Abstract Kazhdan-Lusztig theories

Cline, Edward; Parshall, Brian; Scott, Leonard L.

doi:10.2748/tmj/1178225846

Cited by 75 publications

(142 citation statements)

References 24 publications

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“…Theorem 5.4(b). This result generalizes to all regular dominant weights a theorem of [13] for regular dominant weights inside the Jantzen region, where the modules ∆ red (λ), ∇ red (µ) are irreducible. In one substantive case when λ = pµ for a dominant weight µ, we are even able to establish Theorem 5.4 without assuming the Lusztig conjecture.…”

mentioning

confidence: 86%

“…For (a), see [13,Lemma 2.2], and for part (b), see [13,Lemma 3.2]. However, (b) holds by (a) if M has a filtration by G-modules with ∆-sections (i.e., sections of the form ∆(λ), λ ∈ X + ).…”

Section: (G)t U(g)b-mod)mentioning

confidence: 99%

“…Our setup follows that introduced in [13]. The tracking of degree information in the derived category of rational G-modules provides the main advantage of the enriched Grothendieck groups over the ordinary Grothendieck group of G-mod.…”

Section: Grothendieck Groupsmentioning

confidence: 99%

“…(1) ⇐⇒ (2) follows as in the proof of (1) ⇐⇒ (3) in Lemma 2.1. Next, (2) ⇐⇒ (3), using Lemma 2.1(b) and [13 (6), see [13,Thm. 5.3].…”

Section: The Category E R Is Obtained By Replacing the ∆(λ) Throughoumentioning

confidence: 99%

“…Specifically, let X + reg ⊂ X + be the set of regular dominant weights. Then a well-known theorem [13,Thm. 5.3] states that, given a saturated subset Γ ⊂ X + reg (relative to the ↑ poset structure), L(λ) ∈ E L for all λ ∈ Γ if and only if each L(λ), λ ∈ Γ, has formal character given by Lusztig's famous character formula.…”

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confidence: 99%

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Reduced standard modules and cohomology

Cline

Parshall

Scott

2009

Trans. Amer. Math. Soc.

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Abstract. First cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts, including Wiles's famous paper (1995). Internal to group theory, 1-cohomology plays a role in the general theory of maximal subgroups of finite groups, as developed by Aschbacher and Scott (1985).One can pass to the case where the group acts faithfully and the underlying module is absolutely irreducible. In this case, R. Guralnick (1986) conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups. This paper provides the first general positive results on this conjecture, proving that the generic 1-cohomologyScott, and van der Kallen) (1977) of a finite group G(q) of Lie type, with absolutely irreducible coefficients L (in the defining characteristic of G), is bounded by a constant depending only on the root system. In all cases, we are able to improve this result to a bound on H 1 (G(q), L) itself, still depending only on the root system. The generic H 1 result, and related results for Ext 1 , emerge here as a consequence of a general study, of interest in its own right, of the homological properties of certain rational modules ∆ red (λ), ∇ red (λ), indexed by dominant weights λ, for a reductive group G. The modules ∆ red (λ) and ∇ red (λ) arise naturally from irreducible representations of the quantum enveloping algebra U ζ (of the same type as G) at a pth root of unity, where p > 0 is the characteristic of the defining field for G. Finally, we apply our Ext 1 -bounds, results of Bendel, Nakano, and Pillen (2006), as well as results of

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mentioning

confidence: 86%

Section: (G)t U(g)b-mod)mentioning

confidence: 99%

Section: Grothendieck Groupsmentioning

confidence: 99%

“…(1) ⇐⇒ (2) follows as in the proof of (1) ⇐⇒ (3) in Lemma 2.1. Next, (2) ⇐⇒ (3), using Lemma 2.1(b) and [13 (6), see [13,Thm. 5.3].…”

Section: The Category E R Is Obtained By Replacing the ∆(λ) Throughoumentioning

confidence: 99%

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confidence: 99%

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Reduced standard modules and cohomology

Cline

Parshall

Scott

2009

Trans. Amer. Math. Soc.

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Strong Exact Borel Subalgebras of Quasi-hereditary Algebras and Abstract Kazhdan–Lusztig Theory

König

1999

Advances in Mathematics

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Strong exact Borel subalgebras and strong 2-subalgebras are shown to exist for quasi-hereditary algebras which possess exact Borel subalgebras and 2-subalgebras. This implies that the algebras associated with blocks of category O have strong exact Borel subalgebras and strong 2-subalgebras. The structure of these subalgebras is shown to be closely related to abstract Kazhdan Lusztig theory. The main technical tool in this paper is a construction which has an exact Borel subalgebras (of a given quasi-hereditary algebra) as input and a strong exact Borel subalgebra as output. From this result, Morita invariance of the existence of exact Borel subalgebras is derived.

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Highest Weight Modules for Algebraic Groups Arising from Quantum Groups

Lin

1998

Journal of Algebra

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Abstract Kazhdan-Lusztig theories

Cited by 75 publications

References 24 publications

Reduced standard modules and cohomology

Reduced standard modules and cohomology

Strong Exact Borel Subalgebras of Quasi-hereditary Algebras and Abstract Kazhdan–Lusztig Theory

Highest Weight Modules for Algebraic Groups Arising from Quantum Groups

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