Supergroup $OSP(2,2n)$ and super Jacobi polynomials

Abstract

Coefficients of super Jacobi polynomials of type $B(1,n)$ are rational functions in three parameters $k,p,q$. At the point $(-1,0,0)$ these coefficient may have poles. Let us set $q=0$ and consider pair $(k,p)$ as a point of ${\mathbb{A}}^2$. If we apply blow up procedure at the point $(-1,0)$ then we get a new family of polynomials depending on parameter $t\in \mathbb{P}$. If $t=\infty$ then we get supercharacters of Kac modules for Lie supergroup $OSP(2,2n)$ and supercharacters of irreducible modules can be obtained for nonnegative integer $t$ depending on highest weight. Besides we obtained supercharcters of projective covers as specialisation of some nonsingular modification of super Jacobi polynomials.

19 pages, extended version