I will report on a joint work in progress with Dan Ciubotaru,
David Kazhdan and Yakov Varshavsky. The first result generalizes a
result of Aubert, Ciubotaru and Romano providing a description of the
unipotent cocenter of a p-adic group in terms of the Langlands dual group.
Time permitting I will discuss the relation of this result to characters
of representations and trace of Frobenius of affine character sheaves.