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MR2441254 (2010d:22026) 22E50 (11F85 14M17)
Sakellaridis, Yiannis (IL-TLAV)
On the unramified spectrum of spherical varieties over p-adic fields. (English summary)
Compos. Math. 144 (2008), no. 4, 9781016.
The notion of a distinguished representation acquires importance primarily because distinguished
representations often characterize images of Langlands functorial lifts. Questions related to distin-
guished representations include computing the multiplicity of the space of invariant linear forms,
investigating whether or not the period integral is factorizable, and how distinction affects arith-
metic invariants like L-functions. These and similar questions have been extensively studied in the
past two decades or so.
However, what is not very satisfactory is the lack of a general picture, with most of the work in
the area focusing on special cases. Of late, there have been some serious attempts to understand
the general picture in this context, especially the work of Sakellaridis and Sakellaridis-Venkatesh.
The essence of their work is to do harmonic analysis on spherical varieties over p-adic fields and
to understand this in terms of a dual group that appears in the geometric Langlands programme.
In the automorphic setting, these studies have found applications in the Gross-Prasad conjecture