Ghost Distributions on Supersymmetric Spaces I: Koszul Induced Superspaces, Branching, and the Full Ghost Centre

Abstract Given a Lie superalgebra $$\mathfrak {g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> , Gorelik defined the anticentre $$\mathcal {A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> of its enveloping algebra, which consists of certain elements that square to the center. We seek to generalize and enrich the anticentre to the context of supersymmetric pairs $$(\mathfrak {g},\mathfrak {k})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , or more generally supersymmetric spaces G / K . We define certain invariant distributions on G / K , which we call ghost distributions, and which in some sense are induced from invariant distributions on $$G_0/K_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> . Ghost distributions, and in particular their Harish-Chandra polynomials, give information about branching from G to a symmetric subgroup $$K'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> which is related (and sometimes conjugate) to K . We discuss the case of $$G\times G/G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>×</mml:mo> <mml:mi>G</mml:mi> <mml:mo>/</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> for an arbitrary quasireductive supergroup G , where our results prove the existence of a polynomial which determines projectivity of irreducible G -modules. Finally, a generalization of Gorelik’s ghost centre is defined which we call the full ghost centre, $$\mathcal {Z}_{full}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mi>full</mml:mi> </mml:mrow> </mml:msub> </mml:math> . For type I basic Lie superalgebras $$\mathfrak {g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> we fully describe $$\mathcal {Z}_{full}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mi>full</mml:mi> </mml:mrow> </mml:msub> </mml:math> , and prove that if $$\mathfrak {g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> contains an internal grading operator, $$\mathcal {Z}_{full}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mi>full</mml:mi> </mml:mrow> </mml:msub> </mml:math> consists exactly of those elements in $$\mathcal {U}\mathfrak {g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mi>g</mml:mi> </mml:mrow> </mml:math> acting by $$\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> -graded constants on every finite-dimensional irreducible representation.