Convexity and tightness for restrictions of Hamiltonian functions to fixed point sets of an antisymplectic involution

Abstract

The Kostant convexity theorem for real flag manifolds is generalized to a Hamiltonian framework. More precisely, it is proved that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the momentum mapping for a Hamiltonian torus action on a symplectic manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q"> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding="application/x-tex">Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the fixed point set of an antisymplectic involution of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> leaving <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> invariant, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis upper Q right-parenthesis equals f left-parenthesis upper M right-parenthesis equals"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(Q) = f(M) =</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a convex polytope. Also it is proved that the coordinate functions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are tight, using "half-turn" involutions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q"> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding="application/x-tex">Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .