On multiplication of double cosets for $\GL(\infty)$ over a finite field

Abstract

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.

arXiv comment7pp

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