B–Orbits in Abelian Nilradicals of Types B, C and D: Towards a Conjecture of Panyushev

Abstract

Abstract. Let B be a Borel subgroup of a semisimple algebraic group G and let m be an abelian nilradical in b = Lie(B). Using subsets of strongly orthogonal roots in the subset of positive roots corresponding to m, D. Panyushev [5] gives in particular classification of B−orbits in m and m * and states general conjectures on the closure and dimensions of the B−orbits in both m and m * in terms of involutions of the Weyl group. Using Pyasetskii correspondence between B−orbits in m and m * he shows the equivalence of these two conjectures. In this Note we prove his conjecture in types B n , C n and D n for adjoint case.1. Abelian nilradicals and Panyushev's conjecture 1.1. Minimal nilradicals. Let G be a semisimple linear algebraic group over C and let g be its Lie algebra. Let B be its Borel subgroup and b = Lie(B). Let g = n ⊕ h ⊕ n − be its corresponding triangular decomposition, where b = n ⊕ h. B acts adjointly on n. For x ∈ n let B.x denote its orbit.Since the description of B−orbits in n immediately reduces to simple Lie algebras in what follows we assume that g is simple.Let R be the root system of g and W its Weyl group. For α ∈ R let s α be the corresponding reflection in W .Let R + (resp. R − ) denote the subset of positive (resp. negative) roots. For α ∈ R let X α denote the standard root vector in g so that n =+ be a set of simple roots. Let θ be the maximal root in R + . Recall that any standard parabolic subgroup P of G is of the form P = L ⋉ M where L is a standard Levy subgroup and M is the unipotent radical of P. If R L is the root system of l = Lie(L) then ∆ L = ∆ ∩ R L . Let W P denote Weyl group of l. Let w be the longest element of W P .P is maximal if and only if ∆ L = ∆ \ {α i }. We will write

The Bruhat order on Hermitian symmetric varieties and on abelian nilradicals

“…When G is of type A or C, it can be deduced from more general results on the adjoint and coadjoint B-orbits of nilpotency order 2 and their Bruhat order, studied by Ignatyev [9], Melnikov [12], and Barnea and Melnikov [1]. For orthogonal groups this formula was proved by Barnea and Melnikov [2], so only the exceptional cases remained to be proved. However our proof is not based on a case-by-case analysis, and it does not rely on such results.…”