We consider a group $GL(\infty)$ and its parabolic subgroup $B$ corresponding to partition $\infty=\infty+m+\infty$. Denote by $P$ the kernel of the natural homomorphism $B\to GL(m)$. We show that the space of double cosets of $GL(\infty)$ 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 $GL(\infty)$ over finite field.
The space $L^2$ on semi-infinite Grassmannian over finite field
The Ismagilov conjecture over a finite field ${\mathbb F}_p$
Groups $GL(\infty)$ over finite fields and multiplications of double cosets
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