# Details

## Linearity for actions on vector groups

### Abstract:

Let $$k$$ be an arbitrary field, let $$G$$ be a (smooth) linear algebraic group over $$k$$, and let $$U$$ be a vector group over $$k$$ on which $$G$$ acts by automorphisms of algebraic groups. The action of $$G$$ on $$U$$ is said to be linear if there is a $$G$$-equivariant isomorphism of algebraic groups $$U \simeq$$ Lie$$(U)$$.

Suppose that $$G$$ is connected and that the unipotent radical of $$G$$ is defined over $$k$$. If the $$G$$-module Lie$$(U)$$ is a simple, we show that the action of $$G$$ on $$U$$ is linear. If $$G$$ acts by automorphisms on a connected, split unipotent group $$U$$, we deduce that $$U$$ has a filtration by $$G$$-invariant closed subgroups for which the successive factors are vector groups with a linear action of $$G$$. This verifies for such $$G$$ an assumption made in earlier work of the author on the existence of Levi factors.

On the other hand, for any field $$k$$ of positive characteristic we show that if the category of representations of $$G$$ is not semisimple, there is an action of $$G$$ on a suitable vector group $$U$$ which is not linear.

## Some good-filtration subgroups of simple algebraic groups

### Abstract:

Let $$G$$ be a connected and reductive algebraic group over an algebraically closed field of characteristic $$p>0$$. An interesting class of representations of $$G$$ consists of those $$G$$-modules having a good filtration – i.e. a filtration whose layers are the standard highest weight modules obtained as the space of global sections of $$G$$-linearized line bundles on the flag variety of $$G$$. Let $$H \subset G$$ be a connected and reductive subgroup of $$G$$. One says that $$(G,H)$$ is a Donkin pair, or that $$H$$ is a good filtration subgroup of $$G$$, if whenever the $$G$$-module $$V$$ has a good filtration, the $$H$$-module $$\text{res}^G_H V$$ has a good filtration.

In this paper, we show when $$G$$ is a classical group'' that the optimal $$\text{SL}_2$$-subgroups of $$G$$ are good filtration subgroups. We also consider the cases of subsystem subgroups in all types and determine some primes for which they are good filtration subgroups.

## Levi factors of the special fiber of a parahoric group scheme and tame ramification

### Abstract:

Let $$\mathcal{A}$$ be a Henselian discrete valuation ring with fractions $$K$$ and with perfect residue field $$k$$ of characteristic $$p>0$$. Let $$G$$ be a connected and reductive algebraic group over $$K$$, and let $$\mathcal{P}$$ be a parahoric group scheme over $$\mathcal{A}$$ with generic fiber $$\mathcal{P}_{/K} = G$$. The special fiber $$\mathcal{P}_{/k}$$ is a linear algebraic group over $$k$$.

If $$G$$ splits over an unramified extension of $$K$$, we proved in some previous work that the special fiber $$\mathcal{P}_{/k}$$ has a Levi factor, and that any two Levi factors of $$\mathcal{P}_{/k}$$ are geometrically conjugate. In the present paper, we extend a portion of this result. Following a suggestion of Gopal Prasad, we prove that if $$G$$ splits over a tamely ramified extension of $$K$$, then the geometric special fiber $$\mathcal{P}_{/K}$$ has a Levi factor, where $$K$$ is an algebraic closure of $$k$$.

## On the descent of Levi factors

### Abstract:

Let $$G$$ be a linear algebraic group over a field $$k$$ of characteristic $$p>0$$, and suppose that the unipotent radical $$R$$ of $$G$$ is defined and split over $$k$$. By a Levi factor of $$G$$, one means a closed subgroup $$M$$ which is a complement to $$R$$ in $$G$$. In this paper, we give two results related to the descent of Levi factors.

First, suppose $$\ell$$ is a finite Galois extension of $$k$$ for which the extension degree $$[\ell:k]$$ is relatively prime to $$p$$. If $$G_{/\ell}$$ has a Levi decomposition, we show that $$G$$ has a Levi decomposition. Second, suppose that there is a $$G$$-equivariant isomorphism of algebraic groups $$R \simeq \text{Lie}(R)$$ – i.e. $$R$$ is a vector group with a linear action of the reductive quotient $$G/R$$. If $$G_{/k_{\text{sep}}}$$ has a Levi decomposition for a separable closure $$k_{\text{sep}}$$ of $$k$$, then $$G$$ has a Levi decomposition.

Finally, we give an example of a disconnected, abelian, linear algebraic group $$G$$ for which $$G_{/k_{\text{sep}}}$$ has a Levi decomposition, but $$G$$ itself has no Levi decomposition.

## Levi decompositions of a linear algebraic group

### Dedicatory:

Dedicated to the memory of Vladimir Morozov and to his contributions to mathematics}

### Abstract:

If $$G$$ is a connected linear algebraic group over the field $$k$$, a Levi factor of $$G$$ is a reductive complement to the unipotent radical of $$G$$. If $$k$$ has positive characteristic, $$G$$ may have no Levi factor, or $$G$$ may have Levi factors which are not geometrically conjugate. We give in this paper some sufficient conditions for the existence and the conjugacy of Levi factors of $$G$$.

Let $$\mathcal{A}$$ be a Henselian discrete valuation ring with fractions $$K$$ and with perfect residue field $$k$$ of characteristic $$p>0$$. Let $$G$$ be a connected and reductive algebraic group over $$K$$. Bruhat and Tits have associated to $$G$$ certain smooth $$\mathcal{A}$$-group schemes $$\mathcal{P}$$ whose generic fibers $$\mathcal{P}_{/K}$$ coincide with $$G$$; these are known as parahoric group schemes. The special fiber $$\mathcal{P}_{/k}$$ of a parahoric group scheme is a linear algebraic group over $$k$$. If $$G$$ splits over an unramified extension of $$K$$, we show that $$\mathcal{P}_{/k}$$ has a Levi factor, and that any two Levi factors of $$\mathcal{P}_{/k}$$ are geometrically conjugate.

## Nilpotent subalgebras of semisimple Lie algebras

### Abstract:

Let $$\mathfrak{g}$$ be the Lie algebra of a semisimple linear algebraic group. Under mild conditions on the characteristic of the underlying field, one can show that any subalgebra of $$\mathfrak{g}$$ consisting of nilpotent elements is contained in some Borel subalgebra. In this Note, we provide examples for each semisimple group $$G$$ and for each of the torsion primes for $$G$$ of nil subalgebras not lying in any Borel subalgebra of $$\mathfrak{g}$$.

## Nilpotent centralizers and Springer's isomorphisms

### Abstract:

Let $$G$$ be a semisimple algebraic group over a field $$K$$ whose characteristic is very good for $$G$$, and let $$\sigma$$ be any $$G$$-equivariant isomorphism from the nilpotent variety to the unipotent variety; the map $$\sigma$$ is known as a Springer isomorphism. Let $$y \in G(K)$$, let $$Y \in \text{Lie}(G)(K)$$, and write $$C_y = C_G(y)$$ and $$C_Y= C_G(Y)$$ for the centralizers. We show that the center of $$C_y$$ and the center of $$C_Y$$ are smooth group schemes over $$K$$. The existence of a Springer isomorphism is used to treat the crucial cases where $$y$$ is unipotent and where $$Y$$ is nilpotent.

Now suppose $$G$$ to be quasisplit, and write $$C$$ for the centralizer of a rational regular nilpotent element. We obtain a description of the normalizer $$N_G(C)$$ of $$C$$, and we show that the automorphism of $$\text{Lie}(C)$$ determined by the differential of $$\sigma$$ at zero is a scalar multiple of the identity; these results verify observations of J-P. Serre.

## The centralizer of a nilpotent section

### Dedicatory:

To Toshiaki Shoji, with respect and admiration, on the occasion of his 60th birthday.

### Abstract:

Let $$F$$ be an algebraically closed field and let $$G$$ be a semisimple $$F$$-algebraic group for which the characteristic of $$F$$ is very good. If $$X \in \text{Lie}(G) = \text{Lie}(G)(F)$$ is a nilpotent element in the Lie algebra of $$G$$, and if $$C$$ is the centralizer in $$G$$ of $$X$$, we show that (i) the root datum of a Levi factor of $$C$$, and (ii) the component group $$C/C^o$$ both depend only on the Bala-Carter label of $$X$$; i.e. both are independent of very good characteristic. The result in case (ii) depends on the known case when $$G$$ is (simple and) of adjoint type.

The proofs are achieved by studying the centralizer $$\mathcal{C}$$ of a nilpotent section $$X$$ in the Lie algebra of a suitable semisimple group scheme over a Noetherian, normal, local ring $$\mathcal{A}$$. When the centralizer of $$X$$ is equidimensional on $$\text{Spec}(\mathcal{A})$$, a crucial result is that locally in the étale topology there is a smooth $$\mathcal{A}$$-subgroup scheme $$L$$ of $$\mathcal{C}$$ such that $$L_t$$ is a Levi factor of $$\mathcal{C}_t$$ for each $$t \in \text{Spec}(\mathcal{A})$$.

## Completely reducible SL(2) homomorphisms

### Abstract:

Let $$K$$ be any field, and let $$G$$ be a semisimple group over $$K$$. Suppose the characteristic of $$K$$ is positive and is very good for $$G$$. We describe all group scheme homomorphisms $$\phi:\text{SL}_2 \to G$$ whose image is geometrically $$G$$-completely reducible – or $$G$$-cr – in the sense of Serre; the description resembles that of irreducible modules given by Steinberg's tensor product theorem. In case $$K$$ is algebraically closed and $$G$$ is simple, the result proved here was previously obtained by Liebeck and Seitz using different methods. A recent result shows the Lie algebra of the image of $$\phi$$ to be geometrically $$G$$-cr; this plays an important role in our proof.

## Completely reducible Lie subalgebras

### Abstract:

Let $$G$$ be a connected and reductive group over the algebraically closed field $$K$$. J-P. Serre has introduced the notion of a $$G$$-completely reducible subgroup $$H \subset G$$. In this note, we give a notion of $$G$$-complete reducibility – $$G$$-cr for short – for Lie subalgebras of $$\text{Lie}(G)$$, and we show that if the closed subgroup $$H \subset G$$ is $$G$$-cr, then $$\text{Lie}(H)$$ is $$G$$-cr as well.

## On the centralizer of the sum of commuting nilpotent elements

### Abstract:

Let $$X$$ and $$Y$$ be commuting nilpotent $$K$$-endomorphisms of a vector space $$V$$, where $$K$$ is a field of characteristic $$p \ge 0$$. If $$F=K(t)$$ is the field of rational functions on the projective line $$\mathbf{P}^1_{/K}$$, consider the $$K(t)$$-endomorphism $$A=X+tY$$ of $$V$$. If $$p=0$$, or if $$A^{p-1}=0$$, we show here that $$X$$ and $$Y$$ are tangent to the unipotent radical of the centralizer of $$A$$ in $$\text{GL}(V)$$. For all geometric points $$(a:b)$$ of a suitable open subset of $$\mathbf{P}^1$$, it follows that $$X$$ and $$Y$$ are tangent to the unipotent radical of the centralizer of $$aX + bY$$. This answers a question of J. Pevtsova.

## Optimal SL(2) homomorphisms

### Abstract:

Let $$G$$ be a semisimple group over an algebraically closed field of very good characteristic for $$G$$. In the context of geometric invariant theory, G. Kempf has associated optimal cocharacters of $$G$$ to an unstable vector in a linear $$G$$-representation. If the nilpotent element $$X \in \text{Lie}(G)$$ lies in the image of the differential of a homomorphism $$\text{SL}_2 \to G$$, we say that homomorphism is optimal for $$X$$, or simply optimal, provided that its restriction to a suitable torus of $$\text{SL}_2$$ is optimal for $$X$$ in Kempf's sense.

We show here that any two $$\text{SL}_2$$-homomorphisms which are optimal for $$X$$ are conjugate under the connected centralizer of $$X$$. This implies, for example, that there is a unique conjugacy class of principal homomorphisms for $$G$$. We show that the image of an optimal $$\text{SL}_2$$-homomorphism is a completely reducible subgroup of $$G$$; this is a notion defined recently by J-P. Serre. Finally, if $$G$$ is defined over the (arbitrary) subfield $$K$$ of $$k$$, and if $$X \in \text{Lie}(G)(K)$$ is a $$K$$-rational nilpotent element with $$X^{[p]}=0$$, we show that there is an optimal homomorphism for $$X$$ which is defined over $$K$$.

## Nilpotent orbits over ground fields of good characteristic

### Abstract:

Let $$X$$ be an $$F$$-rational nilpotent element in the Lie algebra of a connected and reductive group $$G$$ defined over the ground field $$F$$. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical of the centralizer of $$X$$ is $$F$$-split. This property has several consequences. When $$F$$ is complete with respect to a discrete valuation with either finite or algebraically closed residue field, we deduce a uniform proof that $$G(F)$$ has finitely many nilpotent orbits in $$\mathfrak{g}(F)$$. When the residue field is finite, we obtain a proof that nilpotent orbital integrals converge. Under some further (fairly mild) assumptions on $$G$$, we prove convergence for arbitrary orbital integrals on the Lie algebra and on the group. The convergence of orbital integrals in the case where $$F$$ has characteristic 0 was obtained by Deligne and Ranga Rao (1972).

## Sub-principal homomorphisms in good characteristic

### Abstract:

Let $$G$$ be a reductive group over an algebraically closed field of characteristic $$p$$, and let $$u \in G$$ be a unipotent element of order $$p$$. Suppose that $$p$$ is a good prime for $$G$$. We show in this paper that there is a homomorphism $$\phi:\text{SL}_{2/k} \to G$$ whose image contains $$u$$. This result was first obtained by D. Testerman (J. Algebra, 1995) using case considerations for each type of simple group (and using, in some cases, computer calculations with explicit representatives for the unipotent orbits).

The proof we give is free of case considerations (except in its dependence on the Bala-Carter theorem). Our construction of $$\phi$$ generalizes the construction of a principal homomorphism made by J.-P. Serre in (Invent. Math. 1996); in particular, $$\phi$$ is obtained by reduction modulo $$\mathfrak{p}$$ from a homomorphism of group schemes over a valuation ring $$\mathcal{A}$$ in a number field. This permits us to show moreover that the weight spaces of a maximal torus of $$\phi(\text{SL}_{2/k})$$ on $$\text{Lie}(G)$$ are the same as in characteristic 0''; the existence of a $$\phi$$ with this property was previously obtained, again using case considerations, by Lawther and Testerman (Memoirs AMS, 1999) and has been applied in some recent work of G. Seitz (Invent. Math. 2000).

### Abstract:

Let $$G$$ be a quasisimple algebraic group over an algebraically closed field of characteristic $$p>0$$. We suppose that $$p$$ is very good for $$G$$; since $$p$$ is good, there is a bijection between the nilpotent orbits in the Lie algebra and the unipotent classes in $$G$$. If the nilpotent $$X \in \text{Lie}(G)$$ and the unipotent $$u \in G$$ correspond under this bijection, and if $$u$$ has order $$p$$, we show that the partitions of $$\text{ad}(X)$$ and $$\text{Ad}(u)$$ are the same. When $$G$$ is classical or of type $$G_2$$, we prove this result with no assumption on the order of $$u$$.

In the cases where $$u$$ has order $$p$$, the result is achieved through an application of results of Seitz concerning good $$A_1$$ subgroups of $$G$$. For classical groups, the techniques are more elementary, and they lead also to a new proof of the following result of Fossum: the structure constants of the representation ring of a 1-dimensional formal group law $$\mathcal{F}$$ are independent of $$\mathcal{F}$$.

## Faithful representations of SL(2) over truncated Witt vectors

### Abstracts:

Let $$\Gamma_2$$ be the six dimensional linear algebraic $$k$$-group $$\text{SL}_2(\mathcal{W}_2)$$, where $$\mathcal{W}_2$$ is the ring of Witt vectors of length two over the algebraically closed field $$k$$ of characteristic $$p>2$$. Then the minimal dimension of a faithful rational $$k$$-representation of $$\Gamma_2$$ is $$p+3$$.

## Component groups of unipotent centralizers in good characteristic

### Abstract:

Let $$G$$ be a connected, reductive group over an algebraically closed field of good characteristic. For $$u \in G$$ unipotent, we describe the conjugacy classes in the component group $$A(u)$$ of the centralizer of $$u$$. Our results extend work of the second author done for simple, adjoint $$G$$ over the complex numbers.

When $$G$$ is simple and adjoint, the previous work of the second author makes our description combinatorial and explicit; moreover, it turns out that knowledge of the conjugacy classes suffices to determine the group structure of $$A(u)$$. Thus we obtain the result, previously known through case-checking, that the structure of the component group $$A(u)$$ is independent of good characteristic.

## Abelian unipotent subgroups of reductive groups

### Abstract:

Let $$G$$ be a connected reductive group defined over an algebraically closed field $$k$$ of characteristic $$p > 0$$. The purpose of this paper is two-fold. First, when $$p$$ is a good prime, we give a new proof of the order formula'' of D. Testerman for unipotent elements in $$G$$; moreover, we show that the same formula determines the $$p$$-nilpotence degree of the corresponding nilpotent elements in the Lie algebra $$\mathfrak{g}$$ of $$G$$.

Second, if $$G$$ is semisimple and $$p$$ is sufficiently large, we show that $$G$$ always has a faithful representation $$(\rho,V)$$ with the property that the exponential of $$d\rho(X)$$ lies in $$\rho(G)$$ for each $$p$$-nilpotent $$X \in \mathfrak{g}$$. This property permits a simplification of the description given by Suslin, Friedlander, and Bendel of the (even) cohomology ring for the Frobenius kernels $$G_d$$, $$d \ge 2$$. The previous authors already observed that the natural representation of a classical group has the above property (with no restriction on $$p$$). Our methods apply to any Chevalley group and hence give the result also for quasisimple groups with exceptional type'' root systems. The methods give explicit sufficient conditions on $$p$$; for an adjoint semisimple $$G$$ with Coxeter number $$h$$, the condition $$p > 2h -2$$ is always good enough.

## The second cohomology of small irreducible modules for simple algebraic groups

### Abstract:

Let $$G$$ be a connected, simply connected, quasisimple algebraic group over an algebraically closed field of characteristic $$p>0$$, and let $$V$$ be a rational $$G$$-module such that $$\dim V \le p$$. According to a result of Jantzen, $$V$$ is completely reducible, and $$H^1(G,V)=0$$. In this paper we show that $$H^2(G,V) = 0$$ unless some composition factor of $$V$$ is a non-trivial Frobenius twist of the adjoint representation of $$G$$.

## Semisimplicity in positive characteristic

### URL: pdf

Page published on 2014-04-06 at 20:13 (EDT).