The $$\ell $$-modular local theta correspondence in type II and partial permutations

Abstract

Abstract In this paper we compute the multiplicities appearing in the $${\overline{\mathbb {F}}_\ell }$$ F ¯ ℓ -modular theta correspondence in type II over a non-archimedean field $$\textrm{F}$$ F , where $$\ell $$ ℓ is a prime not dividing the residue cardinality of $$\textrm{F}$$ F . Unlike for representations with complex coefficients, highly non-trivial multiplicities can emerge. We show that these multiplicities are precisely governed by the action of symmetric groups on the set of partial permutations, and the $${\overline{\mathbb {F}}_\ell }$$ F ¯ ℓ -representation of symmetric groups these give rise to. The problem is thus reduced to certain branching problems in the modular representation theory of symmetric groups. In particular, if d is the order of the residue cardinality of $$\textrm{F}$$ F in $${\overline{\mathbb {F}}_\ell }$$ F ¯ ℓ , and the rank of the involved general linear groups is bounded above by $$d\ell $$ d ℓ , the behavior of the theta correspondence can be predicted via explicit algorithms coming from Pieri’s Formula.