Differential operators on quantized flag manifolds at roots of unity III

AbstractWe formulate a Beilinson-Bernstein-type derived equivalence for a quantized enveloping algebra at a root of 1 as a conjecture. It says that there exists a derived equivalence between the category of modules over a quantized enveloping algebra at a root of 1 with fixed regular Harish-Chandra central character and the category of certain twistedD-modules on the corresponding quantized flag manifold. We show that the proof is reduced to a statement about the (derived) global sections of the ring of differential operators on the quantized flag manifold. We also give a reformulation of the conjecture in terms of the (derived) induction functor.

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Differential operators on quantized flag manifolds at roots of unity

Advances in Mathematics ◽

10.1016/j.aim.2012.04.018 ◽

2012 ◽

Vol 230 (4-6) ◽

pp. 2235-2294 ◽

Cited By ~ 5

Author(s):

Toshiyuki Tanisaki

Keyword(s):

Differential Operators ◽

Flag Manifolds ◽

Roots Of Unity

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Differential operators on quantized flag manifolds at roots of unity, II

Nagoya Mathematical Journal ◽

10.1215/00277630-2402198 ◽

2014 ◽

Vol 214 ◽

pp. 1-52 ◽

Cited By ~ 1

Author(s):

Toshiyuki Tanisaki

Keyword(s):

Differential Operators ◽

Flag Manifold ◽

Flag Manifolds ◽

Roots Of Unity ◽

Central Character ◽

Derived Equivalence ◽

Bernstein Type ◽

Enveloping Algebra ◽

Quantized Enveloping Algebra ◽

Category Of Modules

AbstractWe formulate a Beilinson-Bernstein-type derived equivalence for a quantized enveloping algebra at a root of 1 as a conjecture. It says that there exists a derived equivalence between the category of modules over a quantized enveloping algebra at a root of 1 with fixed regular Harish-Chandra central character and the category of certain twisted D-modules on the corresponding quantized flag manifold. We show that the proof is reduced to a statement about the (derived) global sections of the ring of differential operators on the quantized flag manifold. We also give a reformulation of the conjecture in terms of the (derived) induction functor.

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Singular Perturbations of Differential Operators

10.1017/cbo9780511758904 ◽

2000 ◽

Cited By ~ 203

Author(s):

S. Albeverio ◽

P. Kurasov

Keyword(s):

Singular Perturbations ◽

Differential Operators

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Asymptotics of the spectrum of family functional-differential operators with summable potential

Sibirskii zhurnal chistoi i prikladnoi matematiki ◽

10.33048/pam.2018.18.405 ◽

2018 ◽

Vol 18 (4) ◽

pp. 56-80

Author(s):

S. I. Mitrokhin

Keyword(s):

Differential Operators ◽

Functional Differential ◽

Summable Potential

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The Calder´on-Zygmund Theory I: Ellipticity

10.23943/princeton/9780691162515.003.0001 ◽

2017 ◽

Author(s):

Brian Street

Keyword(s):

Classical Theory ◽

Singular Integral ◽

Singular Integrals ◽

Differential Operators ◽

Integral Operators ◽

Singular Integral Operators ◽

Single Parameter ◽

Partial Differential ◽

Translation Invariant ◽

Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

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