Algebraic Groups and Quantum Groups.

Intro -- Preface -- Program -- W-algebras at the critical level -- 1. Introduction -- 2. Associated graded vertex Poisson algebras -- 3. BRST cohomology of restricted Wakimoto modules -- 4. Proof of Theorems 1.1 and 1.2 -- References -- Highest weight modules at the critical level and noncommutative Springer resolution -- 1. Introduction -- 2. Noncommutative Springer resolution -- 3. Perverse sheaves on the affine flag variety -- 4. Results of Frenkel-Gaitsgory and a proof of Theorem 1 -- References -- An orthogonal form for level two Hecke algebras with applications -- 1. Introduction -- 2. Level one Hecke algebras -- 3. Level two Hecke algebras -- 4. Khovanov's arc algebra -- 5. Category for Grassmannians -- 6. The general linear supergroup -- 7. The walled Brauer algebra -- References -- On the restricted projective objects in the affine category at the critical level -- 1. Introduction -- 2. Affine Kac-Moody algebras -- 3. The deformed category -- 4. Projective objects in ̂ _{ } -- 5. Block decomposition of ̂ _{ } -- 6. The linkage principle -- References -- Remarks on modular representations of finite groups of Lie type in non-defining characteristic -- 1. Introduction -- 2. On modular unipotent representations -- 3. Principal series representations -- 4. On the dimensions of irreducible representations -- References -- Quantum queer superalgebras -- Introduction -- 1. Queer Lie superalgebra ( ) -- 2. Highest weight modules over ( ) -- 3. Quantum queer superalgebra _{ }( ( )) -- 4. Representation Theory of _{ }( ( )) -- 5. Crystal Bases -- 6. Semistandard decomposition tableaux -- References -- Quantum group actions on rings and equivariant K-theory -- 1. Introduction -- 2. Equivariant K-theory of quantum group actions -- 3. Categories of equivariant modules -- 4. Equivariant K-theory of filtered module algebras.

5. Quantum symmetric algebras -- 6. Quantum homogeneous spaces -- Acknowledgements -- References -- Toward Berenstein-Zelevinsky data in affine type , Part I: Construction of the affine analogs -- 1. Introduction -- 2. Berenstein-Zelevinsky data of type _{ } -- 3. Berenstein-Zelevinsky data of type _{∞} -- 4. Berenstein-Zelevinsky data of type _{ℓ}⁽¹⁾ -- 5. Appendix -- References -- Toward Berenstein-Zelevinsky data in affine type , Part II: Explicit description -- 1. Introduction -- Acknowledgement -- 2. Review on Berenstein-Zelevinsky data associated to finite intervals -- 3. Berenstein-Zelevinsky data associated to ℤ -- 4. Berenstein-Zelevinsky data of type _{ -1}⁽¹⁾ -- 5. Lusztig data of infinite size and a crystal structure on them -- 6. BZ data arising from Lusztig data of Infinite size -- OT1OT1cmrcmrmmnnmbxAppendix A. Another explicit description -- References -- On tropical dualities in cluster algebras -- 1. Introduction -- 2. Proofs of Proposition 1.3 and identity (1.11) -- 3. Proofs of Proposition 1.4 and identity (1.12) -- 4. Some corollaries -- References -- KKR type bijection for the exceptional affine algebra ₆⁽¹⁾ -- 1. Introduction -- 2. Quantum affine algebra and crystal -- 3. Rigged configuration and the bijection -- 4. The bijection -- 5. Proof of Theorem 4 -- References -- Iwahori's question for affine Hecke algebras -- 1. Affine Hecke algebras -- 2. Group algebra ℂ[ ] of -- 3. Some particular semisimple elements in -- Proof of Theorem 1.2 -- References -- On Weyl modules of cyclotomic -Schur algebras -- 0. Introduction -- 1. Review on cyclotomic -Schur algebras -- 2. _{ }( )-crystal structure on Weyl module ( ) of _{ , } -- 3. Some properties of the number _{ } -- 4. Characters of the Weyl modules and symmetric functions -- 5. Decomposition matrices of cyclotomic -Schur algebras.

6. The Ariki-Koike algebra as a subalgebra of _{ , } -- References.

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