Classification of Degenerate Verma Modules for E(5, 10)

Abstract

<jats:title>Abstract</jats:title><jats:p>Given a Lie superalgebra <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathfrak {g}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> with a subalgebra <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathfrak {g}}_{\ge 0}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula>, and a finite-dimensional irreducible <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathfrak {g}}_{\ge 0}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula>-module <jats:italic>F</jats:italic>, the induced <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathfrak {g}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>-module <jats:inline-formula><jats:alternatives><jats:tex-math>$$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>U</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mo>⊗</mml:mo> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:msub> <mml:mi>F</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathfrak {g}}=E(5,10)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mi>E</mml:mi> <mml:mo>(</mml:mo> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mn>10</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> with the subalgebra <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathfrak {g}}_{\ge 0}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> of minimal codimension. This is done via classification of all singular vectors in the modules <jats:italic>M</jats:italic>(<jats:italic>F</jats:italic>). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for <jats:italic>E</jats:italic>(5, 10).</jats:p>