Abstract: As is well known, every rational representation of a finite group $G$ can be realized over $\mathbb{Z}$, that is, the corresponding $\mathbb{Q}G$-module $V$ admits a $\mathbb{Z}$-form. Although $\mathbb{Z}$-forms are usually far from being unique, the famous Jordan--Zassenhaus Theorem shows that there are only finitely many $\mathbb{Z}$-forms of any given $\mathbb{Q}G$-module, up to isomorphism. Determining the precise number of these isomorphism classes or even explicit representatives is, however, a hard task in general. In this talk we shall be concerned with the case where $G$ is the symmetric group $\mathfrak{S}_n$ and $V$ is a simple $\mathbb{Q}\mathfrak{S}_n$-module labelled by a hook partition. Building on work of Plesken and Craig we shall present some results as well as open problems concerning the construction of the

integral forms of these modules. This is joint work with Tommy Hofmann from Kaiserslautern.

# Past Representation Theory Seminar

I will review the equivalence of categories of a Bernstein component of a p-adic classical group with the category of right modules over a certain affine Hecke algebra (with parameters) that I obtained previously. The parameters can be made explicit by the parametrization of supercuspidal representations of classical groups obtained by C. Moeglin, using methods of J. Arthur. Via this equivalence, I can show that the category of smooth complex representations of a quasisplit $p$-adic classical group and its pure inner forms is naturally decomposed into subcategories that are equivalent to the tensor product of categories of unipotent representations of classical groups (in the sense of G. Lusztig). All classical groups (general linear, orthogonal, symplectic and unitary groups) appear in this context.

According to the Harish-Chandra philosophy, cuspidal representations are the basic building blocks in the representation theory of finite reductive groups. Similarly for supercuspidal representations of p-adic groups. Self-dual representations play a special role in the study of parabolic induction. Thus, it is of interest to know whether self-dual (super)cuspidal representations exist. With a few exceptions involving some small fields, I will show precisely when a finite reductive group has irreducible cuspidal representations that are self-dual, of Deligne-Lusztig type, or both. Then I will look at implications for the existence of irreducible, self-dual supercuspidal representations of p-adic groups. This is joint work with Manish Mishra.