Lusztig Correspondence and Howe Correspondence for Finite Reductive Dual Pairs

Abstract

Let $(G,G^{\prime })$ be a reductive dual pair of a symplectic group and an orthogonal group over a finite field of odd characteristic. The Howe correspondence establishes a correspondence between a subset of irreducible characters of $G$ and a subset of irreducible characters of $G^{\prime }$. The Lusztig correspondence is a bijection between the Lusztig series indexed by the conjugacy class of a semisimple element $s$ in the connected component $(G^{\ast }{)}^0$ of the dual group of $G$ and the set of irreducible unipotent characters of the centralizer of $s$ in $G^{\ast }$. In this paper, we prove the commutativity (up to a twist of the sign character) between these two correspondences. As a consequence, the Howe correspondence can be explicitly described in terms of Lusztig's parametrization for classical groups.

References in corpus (2)

• Howe Correspondence of Unipotent Characters for a Finite Symplectic/Even-orthogonal Dual Pair
• Remarks on the theta correspondence over finite fields

Cited by in corpus (9)

• Howe Correspondence of Unipotent Characters for a Finite Symplectic/Even-orthogonal Dual Pair
• On Theta and Eta Correspondences for Finite Symplectic/Orthogonal Dual Pairs
• On Theta Ranks of Irreducible Characters of a Finite Classical Group
• On Theta and Eta Correspondences for Finite Unitary Dual Pairs
• Decomposition of the Uniform Projection of the Weil Character
• On the Gan-Gross-Prasad problem for finite classical groups
• Degree Differences in the Eta Correspondences
• Correspondences between affine Hecke algebras and applications
• Lusztig correspondence and the Gan-Gross-Prasad problem