Crossed homomorphisms and low dimensional representations of mapping class groups of surfaces

We continue the study of low dimensional linear representations of mapping class groups of surfaces initiated by Franks--Handel and Korkmaz. We consider $(2g+1)$-dimensional complex linear representations of the pure mapping class groups of compact orientable surfaces of genus $g$. We give a complete classification of such representations for $g \geq 7$ up to conjugation, in terms of certain twisted $1$-cohomology groups of the mapping class groups. A new ingredient is to use the computation of a related twisted $1$-cohomology group by Morita. The classification result implies in particular that there are no irreducible linear representations of dimension $2g+1$ for $g \geq 7$, which marks a contrast with the case $g=2$.