Alphabeta MathArakawa–Suzuki Functors and Higher-Level Schur–Weyl Duality
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1 · Prerequisites
Abstract Algebra
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- The Fundamental Group
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- Uniform Spaces: the Three Definitions
2 · Logical flowchart
3 · Items
Definition
The degenerate affine Hecke algebra
Lemma
Type-A nilpotent orbit closures are normal via chain varieties
Definition
Arakawa-Suzuki Casimir operators on tensor functors
Lemma
Nilpotent-orbit tensor covariants extend and polarize
Definition
The type-A nilpotent pyramid, good grading and Whittaker character
Theorem
The Casimir and place-permutation operators satisfy the degenerate affine Hecke relations
Definition
The type-A finite W-algebra and the Whittaker category
Lemma
The regular coset-dominant Harish–Chandra tensor comparison
Theorem
Vust's graded double centralizer for the type-A nilpotent centralizer
Definition
The shifted Yangian and the pyramid generators of the type-A finite W-algebra
Lemma
Gan-Ginzburg and Skryabin inputs for the type-A pyramid good grading
Theorem
Ordered monomial PBW basis for the shifted Yangian
Theorem
Skryabin's equivalence for the type-A Whittaker pair
Theorem
The loop filtration on the type-A finite W-algebra has associated graded U(gᵉ)
Definition
The degenerate cyclotomic Hecke algebra
Definition
The deformed tensor space for the type-A finite W-algebra
Lemma
The parabolic highest-weight truncation factors through the cyclotomic quotient
Lemma
Separated integral cosets give typed parabolic category equivalences
Definition
The Whittaker functor from the relevant parabolic category-O summand
Theorem
The finite W and degenerate cyclotomic Hecke actions satisfy a double centralizer
Theorem
The Whittaker functor is exact and fully faithful on the required projectives
Definition
The higher-level Schur-Weyl projective generator and Schur functor
Definition
Stable range and block idempotent corner for higher-level Schur-Weyl duality
Definition
Hecke induction and restriction functors on degenerate cyclotomic blocks
Theorem
The higher-level Schur-Weyl endomorphism isomorphism
Theorem
The higher-level Schur-Weyl double centralizer and Schur quotient
Theorem
Schur functor images of standard, simple, projective and tilting objects
Theorem
Category-O translation intertwines residue induction and restriction
Sources
- Jonathan Brundan and Alexander Kleshchev, Schur-Weyl duality for higher levels, Selecta Math. (N.S.) 14 (2008) 1-57 (complete 50-page arXiv article)
- Jonathan Brundan and Alexander Kleshchev, The degenerate analogue of Ariki's categorification theorem, Math. Z. 266 (2010) 877-919 (complete 44-page arXiv article)
- Jonathan Brundan and Alexander Kleshchev, Representations of shifted Yangians and finite W-algebras, Mem. Amer. Math. Soc. 196 (2008), no. 918 (complete 109-page arXiv memoir)
- Hanspeter Kraft and Claudio Procesi, Closures of conjugacy classes of matrices are normal, Inventiones Math. 53 (1979) 227-247
- W. Borho and H. Kraft, Ueber Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen, Comment. Math. Helv. 54 (1979) 61-104
- Tomoyuki Arakawa and Takeshi Suzuki, Lie algebras and degenerate affine Hecke algebras of type A, J. Algebra 209 (1998) 288-304 (complete 16-page arXiv preprint of the origin paper)
- Tomoyuki Arakawa and Takeshi Suzuki, Lie algebras and degenerate affine Hecke algebras of type A, J. Algebra 209 (1998) 288-304 (complete 16-page arXiv preprint)
- Bernstein and Gelfand, Tensor products of finite and infinite dimensional representations of semisimple Lie algebras
- T. Vust, Sur la contraction des espaces de tenseurs, C. R. Acad. Sci. Paris 289 (1979) A141-A143
- Jonathan Brundan and Alexander Kleshchev, Shifted Yangians and finite W-algebras, arXiv:math/0407012 (49 pp.); Parabolic presentations of the Yangian Y(gl_n), arXiv:math/0407011 (28 pp.)
- Wee Liang Gan and Victor Ginzburg, Quantization of Slodowy slices, Int. Math. Res. Not. 2002, no. 5, 243-255 (complete 10-page arXiv article)
- Jonathan Brundan, Simon M. Goodwin and Alexander Kleshchev, Highest weight theory for finite W-algebras, Int. Math. Res. Not. 2008, no. 15 (complete 38-page arXiv article)
- Jonathan Brundan and Alexander Kleshchev, Shifted Yangians and finite W-algebras (arXiv:math/0407012) and Parabolic presentations of the Yangian Y(gl_n) (arXiv:math/0407011)
- Jonathan Brundan and Alexander Kleshchev, Shifted Yangians and finite W-algebras
- Brundan and Kleshchev, The degenerate analogue of Ariki's categorification theorem
- Joseph Bernstein and Sergei Gelfand, Tensor products of finite and infinite dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980) 245-285 (complete 42-page NUMDAM article)
- Brundan and Kleshchev, Representations of shifted Yangians and finite W-algebras
- Alexander Kleshchev, Linear and Projective Representations of Symmetric Groups, Cambridge Tracts in Mathematics 163, Lemma 8.2.2 (as cited by [BKAR, (3.19)])