On multiplication of double cosets for (∞) over a finite field

الأصل · EN

We consider a group GL(∞) and its parabolic subgroup B corresponding to partition ∞=∞+m+∞. Denote by P the kernel of the natural homomorphism B→ GL(m). We show that the space of double cosets of GL(∞) by P admits a natural structure of a semigroup. In fact this semigroup acts in subspaces of P-fixed vectors of some unitary representations of GL(∞) over finite field.