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269
Koszul duality patterns in representation theory
 Jour. Amer. Math. Soc
, 1996
"... 2. Koszul rings 479 3. Parabolicsingular duality and Koszul duality 496 ..."
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Cited by 212 (14 self)
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2. Koszul rings 479 3. Parabolicsingular duality and Koszul duality 496
Branching rules for modular representations of symmetric groups
 J. London Math. Soc
, 1995
"... Let K be a field of characteristic p> 0, Era the symmetric group on n letters, Sn_1 < Lra the subgroup consisting of the permutations of the first « — 1 letters, and D k the irreducible ATnmodule corresponding to a (/^regular) partition X of n. In [9] we described the socle of the restriction D x ..."
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Cited by 64 (14 self)
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Let K be a field of characteristic p> 0, Era the symmetric group on n letters, Sn_1 < Lra the subgroup consisting of the permutations of the first « — 1 letters, and D k the irreducible ATnmodule corresponding to a (/^regular) partition X of n. In [9] we described the socle of the restriction D x [T and obtained a number of other results
COHOMOLOGY OF FINITE GROUP SCHEMES OVER A FIELD
"... A finite group scheme G over a field k is equivalent to its coordinate algebra, a finite dimensional commutative Hopf algebra k[G] over k. In many contexts, it is natural to consider the rational (or Hochschild) cohomology of G with coefficients in a k[G]comodule M. This is naturally isomorphic to ..."
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Cited by 53 (10 self)
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A finite group scheme G over a field k is equivalent to its coordinate algebra, a finite dimensional commutative Hopf algebra k[G] over k. In many contexts, it is natural to consider the rational (or Hochschild) cohomology of G with coefficients in a k[G]comodule M. This is naturally isomorphic to the cohomology of the dual cocommutative Hopf algebra k[G] # with coefficients in the k[G] #module M. In this latter formulation, we encounter familiar examples of the cohomology of group algebras kπ of a finite groups π and of restricted enveloping algebras V (g) of finite dimensional restricted Lie algebras g. In recent years, the representation theory of the algebras kπ and V (g) has been studied by considering the spectrum of the cohomology algebra with coefficients in the ground field k and the support in this spectrum of the cohomology with coefficients in various modules. This approach relies on the fact that H ∗ (π, k) and H ∗ (V (g), k) are finitely generated kalgebras as proved in [G], [E], [V], [FP2]. Rational representations of algebraic groups in positive characteristic correspond to representations of a hierarchy of finite group schemes. In order to begin the process of introducing geometric methods to the study of these other group schemes, finite generation must be proved. Such a proof has proved surprisingly elusive (though partial results can be found in [FP2]). The main theorem of this paper is the following: Theorem 1.1. Let G be a finite group scheme and M a finite dimensional rational Gmodule. Then H ∗ (G, k) is a finitely generated kalgebra and H ∗ (G, M) is a finite H ∗ (G, k)module. Work in progress by C. Bendel (and the authors) reveals that Theorem 1.1 and its proof will provide interesting theorems of a geometric nature concerning the representation theory of finite group schemes. In a sense that is made explicit in section 1, our proof of finite generation is quite constructive. We embed G in some general linear group GLn and establish the existence of universal extension classes for GLn of specified degrees. In a direct manner, these classes provide the generators of H ∗ (G, k). In order to construct these universal extension classes, we follow closely the approach of V. Franjou, J. Lannes, and L. Schwartz [FLS]. This entails the study of
KazhdanLusztig polynomials and character formulae for the Lie superalgebra gl(mn
 J. AMS
"... The problem of computing the characters of the finite dimensional irreducible representations of the Lie superalgebra gl(mn) over C was solved a few years ago by V. Serganova [19]. In this article, we present an entirely different approach. We also formulate a precise ..."
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Cited by 43 (5 self)
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The problem of computing the characters of the finite dimensional irreducible representations of the Lie superalgebra gl(mn) over C was solved a few years ago by V. Serganova [19]. In this article, we present an entirely different approach. We also formulate a precise
Abstract KazhdanLusztig theories
 TÔHOKU MATH. J
, 1993
"... In this paper, we prove two main results. The first establishes that Lusztig's conjecture for the characters of the irreducible representations of a semisimple algebraic group in positive characteristic is equivalent to a simple assertion that certain pairs of irreducible modules have nonsplit ext ..."
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Cited by 34 (17 self)
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In this paper, we prove two main results. The first establishes that Lusztig's conjecture for the characters of the irreducible representations of a semisimple algebraic group in positive characteristic is equivalent to a simple assertion that certain pairs of irreducible modules have nonsplit extensions. The pairs of irreducible modules in question are those with regular dominant weights which are mirror images of each other in adjacent alcoves (in the Jantzen region). Secondly, we establish that the validity of the Lusztig conjecture yields a complete calculation of all Yoneda Ext groups between irreducible modules having regular dominant weights in the Jantzen region. These results
Infinitesimal 1parameter subgroups and cohomology
 J. Amer. Math. Soc
, 1997
"... This is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p> 0. Whereas [SFB] is concerned with detection of cohomology classes, the present paper introduces the graded algebra k[Vr(G)] of functions on ..."
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Cited by 33 (15 self)
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This is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p> 0. Whereas [SFB] is concerned with detection of cohomology classes, the present paper introduces the graded algebra k[Vr(G)] of functions on the scheme of infinitesimal 1parameter subgroups of height ≤ r on an affine group scheme G and demonstrates that this algebra is essentially a retract of H ev (G, k) provided that G is an infinitesimal group scheme of height ≤ r. This work is a continuation of [FS] in which the existence of certain universal extension classes was established, thereby enabling the proof of finite generation of H ∗ (G, k) for any finite group scheme G over k. The role of the scheme of infinitesimal 1parameter subgroups of G was foreshadowed in [FP] where H ∗ (G (1), k) was shown to be isomorphic to the coordinate algebra of the scheme of pnilpotent elements of g = Lie(G) for G a smooth reductive group, G (1) the first Frobenius kernel of G, and p = char(k) sufficiently large. Indeed, pnilpotent elements of g correspond precisely to infinitesimal 1parameter subgroups of G (1). Much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1parameter subgroups. In section 1, we construct the affine scheme Vr(G) of homomorphisms from G a(r) to G which we call the scheme of infinitesimal 1parameter subgroups of height ≤ r in G. In the special case r = 1, this is the scheme of pnilpotent elements of the prestricted Lie algebra Lie(G); for various classical groups G, Vr(G) is the scheme of rtuples of pnilpotent, pairwise commuting elements of Lie(G). More generally, an embedding G ⊂ GLn determines a closed embedding of Vr(G) into the scheme of rtuples of pnilpotent, pairwise commuting elements of gln = Lie(GLn). The relationship between k[Vr(G)], the coordinate algebra of Vr(G), and H ∗ (G, k) is introduced in Theorem 1.14: a natural homomorphism of graded kalgebras ψ: H ev (G, k) → k[Vr(G)] is constructed, a map which we show in [SFB] to induce a bijective map on associated schemes. The universal classes er ∈ H2pr−1 characteristic classes er(G, V) for any affine group scheme G provided with a finite
Integral Canonical Models for Shimura Varieties of Preabelian Type
, 2003
"... We prove the existence of integral canonical models of Shimura varieties of preabelian type with respect to primes of characteristic at least 5. ..."
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Cited by 29 (19 self)
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We prove the existence of integral canonical models of Shimura varieties of preabelian type with respect to primes of characteristic at least 5.
Localization of modules for a semisimple Lie algebra in prime characteristic
, 2006
"... Abstract. We observe that on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of characteristic p> 2h − 2 (where h is the Coxeter number), with a given (generalized) central character are the same as the coherent sheaves on the correspo ..."
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Cited by 27 (8 self)
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Abstract. We observe that on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of characteristic p> 2h − 2 (where h is the Coxeter number), with a given (generalized) central character are the same as the coherent sheaves on the corresponding Springer fiber. The first step is to observe that the derived functor of global sections provides an equivalence between the derived category of Dmodules (with no divided powers) on