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

## TL;DR

This paper explores super Jacobi polynomials related to the Lie supergroup OSP(2,2n), revealing how their coefficients behave under parameter transformations and connecting them to supercharacters of various modules.

## Contribution

It introduces a new family of super Jacobi polynomials obtained via blow-up procedures, linking them to supercharacters of Kac modules, irreducible modules, and projective covers.

## Key findings

- Polynomials depend rationally on parameters k,p,q
- Blow-up at (-1,0) yields a family parameterized by t
- Supercharacters are obtained as special cases for specific t values

## 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 $\Bbb A^2$. If we apply blow up procedure at the point $(-1,0)$ then we get a new family of polynomials depending on parameter $t\in \Bbb P$. 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.