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 ) ).