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  "body": "\n## Summary\n\nA discrete series representation of a locally compact group $G$ is an irreducible unitary representation.\nIt occurs as a direct summand of the regular representation on $L^2(G)$.\nThese are the representations that contribute point masses to the Plancherel measure.\nLet $G$ be a connected semisimple Lie group with finite center.\nHarish-Chandra showed that $G$ has discrete series exactly when it has a compact Cartan subgroup, and he classified them.\nThey are the building blocks from which the [[Langlands classification|YHFP9HGBTT]] constructs all irreducible admissible representations.\n\n## Explanations\n\n### L1 human\n\nThe functions on a group can be broken into pieces on which the group acts irreducibly, much as a sound breaks into pure tones.\nFor the circle or the real line, all pieces come in continuous families, like frequencies.\nSome groups, such as $\\mathrm{SL}_2(\\R)$, also have isolated pieces that appear on their own, with positive \"weight\".\nThese isolated pieces are the discrete series.\nFor $\\mathrm{SL}_2(\\R)$, they are realised on spaces of holomorphic functions on the upper half-plane and are closely linked to modular forms.\n",
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