Generalized Bruhat Cells and Completeness of Hamiltonian Flows of Kogan-Zelevinsky Integrable Systems

Abstract

Let G be any connected and simply connected complex semisimple Lie group, equipped with a standard holomorphic multiplicative Poisson structure. We show that the Hamiltonian flows of all the Fomin-Zelevinsky twisted generalized minors on every double Bruhat cell of G are complete in the sense that all the integral curves of their Hamiltonian vector fields are defined on C. It follows that the Kogan-Zelevinsky integrable systems on G have complete Hamiltonian flows, generalizing the result of Gekhtman and Yakimov for the case of SL(n, C). We in fact construct a class of integrable systems with complete Hamiltonian flows associated to generalized Bruhat cells which are defined using arbitrary sequences of elements in the Weyl group of G, and we obtain the results for double Bruhat cells through the so-called open Fomin-Zelevinsky embeddings of (reduced) double Bruhat cells in generalized Bruhat cells. The Fomin-Zelevinsky embeddings are proved to be Poisson, and they provide global coordinates on double Bruhat cells, called Bott-Samelson coordinates, in which all the Fomin-Zelevinsky minors become polynomials and the Poisson structure can be computed explicitly.

Configuration Poisson groupoids of flags

“…It is easy to see that (X × T, π X ⊲⊳ 0) is a T-Poisson manifold with respect to the diagonal T-action. The following fact is proved in [16,Lemma 2.23]. Lemma 1.4.…”

“…, π 2n ), we can apply results in [5,16] on arbitrary generalized Schubert cells and their T -extensions. In particular, the model in (1.7) allows us to describe all the symplectic leaves of (Γ (u,u −1 ) , π 2n ), thereby proving that the symplectic leaf of (Γ (u,u −1 ) , π 2n ) through the section of units of the groupoid Γ…”

“…Proof. Symplectic leaves of (Γ w , π n ) for any n ≥ 1 and any w ∈ W n are described in Theorem D. 16, and the case of w = (u, u −1 ) ∈ W 2n is given in Example D.18. More concretely, for γ ∈ Γ (u,u −1 ) in (5.1), let (c n+1 , .…”

“…where µ * : T (q,t) (O w e × T ) → T µ(q,t) T ∼ = h is the differential of µ at (q, t). Recall that O w e × T is a single T -leaf of π, and that by [16,Proposition 2.24], the co-rank of π in O w e × T is equal to the co-dimension of w in T . Thus the two vector spaces on the two sides of (D.13) have the same dimension, and it is enough to show that Im π # (q,t) ⊂ µ −1 * (Im(1 − w)).…”

“…For any regular function f on O w e × T that is a weight vector with weight χ f ∈ X * (T ) for the diagonal action of T on O w e × T , one has by [16,Corollary 2.15] and the definition π that {µ χ , f } π = χ − wχ, χ f µ χ f, (D. 14) where { , } π is the Poisson bracket on the coordinate ring of O w e × T defined by π. For χ ∈ h * , let χ be the left invariant 1-form on T with value χ at the identity element.…”