Schur 𝑄-functions and the Capelli eigenvalue problem for the Lie superalgebra \ger𝑞(𝑛)

Let <inline-formula content-type="math/tex"> <tex-math>\ger l\defi \ger q(n)\times \ger q(n)</tex-math> </inline-formula> , where <inline-formula content-type="math/tex"> <tex-math>\ger q(n)</tex-math> </inline-formula> denotes the queer Lie superalgebra. The associative superalgebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Q(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a left and right action of <inline-formula content-type="math/tex"> <tex-math>\ger q(n)</tex-math> </inline-formula> , and hence is equipped with a canonical <inline-formula content-type="math/tex"> <tex-math>\ger l</tex-math> </inline-formula> -module structure. We consider a distinguished basis <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper D Subscript lamda Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mi> λ </mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{D_\lambda \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the algebra of <inline-formula content-type="math/tex"> <tex-math>\ger l</tex-math> </inline-formula> -invariant super-polynomial differential operators on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which is indexed by strict partitions of length at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that the spectrum of the operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript lamda"> <mml:semantics> <mml:msub> <mml:mi>D</mml:mi> <mml:mi> λ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">D_\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , when it acts on the algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P left-parenthesis upper V right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr P(V)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of super-polynomials on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , is given by the factorial Schur <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> -functions of Okounkov and Ivanov. As an application, we show that the radial projections of the spherical super-polynomials (corresponding to the diagonal symmetric pair <inline-formula content-type="math/tex"> <tex-math>(\ger l,\ger m)</tex-math> </inline-formula> , where <inline-formula content-type="math/tex"> <tex-math>\ger m\defi \ger q(n)</tex-math> </inline-formula> ) of irreducible <inline-formula content-type="math/tex"> <tex-math>\ger l</tex-math> </inline-formula> -submodules of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P left-parenthesis upper V right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathscr P(V)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are the classical Schur <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> -functions. As a further application, we compute the Harish-Chandra images of the Nazarov basis <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper C Subscript lamda Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi> λ </mml:mi> </mml:msub>

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