Séminaire Lotharingien de Combinatoire, 84B.5 (2020), 12 pp.

Jimmy He

A Characteristic Map for the Symmetric Space of Symplectic Forms over a Finite Field

Abstract. The characteristic map for the symmetric group is an isomorphism relating the representation theory of the symmetric group to symmetric functions. An analogous isomorphism is constructed for the symmetric space of symplectic forms over a finite field, with the spherical functions being sent to Macdonald polynomials with parameters (q,q²). An analogue of parabolic induction is interpreted as a certain multiplication of symmetric functions. Applications are given to Schur-positivity of skew Macdonald polynomials with parameters (q,q²).

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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