Spherical functions of the symmetric space 𝐺(𝔽_{𝕢²})/𝔾(𝔽_{𝕢})

We apply Lusztig’s theory of character sheaves to the problem of calculating the spherical functions of G ( F q 2 ) / G ( F q ) G(\mathbb {F}_{q^{2}})/G(\mathbb {F}_q) , where G G is a connected reductive algebraic group. We obtain the solution for generic spherical functions for any G G , and for all spherical functions when G = G L n G=GL_n . The proof includes a result about convolution of character sheaves and its interaction with the associated two-sided cells.