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Representationtheoretic support spaces for finite group schemes
 American Journal of Math
"... Abstract. We introduce the space P (G) of abelian ppoints of a finite group scheme over an algebraically closed field of characteristic p> 0. We construct a homeomorphism ΨG: P (G) → Proj G  from P (G) to the projectivization of the cohomology variety for any finite group G. For an elementary ..."
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Abstract. We introduce the space P (G) of abelian ppoints of a finite group scheme over an algebraically closed field of characteristic p> 0. We construct a homeomorphism ΨG: P (G) → Proj G  from P (G) to the projectivization of the cohomology variety for any finite group G. For an elementary abelian pgroup (respectively, an infinitesimal group scheme), P (G) can be identified with the projectivization of the variety of cyclic shifted subgroups (resp., variety of 1parameter subgroups). For a finite dimensional Gmodule M, ΨG restricts to a homeomorphism P (G)M → Proj GM, thereby giving a representationtheoretic interpretation of the cohomological support variety. Even though the cohomology groups H i (G, k) of a finite group are typically difficult to compute, D. Quillen in his seminal papers [17] gave a general description of the maximal ideal spectrum G  of the commutative kalgebra H ev (G, k) in terms
Πsupports for modules for finite group schemes
"... Abstract. We introduce the space Π(G) of equivalence classes of πpoints of a finite group scheme G, and associate a subspace Π(G)M to any Gmodule M. Our results extend to arbitrary finite group schemes G over arbitrary fields k of positive characteristic and to arbitrarily large Gmodules the basi ..."
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Cited by 26 (8 self)
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Abstract. We introduce the space Π(G) of equivalence classes of πpoints of a finite group scheme G, and associate a subspace Π(G)M to any Gmodule M. Our results extend to arbitrary finite group schemes G over arbitrary fields k of positive characteristic and to arbitrarily large Gmodules the basic results about “cohomological support varieties ” and their interpretation in terms of representation theory. In particular, we prove that the projectivity of any (possibly infinite dimensional) Gmodule can be detected by its restriction along πpoints of G. Unlike the cohomological support variety of a Gmodule M, the invariant M ↦ → Π(G)M satisfies good properties for all modules, thereby enabling us to determine the thick, tensorideal subcategories of the stable
Abelian unipotent subgroups of reductive groups
 J. Pure Appl. Algebra
"... 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 twofold. 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 ..."
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Cited by 23 (4 self)
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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 twofold. 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 pnilpotence degree of the corresponding nilpotent elements in the Lie algebra g of G. Second, if X is a pnilpotent element (an element of pnilpotence degree 1) in g, we show that G always has a representation V for which the exponential homomorphism Ga → GL(V) determined by X factors through the action of G. This property permits a simplification of the description given by Suslin, Friedlander, and Bendel of the (even) cohomology ring for the higher Frobenius kernels Gd, d ≥ 2. 1.
Support varieties for modules over Chevalley groups and classical Lie algebras
, 2005
"... Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p> 0, G1 be the first Frobenius kernel, and G(Fp) be the corresponding finite Chevalley group. Let M be a rational Gmodule. In this paper we relate the support variety of M over the first Froben ..."
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Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p> 0, G1 be the first Frobenius kernel, and G(Fp) be the corresponding finite Chevalley group. Let M be a rational Gmodule. 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(Fp). This provides an answer to a question of Parshall. Applications of our new techniques are presented which allow us to extend results of AlperinMason and JaniszczakJantzen, and to calculate the dimensions of support varieties for finite Chevalley groups.
Cohomology of quantum groups via the geometry of the nullcone
, 2007
"... Let ζ be a complex ℓth root of unity for an odd integer ℓ> 1. For any complex simple Lie algebra g, let uζ = uζ(g) be the associated “small ” quantum enveloping algebra. This algebra is a finite dimensional Hopf algebra which can be realized as a subalgebra of the Lusztig (divided power) quantu ..."
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Cited by 18 (4 self)
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Let ζ be a complex ℓth root of unity for an odd integer ℓ> 1. For any complex simple Lie algebra g, let uζ = uζ(g) be the associated “small ” quantum enveloping algebra. This algebra is a finite dimensional Hopf algebra which can be realized as a subalgebra of the Lusztig (divided power) quantum enveloping algebra Uζ and as a quotient algebra of the De Concini–Kac quantum enveloping algebra Uζ. It plays an important role in the representation theories of both Uζ and Uζ in a way analogous to that played by the restricted enveloping algebra u of a reductive group G in positive characteristic p with respect to its distribution and enveloping algebras. In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when l (resp., p) is smaller than the Coxeter number h of the underlying root system. For example, Lusztig’s Conjecture concerning the irreducible modules can only be formulated when p ≥ h. The main result in this paper provides a surprisingly uniform answer for the cohomology algebra H • (uζ, C) of the small quantum group. When ℓ> h, this cohomology algebra has been calculated by Ginzburg and Kumar [GK]. Our result requires powerful tools from complex geometry and a detailed knowledge of the geometry of the nullcone of g. In this way, the methods point out difficulties present in obtaining similar results for the restricted enveloping algebra u in small characteristics, though they do provide some clarification of known results there also. Finally, we establish that if M is a finite dimensional uζmodule, then H • (uζ, M) is a finitely generated H • (uζ, C)module, and we obtain new results on the theory of support varieties for uζ.
On Cocommutative Hopf Algebras Of Finite Representation Type
 Universitat Bielefeld, SFB Preprint
"... Let G be a finite algebraic group, defined over an algebraically closed field k of characteristic p ? 0. Such a group decomposes into a semidirect product G = G 0 \Theta G red with a constant group G red and a normal infinitesimal subgroup G 0 . If the principal block B 0 (G) of the group alge ..."
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Let G be a finite algebraic group, defined over an algebraically closed field k of characteristic p ? 0. Such a group decomposes into a semidirect product G = G 0 \Theta G red with a constant group G red and a normal infinitesimal subgroup G 0 . If the principal block B 0 (G) of the group algebra H(G) has finite representation type, then both constituents have the same property, with at least one of them being semisimple. We determine the structure of the infinitesimal constituent G 0 up to the classification of Vuniserial groups. 0. Introduction This paper is concerned with the representation theory of finitedimensional cocommutative Hopf algebras over algebraically closed fields of positive characteristic. As is wellknown, such an algebra can be viewed as the group algebra of a finite algebraic kgroup G. Special cases are the Hopf algebras associated to constant groups, i.e., the modular group algebras, as well as those of the infinitesimal groups of height 1, that is, ...
VARIETIES FOR MODULES OF QUANTUM ELEMENTARY ABELIAN GROUPS
"... Abstract. We define a rank variety for a module of a noncocommutative Hopf algebra A = Λ ⋊ G where Λ = k[X1,..., Xm]/(X ℓ 1,..., X ℓ m), G = (Z/ℓZ) m, and char k does not divide ℓ, in terms of certain subalgebras of A playing the role of “cyclic shifted subgroups”. We show that the rank variety of a ..."
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Abstract. We define a rank variety for a module of a noncocommutative Hopf algebra A = Λ ⋊ G where Λ = k[X1,..., Xm]/(X ℓ 1,..., X ℓ m), G = (Z/ℓZ) m, and char k does not divide ℓ, in terms of certain subalgebras of A playing the role of “cyclic shifted subgroups”. We show that the rank variety of a finitely generated module M is homeomorphic to the support variety of M defined in terms of the action of the cohomology algebra of A. As an application we derive a theory of rank varieties for the algebra Λ. When ℓ = 2, rank varieties for Λmodules were constructed by Erdmann and Holloway using the representation theory of the Clifford algebra. We show that the rank varieties we obtain for Λmodules coincide with those of Erdmann and Holloway. 1.
The restricted nullcone
 Cont. Math
"... Abstract. Let (g, [p]) be a restricted Lie algebra over an algebraically closed field of characteristic p> 0. The restricted nullcone of g, denoted by N1(g), consists of those x ∈ g such that x [p] = 0. In this paper the authors provide a concrete description of this variety (via closures of Ric ..."
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Cited by 14 (5 self)
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Abstract. Let (g, [p]) be a restricted Lie algebra over an algebraically closed field of characteristic p> 0. The restricted nullcone of g, denoted by N1(g), consists of those x ∈ g such that x [p] = 0. In this paper the authors provide a concrete description of this variety (via closures of Richardson orbits) when g is the Lie algebra of a reductive group G and p a good prime. Various applications to representation theory and cohomology theory are provided. 1.
Generic and Maximal Jordan types
"... Abstract. For a finite group scheme G over a field k of characteristic p> 0, we associate new invariants to a finite dimensional kGmodule M. Namely, for each generic point of the projectivized cohomological variety Proj H • (G, k) we exhibit a “generic Jordan type ” of M. In the very special cas ..."
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Abstract. For a finite group scheme G over a field k of characteristic p> 0, we associate new invariants to a finite dimensional kGmodule M. Namely, for each generic point of the projectivized cohomological variety Proj H • (G, k) we exhibit a “generic Jordan type ” of M. In the very special case in which G = E is an elementary abelian pgroup, our construction specializes to the nontrivial observation that the Jordan type obtained by restricting M via a generic cyclic shifted subgroup does not depend upon a choice of generators for E. Furthermore, we construct the nonmaximal support variety Γ(G)M, a closed subset of Proj H • (G, k) which is proper even when the dimension of M is not divisible by p. Elementary abelian psubgroups of a finite group G capture significant aspects of the cohomology and representation theory of G. For example, if k is a field of characteristic p> 0, then a theorem of D. Quillen [18] asserts that the Krull dimension of the cohomology algebra H • (G, k) is equal to the maximum of the ranks of
Constructions for infinitesimal group schemes
, 2010
"... Let G be an infinitesimal group scheme over a field k of characteristic p> 0. We introduce the global pnilpotent operator ΘG: k[G] → k[V (G)], where V (G) is the scheme which represents 1parameter subgroups of G. This operator ΘG applied to M encodes the local Jordan type of M, and leads to c ..."
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Cited by 12 (8 self)
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Let G be an infinitesimal group scheme over a field k of characteristic p> 0. We introduce the global pnilpotent operator ΘG: k[G] → k[V (G)], where V (G) is the scheme which represents 1parameter subgroups of G. This operator ΘG applied to M encodes the local Jordan type of M, and leads to computational insights into the representation theory of G. For certain kGmodules M (including those of constant Jordan type), we employ ΘG to associate various algebraic vector bundles on P(G), the projectivization of V (G). These vector bundles not only distinguish certain representations with the same local Jordan type, but also provide a method of constructing algebraic vector bundles on P(G).