Related papers: Eigenvalue Coincidences and Multiplicity Free Sphe…

Let $r$ be a positive integer and let $G_n$ be the reflection group of $n \times n$ monomial matrices whose entries are $r^{th}$ complex roots of unity and let $k \leq n$. We define and study two new graded quotients $R_{n,k}$ and $S_{n,k}$…

Let G be a locally compact group and let K be a compact subgroup of Aut(G), the group of automorphisms of G. The pair $(G, K )$ is a Gelfand pair if the algebra $L^{1}_{K}(G)$ of K-invariant integrable functions on G is commutative under…

For an irreducible complex reflection group $W$ of rank $n$ containing $N$ reflections, we put $g=2N/n$ and construct a $(g+1)^n$-dimensional irreducible representation of the Cherednik algebra which is (as a vector space) a quotient of the…

Let g be a semisimple Lie algebra over an algebraically closed field k of characteristic 0. Let V be a simple finite-dimensional g-module and let y\in V be a highest weight vector. It is a classical result of B. Kostant that the algebra of…

We provide spectral Lie algebras with enveloping algebras over the operad of little $G$-framed $n$-dimensional disks for any choice of dimension $n$ and structure group $G$, and we describe these objects in two complementary ways. The first…

We consider $\rm R$-matrix realization of the quantum deformations of the loop algebras $\tilde{\mathfrak{g}}$ corresponding to non-exceptional affine Lie algebras of type $\hat{\mathfrak{g}}=A^{(1)}_{N-1}$, $B^{(1)}_n$, $C^{(1)}_n$,…

We study the operational bivariant theory associated to the covariant theory of Grothendieck groups of coherent sheaves, and prove that it has many geometric properties analogous to those of operational Chow theory. This operational…

Fix an integral semisimple element $\lambda$ in the Lie algebra $\mathfrak{g}$ of a complex reductive algebraic group $G$. Let $L$ denote the centralizer of $\lambda$ in $G$ and let $\mathfrak{g}(-1)$ denote the $-1$ eigenspace of…

We consider actions of reductive groups on a varieties with finitely generated Cox ring, e.g., the classical case of a diagonal action on a product of projective spaces. Given such an action, we construct via combinatorial data in the Cox…

Let $\mathfrak g$ be a complex simple Lie algebra and $\mathfrak b=\mathfrak t\oplus\mathfrak u^+$ a fixed Borel subalgebra. Let $\Delta^+$ be the set of positive roots associated with $\mathfrak u^+$ and $\mathcal K\subset\Delta^+$ the…

In this paper, we reconstruct explicitly the generating function of genus-zero K-theoretic permutation-invariant Gromov-Witten invariants, known as the big $\mathcal{J}$-function, for any partial flag variety. The reconstruction may start…

A full set of (higher order) Casimir invariants for the Lie algebra $gl(\infty )$ is constructed and shown to be well defined in the category $O_{FS}$ generated by the highest weight (unitarizable) irreducible representations with only a…

We give a description of operator algebras of free wreath products in terms of fundamental algebras of graphs of operator algebras as well as an explicit formula for the Haar state. This allows us to deduce stability properties for certain…

The Orlik-Solomon algebra ${\cal A}(G)$ of a matroid $G$ is the free exterior algebra on the points, modulo the ideal generated by the circuit boundaries. On one hand, this algebra is a homotopy invariant of the complement of any complex…

We construct a Gelfand-Zeitlin system on a one-parameter family of $G_2$ coadjoint orbits that are multiplicity-free Hamiltonian $SU(3)$-spaces. Using this system we prove a lower bound for the Gromov width of these orbits. This lower bound…

Let G be the split special orthogonal group of degree 2n+1 over a field F of char F \ne 2. Then we describe G-orbits on the triple flag varieties G/P\times G/P\times G/P and G/P\times G/P\times G/B with respect to the diagonal action of G…

We introduce an equivariant algebraic cobordism theory \Omega^G for algebraic varieties with G-action, where G is a split diagonalizable group scheme over a field k. It is done by combining the construction of the algebraic cobordism theory…

The variety of bicommutative algebras consists of all nonassociative algebras satisfying the polynomial identities of right- and left-commutativity $(x_1x_2)x_3=(x_1x_3)x_2$ and $x_1(x_2x_3)=x_2(x_1x_3)$. Let $F_d$ be the free $d$-generated…

This thesis studies the symplectic structure of holomorphic coadjoint orbits, and their projections. A holomorphic coadjoint orbit O is an elliptic coadjoint orbit which is endowed with a natural invariant K\"ahlerian structure. These…

Let $G$ be an algebraic group and let $X$ be a smooth $G$-variety with two orbits: an open orbit and a a closed orbit of codimension $1$. We give an algebraic description of the category of $G$-equivariant vector bundles on $X$ under a mild…