# Sturm's operator acting on vector valued $K$-types

## TL;DR

This paper investigates Sturm's operator on vector valued Siegel modular forms, providing explicit descriptions of holomorphic projections for large weights and revealing phantom terms for small weights, extending previous scalar form results.

## Contribution

It introduces Sturm's operator for vector valued forms and characterizes its behavior across different weight ranges, including the emergence of phantom terms for small weights.

## Key findings

- Explicit description of holomorphic projection for large weights
- Identification of phantom terms for small weights
- Extension of scalar form results to vector valued forms

## Abstract

We define Sturm's operator on vector valued Siegel modular forms obtaining an explicit description of their holomorphic projection in case of large absolute weight. However, for small absolute weight, Sturm's operator produces phantom terms in addition. This confirms our earlier results for scalar Siegel modular forms.