Geometric aspects of representation theory of supergroups

The representation theory of Lie superalgebras began in the end of 1970-s following Kac's classification of simple Lie superalgebras. In the early 1980-s, Manin and his students Penkov, Skornyakov and Voronov studied geometry of flag supervarieties $G/B$. In particular, several important results towards super Borel--Weil--Bott theorem were obtained by Penkov. A natural next step would be to generalize the Beilinson--Bernstein localization theorem.

However, a satisfactory approach is still unknown.

Even the cohomology groups of an arbitrary $G$-equivariant line bundle on $G/B$ remain largely unknown for a classical supergroup $G$.

In this talk I will focus on the category $\operatorname{Rep} G$ of representations of an algebraic supergroup $G$ with reductive underlying group $G_0$. Although this category is not as complicated as the category $\mathcal O$, it seems to capture the essential difficulties in passing from $G_0$ to $G$ and is still challenging to study.

The tensor category $\operatorname{Rep} G$ is Frobenius. By this reason one can apply certain techniques developed in the modular representation theory of finite groups.  We generalize the notions of $p$-subgroups and Sylow subgroups, as well as support and rank varieties.

We also formulate a conjecture about the Balmer spectrum of the stable category $\operatorname{St} G$, verified in several cases.

Finally, we discuss new results on homogeneous supervarieties and odd vector fields, which play a crucial role in this approach.