# Details

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## Table of Contents

- Central subalgebras of the centralizer of a nilpotent element
- Linearity for actions on vector groups
- Some good-filtration subgroups of simple algebraic groups
- Levi factors of the special fiber of a parahoric group scheme and tame ramification
- On the descent of Levi factors
- Levi decompositions of a linear algebraic group
- Nilpotent subalgebras of semisimple Lie algebras
- Nilpotent centralizers and Springer's isomorphisms
- The centralizer of a nilpotent section
- Completely reducible SL(2) homomorphisms
- Completely reducible Lie subalgebras
- On the centralizer of the sum of commuting nilpotent elements
- Optimal SL(2) homomorphisms
- Nilpotent orbits over ground fields of good characteristic
- Sub-principal homomorphisms in good characteristic
- Adjoint Jordan blocks
- Faithful representations of SL(2) over truncated Witt vectors
- Component groups of unipotent centralizers in good characteristic
- Abelian unipotent subgroups of reductive groups
- The second cohomology of small irreducible modules for simple algebraic groups
- Filtrations and positive characteristic Howe duality
- Semisimplicity of exterior powers of semisimple representations of groups
- Semisimple modules for finite groups of Lie type
- Dimensional criteria for semisimplicity of representations
- Semisimplicity in positive characteristic

## Central subalgebras of the centralizer of a nilpotent element

### Authors: George J. McNinch and Donna Testerman

### Publication status: pre-print (submitted for publication)

### abstract:

Let \(G\) be a connected, semisimple algebraic group over a field \(k\)
whose characteristic is **very good** for \(G\). In a canonical manner,
one associates to a nilpotent element \(X \in\) Lie\((G)\)
a parabolic subgroup \(P\) – in characteristic zero, \(P\) may be
described using an \(\mathfrak{sl}_2\)-triple containing \(X\); in
general, \(P\) is the ``instability parabolic'' for \(X\) as in
geometric invariant theory.

In this setting, we are concerned with the center \(Z(C)\) of the
centralizer \(C\) of \(X\) in \(G\). Choose a Levi factor \(L\) of \(P\),
and write \(d\) for the dimension of the center \(Z(L)\). Finally,
assume that the nilpotent element \(X\) is **even**. In this case, we
can **deform** Lie\((L)\) to Lie\((C)\), and our deformation produces a
\(d\)-dimensional subalgebra of Lie\((Z(C))\). Since \(Z(C)\) is a
smooth group scheme, it follows that dim \(Z(C) \ge d =\) dim \(Z(L)\).

In fact, Lawther and Testerman have proved that dim \(Z(C) =\) dim \(Z(L)\). Despite only yielding a partial result, the interest in the method found in the present work is that it avoids the extensive case checking carried out by Lawther-Testerman in the memoir [LT 11].

## Linearity for actions on vector groups

### Author(s): George J. McNinch

### Publication: Journal of Algebra (2014), 397, p. 666-688.

### URL: version of Sept 2013

### 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.

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## Some good-filtration subgroups of simple algebraic groups

### Authors : Charles Hague and George J. McNinch

### Publication: **Journal of Pure and Applied Algebra** 217 (2013), no. 12, 2400\u20132413

### (DOI 10.1016/j.jpaa.2013.04.005)

### 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.

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

### Author(s): George J. McNinch

### Publication: **Algebras and Representation Theory** vol. 17 (2014), no. 2, p. 469-479.

### DOI: 10.1007/s10468-013-9404-4

### URL: version of Jan 2013 / final publication available at springerlink

### 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 \(k\).

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## On the descent of Levi factors

### Author(s): George J. McNinch

### Publication: Archiv der Mathametik (2013), vol. 100, pp. 7-24.

### 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.

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## Levi decompositions of a linear algebraic group

### Author: George J. McNinch

### Publication: Transformation Groups (Morozov centennial issue) 15 (2010), 937–964

### Dedicatory:

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

### URL: arXiv.math/1007.2777

### 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.

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## Nilpotent subalgebras of semisimple Lie algebras

### Author(s): Paul Levy, George J. McNinch, and Donna M. Testerman

### URL: manuscript

### Publication: C. R. Acad. Sci. Paris, Ser. I 347 (2009) 477–482

### 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 \(\mathfrak{g}\).

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## Nilpotent centralizers and Springer's isomorphisms

### Author(s): George J. McNinch and Donna M. Testerman

### Publication: J Pure and Appl Alg 213 (2009) 1346 – 1363

### URL: arXiv:0805.2574

### 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.

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## The centralizer of a nilpotent section

### Author(s): George J. McNinch

### Publication: Nagoya Math. J. 190 (2008), 129–181.

### 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 \(t \in \text{Spec}(\mathcal{A})\).

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## Completely reducible SL(2) homomorphisms

### Publication: Transactions AMS 2007 (359) 4489 – 4510

### Author(s): George J. McNinch and Donna M. Testerman

### 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.

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## Completely reducible Lie subalgebras

### Author: George J. McNinch

### Publication: Transf. Groups 2007 (12) 127–135

### 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.

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

### Author: George J. McNinch

### Publication: J. Pure Appl Alg (Friedlander birthday volume) 2006 (206) 123–140

### Dedicatory: To Eric Friedlander, on his 60th birthday.

### 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.

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## Optimal SL(2) homomorphisms

### Author: George J. McNinch

### Publication: Comment. Math. Helv., 2005 (80), 391-426

### 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 \(K\).

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## Nilpotent orbits over ground fields of good characteristic

### Author: George J. McNinch

### Publication: Mathematische Annalen, 2004 (329), 49-85

### URL: arXiv:math/0209151

### 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).

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## Sub-principal homomorphisms in good characteristic

### Author: George J. McNinch

### URL: arXiv:math/0108.5140

### Publication: Mathematische Zeitschrift, 2003 (244), 433-455.

### 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).

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## Adjoint Jordan blocks

### Author: George J. McNinch

### 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 \(\mathcal{F}\).

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

### Author: George J. McNinch

### Publication: Journal of Algebra, 2003 (265), 606-618.

### 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 \(p+3\).

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## Component groups of unipotent centralizers in good characteristic

### Authors: George J. McNinch and Eric Sommers

### Publication: J. Algebra (Steinberg 80th b'day vol), 2003 (260), 323-337

### Dedicatory: To Robert Steinberg, on his 80th birthday.

### 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.

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## Abelian unipotent subgroups of reductive groups

### Author: George J. McNinch

### Publication: J. Pure Appl Alg, 2002 (167), 269-300.

### 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.

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

### Author: George J. McNinch

### Publication: Pacific Journal of Math, 2002 (204), 459-472.

### 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 \(G\).

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