Connected Components of Real Double Bruhat Cells

The main geometric objects of study in this paper are double Bruhat cells G = BuB∩B−vB− in a simply-connected connected complex semisimple group G; here B and B− are two opposite Borel subgroups in G, and u and v any two elements of the Weyl group W . Double Bruhat cells were introduced and studied in [5] as a geometric framework for the study of total positivity in semisimple groups; they are also closely related to symplectic leaves in the corresponding Poisson-Lie groups (see [4, 6]). It will be convenient for us to replaceG with a reduced double Bruhat cell L introduced in [3]. The variety L can be identified with the quotient of G modulo the left (or right) action of the maximal torus H = B ∩B−. As shown in [5, 3], an algebraic variety L is biregularly isomorphic to a Zariski open subset of an affine space of dimension m = l(u)+l(v), where l(u) is the length of u in the Coxeter group W . However, the smooth topology of L can be quite complicated. A first step towards understanding this topology is enumerating the connected components of the real part L(R). In the case when G is simply-laced, a conjectural answer was given in [14, Conjecture 4.1]. Here we prove this conjecture and extend the result to an arbitrary semisimple group G. The answer is given in the following terms: as shown in [14] for G simply-laced, every reduced word i of (u, v) ∈ W×W gives rise to a subgroup Γi(F2) ⊂ GLm(F2) generated by symplectic transvections (here F2 is the 2-element field). We extend the construction of Γi(F2) to an arbitrary G (it is still generated by transvections but not necessarily by symplectic ones). Extending [14, Conjecture 4.1], we show that the connected components of L(R) are in a natural bijection with the Γi(F2)-orbits in F m 2 . As explained in [14], this provides a far-reaching generalization of the results in [12, 13]; this also refines and generalizes results in [10, 11]. Our proof uses methods and results developed in [5, 3]. First, it was shown there that every reduced word i of (u, v) ∈ W ×W gives rise to a biregular isomorphism between the complex torus C6=0 and a Zariski open subset Ui ⊂ L . We refine this result by showing that the complement L − Ui is the union of m divisors {Mk,i = 0}, where M1,i, . . . ,Mm,i are some irreducible regular functions on L. We further show that every i-bounded index n ∈ [1,m] (see Section 2 for the definition) gives rise to a regular function M ′ n,i on L u,v such that replacing the divisor {Mn,i = 0} with {M ′ n,i = 0} leads to another “toric chart” Un,i in L . Then we prove that the connected components of the real part of the union of charts Ui ⋃ ⋃