Counting conjugacy classes in symmetric spaces over Fq

AbstractIn this paper we verify a prediction of the Langlands-Lusztig program in the special case of the algebra of double cosets (K(Fq)β GLn(Fq)K(Fq)), where K is the fixed point subgroup of an involution on GLn (see Conjecture 3.6 and Example 7 of [G]). We calculate the dimension of the algebra by computing the number of K(Fq)-conjugacy classes in the space GLn(Fq))K(Fq). We compare this dimension with the size of the set parametrizing representations of the double coset algebra, defined in terms of perverse sheaves on the flag variety. These numbers turn out to be the same