroup of \({{\,\mathrm{SO}\,}}_9\cong {{\,\mathrm{Spin}\,}}_9/Z({{\,\mathrm{Spin}\,}}_9)\) if \(p\ne 2\); so L satisfies (ii) in that case, and if \(p=2\), then L is listed in (iii). It remains to deal with the case \(G'=E_6\). Using Bourbaki notation [3], we take L to be the \(A_5\) Levi subgroup corresponding to the nodes \(\{1,3,4,5,6\}\) of the Dynkin diagram explicitly. Assume first \(G'\) is simply connected. Then \({\mathscr {D}}(L)\) is simply connected. Since L centralises the \({{\,\mathrm{SL}\,}}_2\) subgroup generated by the \(\pm {{\tilde{\alpha }}}\) root groups, where \({{\tilde{\alpha }}}\) is the longest root, the central torus C of L is the image of the cocharacter \(h_{\tilde{\alpha }}:{{\,\mathrm{{\mathbb {G}}_m}\,}}\rightarrow G', t\mapsto h_{\tilde{\alpha }}(t)\). One calculates \(h_{\tilde{\alpha }}(t)=h_{\alpha _1}(t)h_{\alpha _2}(t^2)h_{\alpha _3}(t^2)h_{\alpha _4}(t^3)h_{\alpha _5}(t^2)h_{\alpha _6}(t)\). This is inside \({\mathscr {D}}(L)\cong {{\,\mathrm{SL}\,}}_6\) iff \(t^2=1\); therefore \({\mathscr {D}}(L)\cap C\cong \mu _2\)—so L is not a skewed \({{\,\mathrm{GL}\,}}_r\). However, if \(p\ne 3\), then there is an étale isogeny from L to \(L/Z\cong {{\,\mathrm{GL}\,}}_6\) where Z is a diagonal copy of \(\mu _3\) in both factors \({\mathscr {D}}(L)\) and C of L, so that (ii) holds. Besides which, since \(Z(G')\cong \mu _3\), then if \(p\ne 3\), the quotient by \(Z(G')\) is étale and \(L/Z(G')\) satisfies (ii) for the adjoint group \(G/Z(G')\). Otherwise \(p=3\) and (iv) holds.
We prove the last statement. Suppose (ii) holds. Then we note that if \(\pi :A\rightarrow B\) is an étale central isogeny of reductive groups, then \(\mathrm {R}_{k'/k}(A)\rightarrow \mathrm {R}_{k'/k}(B)\) is an isogeny whose kernel has trivial intersection with the unipotent radical. As in Proposition 2.3(iv), it follows that the induced map \({\mathscr {R}}_u(\mathrm {R}_{k'/k}(A)_{{\bar{k}}})\rightarrow {\mathscr {R}}_u(\mathrm {R}_{k'/k}(B)_{{\bar{k}}})\) is an isomorphism as required. If (i) holds, then L is a skewed \({{\,\mathrm{GL}\,}}_r\)-subgroup and by the remarks preceding the lemma, \(L\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\cong {{\,\mathrm{GL}\,}}_r\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\). Observe \({\mathscr {R}}_u(\mathrm {R}_{k'/k}(L\times {{\,\mathrm{{\mathbb {G}}_m}\,}})_{{\bar{k}}})\cong {\mathscr {R}}_u(\mathrm {R}_{k'/k}(L)_{{\bar{k}}})\times {\mathscr {R}}_u(\mathrm {R}_{k'/k}({{\,\mathrm{{\mathbb {G}}_m}\,}})_{{\bar{k}}})\); since L already contains \(C\cong {{\,\mathrm{{\mathbb {G}}_m}\,}}\), the exponent of the product is equal to that of the first factor, e(R). The same argument with \({{\,\mathrm{GL}\,}}_r\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\) in place of \(L\times {{\,\mathrm{{\mathbb {G}}_m}\,}}\) implies the claim. \(\square \)
Proof of Corollary 1.3
With the hypotheses on p, by Lemma 5.1, we get a reductive subgroup L of \(G'\), with
By Proposition 2.3(iii), it follows that \(e(R)\ge {{\,\mathrm{E}\,}}(k'/k,r)\). If \(\sum _{i=r+1}^l(p^{e_i}-1)<r-1\), then Weil restricting a faithful representation \(G'\rightarrow {{\,\mathrm{GL}\,}}_n\) from \(k'\) to k gives an embedding of G in \({{\,\mathrm{GL}\,}}_m\) with \(m=[k':k]\cdot n\ge r\). Now by Proposition 2.3(iii) again, \(e(R)\le e({\mathscr {R}}_u(\mathrm {R}_{k'/k}({{\,\mathrm{GL}\,}}_m))_{{\bar{k}}})\) which is equal to \({{\,\mathrm{E}\,}}(k'/k,m)={{\,\mathrm{em}\,}}(k'/k)\) by Theorem 1.1.\(\square \)
References
Bate, M., Martin, B., Röhrle, G., Stewart, D.I.: On unipotent radicals of pseudo-reductive groups. Michigan Math. J. 68(2), 277–299 (2019)
Bourbaki, N.: Algebra I. Chapters 1-3. Translated from the French. Reprint of the 1989 English translation. Elements of Mathematics (Berlin). Springer, Berlin (1998)
Bourbaki, N.: Lie Groups and Lie Algebras. Chapters 4 -6. Translated from the 1968 French original by Andrew Pressley. Elements of Mathematics (Berlin). Springer, Berlin (2002)
Conrad, B., Gabber, O., Prasad, G: Pseudo-Reductive Groups, 2nd edn., New Mathematical Monographs, vol. 26. Cambridge University Press, Cambridge (2015)
Conrad, B., Prasad, G.: Classification of Pseudo-Reductive Groups. Annals of Mathematics Studies, vol. 191. Princeton University Press, Princeton (2016)
Humphreys, J.E.: Linear Algebraic Groups. Graduate Texts in Mathematics, No. 21. Springer, New York (1975)
Jantzen, J.C.: Nilpotent orbits in representation theory. In: Lie Theory, Progress in Mathematics, vol. 228, pp. 1–211. Birkhäuser, Boston (2004)
Kleidman, P.B.: The maximal subgroups of the Chevalley groups \(G_2(q)\) with \(q\) odd, the Ree groups \(^2G_2(q)\), and their automorphism groups. J. Algebra 117(1), 30–71 (1988)
Oesterlé, J.: Nombres de Tamagawa et groupes unipotents en caractéristique \(p\). Invent. Math. 78(1), 13–88 (1984)
Pickert, G.: Eine Normalform für endliche reininseparable Körpererweiterungen. Math. Z. 53, 133–135 (1950)
Rasala, R.: Inseparable splitting theory. Trans. Amer. Math. Soc. 162, 411–448 (1971)
Taylor, J.: The structure of root data and smooth regular embeddings of reductive groups. Proc. Edinb. Math. Soc. (2) 62(2), 523–552 (2019)
Acknowledgements
We thank Michael Bate and Gerhard Röhrle for helpful comments on early versions of the paper. Many thanks to Jay Taylor for discussions about root data. Finally, thanks to the referee for suggesting improvements to the exposition.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Funded by Leverhulme Trust Research Project Grant Number RPG-2021-080.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bannuscher, F., Gruchot, M. & Stewart, D.I. On the exponent of geometric unipotent radicals of pseudo-reductive groups. Arch. Math. 118, 451–464 (2022). https://doi.org/10.1007/s00013-022-01731-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-022-01731-3