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Register a new userThe $mathbb{A}_{q,t}$ algebra and parabolic flag Hilbert schemes
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The earlier work of the first and the third named authors introduced the algebra $mathbb{A}_{q,t}$ and its polynomial representation. In this paper we construct an action of this algebra on the equivariant K-theory of certain smooth strata in the flag Hilbert schemes of points on the plane. In this presentation, the fixed points of torus action correspond to generalized Macdonald polynomials and the the matrix elements of the operators have explicit combinatorial presentation.
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In this paper we study the tangent spaces of the smooth nested Hilbert scheme $ Hil{n,n-1}$ of points in the plane, and give a general formula for computing the Euler characteristic of a $TT^2$-equivariant locally free sheaf on $Hil{n,n-1}$. Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables $q$ and $t$ with non-negative integer coefficients . We call this conjecturally positive polynomial as textsl{the nested $q,t$-Cat alan series}, for it has many conjectural properties similar to that of the $q,t $-Catalan series.
This paper is about the reduced group C*-algebras of real reductive groups, and about Hilbert C*-modules over these C*-algebras. We shall do three things. First we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced C*-algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced C*-algebra to determine the structure of the Hilbert C*-bimodule that represents the functor of parabolic induction. Third, we shall prove that the parabolic induction bimodule admits a secondary inner product, using which we can define a functor of parabolic restriction in tempered representation theory. We shall prove in the sequel to this paper that parabolic restriction is adjoint, on both the left and the right, to parabolic induction.
We show that the generic Hall algebra of nilpotent representations of an oriented cycle specialised at $q=0$ is isomorphic to the generic extension monoid in the sense of Reineke. This continues the work of Reineke.
For a commutative algebra $A$ over $mathbb{C}$,denote $mathfrak{g}=text{Der}(A)$. A module over the smash product $A# U(mathfrak{g})$ is called a jet $mathfrak{g}$-module, where $U(mathfrak{g})$ is the universal enveloping algebra of $mathfrak{g}$.In the present paper, we study jet modules in the case of $A=mathbb{C}[t_1^{pm 1},t_2]$.We show that $A#U(mathfrak{g})congmathcal{D}otimes U(L)$, where $mathcal{D}$ is the Weyl algebra $mathbb{C}[t_1^{pm 1},t_2, frac{partial}{partial t_1},frac{partial}{partial t_2}]$, and $L$ is a Lie subalgebra of $A# U(mathfrak{g})$ called the jet Lie algebra corresponding to $mathfrak{g}$.Using a Lie algebra isomorphism $theta:L rightarrow mathfrak{m}_{1,0}Delta$, where $mathfrak{m}_{1,0}Delta$ is the subalgebra of vector fields vanishing at the point $(1,0)$, we show that any irreducible finite dimensional $L$-module is isomorphic to an irreducible $mathfrak{gl}_2$-module. As an application, we give tensor product realizations of irreducible jet modules over $mathfrak{g}$ with uniformly bounded weight spaces.
The canonical bases of cluster algebras of finite types and rank 2 are given explicitly in cite{CK2005} and cite{SZ} respectively. In this paper, we will deduce $mathbb{Z}$-bases for cluster algebras for affine types $widetilde{A}_{n,n},widetilde{D}$ and $widetilde{E}$. Moreover, we give an inductive formula for computing the multiplication between two generalized cluster variables associated to objects in a tube.
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