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Reductive dual pair
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<p>In the mathematical field of <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a>, a reductive dual pair is a pair of <a href="/facts/Subgroups/r91mBgpa">subgroups</a> (<i>G</i>, <i>G</i>′) of the <a href="/facts/Symplectic_group/G4CKBc7z">isometry group</a> Sp(<i>W</i>) of a <a href="/facts/Symplectic_vector_space/o95tWKBc">symplectic vector space</a> <i>W</i>, such that <i>G</i> is the <a href="/facts/Centralizer/AKhrEUGR">centralizer</a> of <i>G</i>′ in Sp(<i>W</i>) and vice versa, and these groups act <a href="/facts/Completely_reducible/Wz7E1GbT">reductively</a> on <i>W</i>. Somewhat more loosely, one speaks of a dual pair whenever two groups are the mutual centralizers in a larger group, which is frequently a <a href="/facts/General_linear_group/B54gNILS">general linear group</a>. The concept was introduced by <a href="/facts/Roger_Evans_Howe/DQzbE2YJ">Roger Howe</a> in Howe (1979). Its strong ties with <a href="/facts/The_Classical_Groups/8z6UvA42">Classical Invariant Theory</a> are discussed in Howe (1989a). </p>