Towards a cluster structure on trigonometric zastava
This is the third paper in a series which describes a conjectural analogue of the affine Grassmannian for affine Kac-Moody groups (also known as the double affine Grassmannian). The present paper is dedicated to the description of the conjectural analogue of the convolution diagram for the double affine Grassmannian and affine zastava.
Let G be an almost simple simply connected group over C. For a positive element α of the coroot lattice of G let View the MathML source denote the space of maps from P1 to the flag variety B of G sending ∞∈P1 to a fixed point in B of degree α. This space is known to be isomorphic to the space of framed G -monopoles on R3 with maximal symmetry breaking at infinity of charge α. In  a system of (étale, rational) coordinates on Z source is introduced. In this note we compute various known structures on Z source in terms of the above coordinates. As a byproduct we give a natural interpretation of the Gaiotto–Witten superpotential studied in  and relate it to the theory of Whittaker D-modules discussed in .
Let G be an almost simple simply connected group over complex numbers. For a positive element of the coroot lattice of G, we consider the open zastava space of based maps from the projective line to the flag variety of G, having the above prescribed degree. This space is known to be isomorphic to the space of framed eucledian G-monopoles with maximal symmetry breaking at infinity of the above prescribed topological charge. In the previous work of Finkelberg-Kuznetsov-Markarian-Mirkovic, a system of etale rational coordinates on the open zastava was introduced. In this note we compute various known structures on the open zastava in terms of the above coordinates. As a byproduct we give a natural interpretation of the Gaiotto-Witten superpotential and relate it to the theory of Whittaker D-modules developed by D.Gaitsgory.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.