Abstract. Let G be an almost simple simply connected group over C. For a positive element α of the coroot lattice of G letα denote the space of maps from P 1 to the flag variety B of G sending ∞ ∈ P 1 to a fixed point in B of degree α. This space is known to be isomorphic to the space of framed G-monopoles on R 3 with maximal symmetry breaking at infinity of charge α.In [6] a system of (étale, rational) coordinates onα is introduced. In this note we compute various known structures on• Z α in terms of the above coordinates. As a byproduct we give a natural interpretation of the Gaiotto-Witten superpotential studied in [9] and relate it to the theory of Whittaker D-modules discussed in [8].

Abstract. Feigin and Shoikhet conjectured in [FS] that successive quotients Bm(An) of the lower central series filtration of a free associative algebra An have polynomial growth. In this paper we give a proof of this conjecture, using the structure of a representation of Wn, the Lie algebra of polynomial vector fields on C n , on Bm(An) which was defined in [FS]. Moreover, we show that the number of squares in a Young diagram D corresponding to an irreducible Wn-module in the Jordan-Hölder series of Bm(An) is bounded above by the integer (m − 1) 2 + 2[](m − 1), which allows us to confirm the structure of B 3 (A 3 ) conjectured in [FS].

We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety. We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava. 5 µ * 20

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