A Note on a Symplectic Structure on the Space of G -Monopoles

Abstract

Let G be a semisimple complex Lie group with a Borel subgfoup B. Let X = G/B be the flag manifold of GThe moduli space of G-monopoles carries a natural hyperkähler structure, and hence a holomorphic symplectic structure. It was recently explicitly computed by R. Bielawski in case G = SL n . We propose a simple explicit formula for another natural symplectic structure on M b (X, α). It is tantalizingly similar to R. Bielawski's formula, but in general (rank > 1) the two structures do not coincide. Let P ⊃ B be a parabolic subgroup. The construction of the Poisson structure on M b (X, α) generalizes verbatim to the space of based maps M = M b (G/P, β). In most cases the corresponding map T * M → T M is not an isomorphism, i.e. M splits into nontrivial symplectic leaves. These leaves are explicitly described.

Twisted zastava and q-Whittaker functions

“…We will take , and (the sum of boundary divisors with multiplicity one). Recall the symplectic form on constructed in , and let be the corresponding regular nonvanishing section of . According to , has a pole of the first order at each boundary divisor component .…”

“…Recall the symplectic form on constructed in , and let be the corresponding regular nonvanishing section of . According to , has a pole of the first order at each boundary divisor component . Here is an open smooth subvariety with codimension 2 complement formed by all the quasimaps with defect of degree at most a simple coroot.…”

Towards a cluster structure on trigonometric zastava

“…The following diagram commutes: 16. Calculation on C. The differential d C 2 of Corollary 3.15 was computed in [10]. To formulate the result, we introduce homogeneous coordinates z 1 , z 2 on C such that z = z 1 /z 2 , so that z 1 = 0 (resp.…”

“…Let • Z α denote the space of maps C = P 1 → B of degree α sending ∞ ∈ P 1 to B − ∈ B. It is known [10] that this is a smooth symplectic affine algebraic variety, which can be identified with the hyperkähler moduli space of framed G-monopoles on R 3 with maximal symmetry breaking at infinity of charge α [15], [16].…”

Gelfand–Tsetlin algebras and cohomology rings of Laumon spaces

“…Recall the symplectic form on • Q d constructed in[9]. Note that the com-plement Q d \ • Q dequals the union of divisors 1≤i≤n−1 D i .…”