The stable Bernstein center

The Bernstein center of a p-adic reductive group G is a beautiful and explicit commutative ring which acts on “everything” related to the representation theory of G. In recent years, the idea has emerged that this ring contains a canonical subring – the stable Bernstein center – which should be intimately related with the local Langlands correspondence. However, while it is easy to define the stable Bernstein center, it is very difficult to exhibit elements in this subring. On the other hand, recent work of Fargues-Scholze defines another totally canonical subring of the Bernstein center, whose construction uses V. Lafforgue’s theory of excursion operators adapted to the Fargues-Fontaine curve. After reviewing these stories, I’ll sketch a proof that the Fargues-Scholze subring is actually contained in the stable Bernstein center, for all G. In the pretalk, I’ll give a more leisurely introduction to the Bernstein center.