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Special Symplectic Geometry Seminar - Jeremy Lane - Symplectic geometry of extended string cones


Title: Symplectic geometry of extended string cones

Speaker: Dr Jeremy Lane, Univeristy of Geneva

Abstract: Let $G$ be a complex semisimple Lie group with Lie algebra $\mathfrak{g}$, compact real form $K$, maximal torus $T\subset K$, and positive Weyl chamber $\mathfrak{t}_+^*\subset \mathfrak{t}^*$. Let $m = \frac{1}{2}dim(K/T)$. There are two naturally defined symplectic manifolds associated to $G$ as follows.

First, we may consider $K \times (\mathfrak{t}_+^*)^{int}$. This is a symplectic submanifold of the cotangent bundle of $K$ equipped with a Hamiltonian action of $K\times T$. The quotient of this space by the action of $T$ is the subset of regular elements in $\mathfrak{k}^*$. The $T$-reduced spaces are the regular coadjoint orbits.

Second, from the representation theory of the quantum universal enveloping algebra $U_q(\mathfrak{g})$ and Lusztig's theory of canonical bases, we can define (with some choice) extended string cones $C \subset \mathfrak{t} \times \mathbb{R}^m$: convex rational polyhedral cones that fiber naturally over $\mathfrak{t}_+^*$. The choice of $G$ determines a lattice in $\mathfrak{t}\times \mathbb{R}^m$ that defines a canonical symplectic structure on $C^{int} \times T \times \mathbb{T}$, where $\mathbb{T} = \mathbb{R}^m/\mathbb{Z}^m$. By definition, this symplectic manifold is equipped with a completely integrable Hamiltonian action of $\mathbb{T}\times T $.

In this talk I will discuss how we aim to show these spaces are related. This is part of joint work with Anton Alekseev, Benjamin Hoffman, and Yanpeng Li.
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