Geometric Satake, Springer correspondence and small representations

Abstract

Abstract. For a simply-connected simple algebraic group G over C, we exhibit a subvariety of its affine Grassmannian that is closely related to the nilpotent cone of G, generalizing a well-known fact about GLn. Using this variety, we construct a sheaf-theoretic functor that, when combined with the geometric Satake equivalence and the Springer correspondence, leads to a geometric explanation for a number of known facts (mostly due to Broer and Reeder) about small representations of the dual group.

Diagram automorphisms of quiver varieties

“…The (partly conjectural) duality here is that of symplectic dual pairs, in the sense of [2, Remark 1.5]. Along the same lines, the isomorphisms of Theorem 1.2 could be thought of as dual to some of the connections proved in [1] between the small part of the affine Grassmannian and the nilpotent cone, in types other than A. This may explain why one should have to restrict to quiver varieties corresponding to small representations and Slodowy slices to 'big' nilpotent orbits.…”

“…We answer question (1) in complete generality (indeed, with a slightly more general definition of diagram automorphism than has hitherto appeared), and questions (2) and (3) in special cases. When M(v, w) is of type A, the answer to question (1) is that M(v, w) θ is a disconnected union of quiver varieties of type D. As a result, what comes out of (3) are some analogues of Maffei's isomorphisms, this time between 'small' type-D quiver varieties and 'two-row' Slodowy varieties of types C and D.…”

“…In the k < n case, W n = 0 so there is no choice for σ n ; in the k = n case, there are (up to G W -conjugacy) three choices for σ n , namely σ n = id W n , σ n = −id W n , and σ n having eigenvalues 1 and −1; the respective values of (w + , w − ) are (2, 0), (0, 2) and (1,1). Also note that the defining equations (3.6) of Λ A 2n−1 (V, W ) now take a simpler form, since Γ i = Δ i = 0 for i / ∈ {k, 2n − k}.…”

“…What involutions of Slodowy varieties correspond to the diagram involutions of quiver varieties of type A? (3) What information is obtained by combining the answers to (1) and (2)?…”

Weyl group actions on the Springer sheaf

“…Here, Gr λ is the orbit in the affine Grassmannian of G labelled by λ, M is the G-stable locally closed subvariety of the affine Grassmannian defined in [1] (which intersects Gr λ in an open dense subvariety for all small λ), and π : M → N is the G-equivariant finite map defined and described in [1]. (We follow the notation of [1] for this finite map since we no longer need the Grothendieck-Springer map denoted π in previous sections. )…”