We study algebras constructed by quantum Hamiltonian reduction associated with symplectic quotients of symplectic vector spaces, including deformed preprojective algebras, symplectic reection algebras (rational Cherednik algebras), and quantization of hypertoric varieties introduced by Musson and Van den Bergh in [MVdB]. We determine BRST cohomologies associated with these quantum Hamiltonian reductions. To compute these BRST cohomologies, we make use of the method of deformation quantization (DQ-algebras) and F-action studied by Kashiwara and Rouquier in [KR], and Gordon and Losev in [GL].
We consider two types of quotients of the integrable modules of sl<sub>2</sub><sup>∧</sup>. These spaces of coinvariants have dimensions described in terms of the Verlinde algebra of level k. We describe monomial bases for the spaces of coinvariants, which leads to a fermionic description of these spaces. For k=1, we give the explicit formulas for the characters. We also present recursion relations satisfied by the characters and the monomial bases.
We obtain an explicit presentation for the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation for the ordinary cobordism ring. Another application is an equivariant Schubert calculus in cobordism. We also describe the rational equivariant cobordism rings of wonderful symmetric varieties of minimal rank.
We give a criterion of existence of a unipotent group action on the affine cone over a projective variety or, more generally, on the affine quasicone over a variety which is projective over another affine variety.
We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given.
Let k be an arbitrary field of characteristic zero. In this paper we study quotients of k-rational conic bundles over P 1 k by finite groups of automorphisms. We construct smooth minimal models for such quotients. We show that any quotient is birationally equivalent to a quotient of other k-rational conic bundle cyclic group C2 k of order 2k , dihedral group D2 k of order 2k , alternating group A4 of degree 4, symmetric group S4 of degree 4 or alternating group A5 of degree 5 effectively acting on the base of the conic bundle. Also we construct infinitely many examples of such quotients which are not k-birationally equivalent to each other.