Seminar Darstellungstheorie

Das Seminar Darstellungstheorie findet normalerweise dienstags, 14.15 - 15.15 Uhr in G.13.18 statt.

Beteiligte Dozenten: Sascha Orlik, Kay Rülling, Tobias Schmidt, Britta Späth.

Miles: Gluing sheaves along Harder-Narasimhan strata of Bun_2

We explain how to glue (prime-to-p torsion) sheaves corresponding to smooth representations of a p-adic reductive group G (and inner forms of Levi subgroups) along lower- and upper-* functors for the stratification of Fargues--Scholze’s Bun_G given by the Harder-Narasmihan slope decomposition. We then restrict to G = GL_2 and use this method to compute gluing data i^{*}j_{*} for sheaves supported on an open stratum of a connected component to its immediate specialization in Bun_2. The computation ultimately reduces to the étale cohomology of Banach--Colmez spaces (the geometric part) and smooth SL_2(Q_p)-homology via the resolutions of Schneider--Stuhler (the representation-theoretic part). Notably, we compute the gluing data of non-generic unramified principal series, which do not vanish, using purely automorphic methods.

Cotner: On the Fargues--Scholze local Langlands correspondence for Yu's supercuspidals

There are several different approaches to the local Langlands correspondence, which can be roughly grouped as "classical", "explicit", and "geometric". Each of these has its own useful features, and it is expected that they all ultimately yield the same correspondence, but this is far from clear. There has been a lot of work by many people comparing the classical and geometric approaches, and some partial work comparing the other two pairs of approaches. I will discuss recent joint work with Tony Feng which is focused on comparing the explicit and geometric
approaches due to Kaletha and Fargues--Scholze, respectively. The main recent tools making this comparison possible are Feng's modular functoriality and Scholze's independence of \ell.

Sorgdrager: Bound on the Gelfand--Kirillov dimension for GL_2

The Gelfand--Kirillov dimension is an invariant of p-adic representations of p-adic Lie groups. It is a generalization of the Krull dimension to (non-commutative) completed group rings. Precise knowledge of this dimension can lead to interesting applications to the p-adic Langlands program and completed cohomology, as observed by Gee--Newton. In my talk I will introduce the above dimension and then discuss the following result from my thesis: let K be a p-adic field, then any admissible p-adic Banach representation of GL_2K admitting an infinitesimal character has Gelfand--Kirillov dimension at most [K:Q_p]. I have also proved a version in families of the above result. This family version allows one to generalize the Gelfand-Kirillov bound of Breuil--Herzig--Hu--Morra--Schraen for mod p representations beyond K unramified. I hope to sketch the proof in the case of K=Q_p.

• Seminar Darstellungstheorie vergangener Semester