The restricted nullcone

Carlson, Jon; Lin, Zongzhu; Nakano, Daniel K.; Parshall, Brian

doi:10.1090/conm/325/05664

Cited by 22 publications

(31 citation statements)

References 7 publications

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“…Some calculations similar to certain ones here appear in a recent paper [2] of Carlson, Lin, Nakano and Parshall, concerning elements in a restricted Lie algebra in characteristic p which satisfy x…”

supporting

confidence: 60%

“…As C G ad (ĝ) = h∈ker φ φ(G g,h ) and ker φ is finite, Note that an inequality is the best possible result here, as may be seen by considering groups of type A 1 in odd characteristic with r = 2. We have d A1,2 = 1: if G is the adjoint group P GL 2 (K), the involution which is the image of diag(1, −1) is a regular semisimple element, giving codim G [2] = 1 as required by Theorem 3.8. However, if G is the simply-connected group SL 2 (K), the only involution in G is the central element diag (−1, −1), so that codim G [2] = 3.…”

Section: Lemma 31 With the Notation Established Codim Gmentioning

confidence: 99%

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Elements of specified order in simple algebraic groups

Lawther

2004

Trans. Amer. Math. Soc.

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Abstract. In this paper we let G be a simple algebraic group and r be a natural number, and consider the codimension in G of the variety of elements g ∈ G satisfying g r = 1. We shall obtain a lower bound for this codimension which is independent of characteristic, and show that it is attained if G is of adjoint type.Let G be a simple algebraic group over an algebraically closed field K of characteristic p; let Φ be the root system of G, and take r ∈ N. Defineand(where o(g) denotes the order of g); then G [r] and G (r) are both subvarieties of G, and G [r] is the disjoint union of those G (r ) with r dividing r. Our attention here is on the codimension in G of these varieties (if they are non-empty; clearly G [r] = ∅, but the example of G = SL 2 (K), p = 2 and r = 4 shows that G (r) may be empty). It is immediate that if G (r) = ∅ we have codim G (r) ≥ codim G [r] . Our main result may be stated as follows. Statements equivalent to the inequality codim G (r) ≥ |Φ|/r are already known in certain cases. If r = 2 and p = 2, the equivalent statement that, if g ∈ G is an involution, then dim C G (g) ≥ dim(G/B) (where B is a Borel subgroup), is well known; the stronger statement that C G (g) is then spherical, i.e., it has finitely many orbits on the flag variety G/B, was proved by Matsuki in [13] for K = C, and by Springer in [19] for p odd-recently Seitz gave an alternative proof of Springer's result in [16]. In the case r = 3 and p = 3, the result follows from work of Liebeck and Shalev in [10]; this case and that with r = 2 are used in work of Liebeck, Seitz and the author concerning dimensions of fixed point spaces in [9]. More generally, for r an odd prime a result in this direction appears in further work of Liebeck and Shalev in [11], while the results proved here find application in [12] to homomorphisms from Fuchsian groups to finite simple groups.

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confidence: 60%

Section: Lemma 31 With the Notation Established Codim Gmentioning

confidence: 99%

Elements of specified order in simple algebraic groups

Lawther

2004

Trans. Amer. Math. Soc.

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“…If p ≥ h, then (a) is true because all roots α have height at most h − 1. Assume that p < h. For the exceptional groups, the sets J are determined in [CLNP,4.4]. For the reader's convenience, we list them here together with α 0 .…”

Section: 2 §41]mentioning

confidence: 99%

Support varieties for modules over Chevalley groups and classical Lie algebras

Carlson

Lin

Nakano

2007

Trans. Amer. Math. Soc.

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Abstract. Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p > 0, G 1 be the first Frobenius kernel, and G(F p ) be the corresponding finite Chevalley group. Let M be a rational G-module. In this paper we relate the support variety of M over the first Frobenius kernel with the support variety of M over the group algebra kG(F p ). This provides an answer to a question of Parshall. Applications of our new techniques are presented, which allow us to extend results of Alperin-Mason and Janiszczak-Jantzen, and to calculate the dimensions of support varieties for finite Chevalley groups.

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“…[17], [7]). Using this result together with the hypothesis of normality and an investigation of centralizers in G of elements of U J and u J , J. Carlson, Z. Lin, and D. Nakano extend (8) to a G-equivariant isomorphism log : U 1 ∼ → N 1 .…”

Section: Log Mapsmentioning

confidence: 99%

Weil restriction and support varieties

Friedlander¹

2010

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. This paper extends the comparison results for cohomological support varieties for Chevalley groups of the form G( Consider a simple affine algebraic group G over k defined and split over F p with p-restricted Lie algebra Lie(G) = g. In [15], Z. Lin and D. Nakano showed for a rational G-module M that the rate of growth of a projective resolution for M as a kG(F p )-module is dominated by one-half the rate of growth of a projective resolution for M viewed as module for the restricted enveloping algebra for g. This result is illuminated and clarified by recent work of J. Carlson, Z. Lin, and D. Nakano who gave a direct relation between the cohomological varieties of G(F p ) and g [6]. For a restricted class of rational G-modules M , a comparison was also verified between the support variety of M for G and the support variety of M for g.In this paper, we extend the results of [15] and [6], so that an appropriate generalization of the formulation of their results now applies to G(F q ) where q is a prime power (with the same constraint on the prime, depending upon G, as required by [6]).

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The restricted nullcone

Cited by 22 publications

References 7 publications

Elements of specified order in simple algebraic groups

Elements of specified order in simple algebraic groups

Support varieties for modules over Chevalley groups and classical Lie algebras

Weil restriction and support varieties

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