Ennola duality for decomposition of tensor products

Ennola duality relates the character table of the finite unitary group GU n ( 𝔽 q ) to that of GL n ( 𝔽 q ) where we replace q by - q (see [5] for the original observation and [21] for its proof). The aim of this paper is to investigate Ennola duality for the decomposition of tensor products of irreducible characters. It does not hold just by replacing q by - q . The main result of this paper is the construction of a family of two-variable polynomials 𝒯 μ ( u , q ) indexed by triples of partitions of n which interpolates between multiplicities in decompositions of tensor products of unipotent characters for GL n ( 𝔽 q ) and GU n ( 𝔽 q ) . We give a module theoretic interpretation of these polynomials and deduce that they have non-negative integer coefficients. We also deduce that the coefficient of the term of highest degree in u equals the corresponding Kronecker coefficient for the symmetric group and that the constant term in u give multiplicities in tensor products of generic irreducible characters of unipotent type (i.e., unipotent characters twisted by linear characters of GL 1 ( 𝔽 q ) ).