# Number Theory: Covers of reductive groups and functoriality

SEMINARS, NUMBER THEORY

##### Speaker:

Tasho Kaletha *- University of Michigan*

To a connected reductive group G over a local field F we define a compact topological group π_1~(G) and an extension G(F)_∞ of G(F) by π_1~(G). From any character x of π_1~(G) of order n we obtain an n-fold cover G(F)_x of the topological group G(F). We also define an L-group for G(F)_x, which is a usually non-split extension of the Galois group by the dual group of G, and deduce from the linear case a refined local Langlands correspondence between genuine representations of G(F)_x and L-parameters valued in this L-group.

This construction is motivated by Langlands functoriality. We show that a subgroup of the L-group of G of a certain kind naturally leads to a smaller quasi-split group H and a double cover of H(F). Genuine representations of this double cover are expected to be in functorial relationship with representations of G(F). We will present two concrete applications of this, one that gives a characterization of the local Langlands correspondence for supercuspidal L-parameters when p is sufficiently large, and one to the theory of endoscopy.