Lusztig Correspondence and Howe Correspondence for Finite Reductive Dual Pairs Article Swipe
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· 2019
· Open Access
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· DOI: https://doi.org/10.48550/arxiv.1906.01158
· OA: W2948975886
Let $(G,G')$ 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 $G'$. The Lusztig correspondence is a bijection between the Lusztig series indexed by the conjugacy class of a semisimple element $s$ in the connected component $(G^*)^0$ of the dual group of $G$ and the set of irreducible unipotent characters of the centralizer of $s$ in $G^*$. 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.
