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  "body": "\n## Summary\n\nThe Harish-Chandra character extends the [[character of a representation|RMWMYBY8TC]] to infinite-dimensional representations of reductive groups.\nThe operators $\\pi(g)$ have no trace, but their averages $\\int f(g) \\pi(g) \\, dg$ against smooth compactly supported functions $f$ do.\nThe map $f \\mapsto \\operatorname{tr} \\pi(f)$ is a distribution on the group, the character of $\\pi$.\nIt determines an irreducible admissible representation up to infinitesimal equivalence, and it is the basic tool of harmonic analysis on reductive groups.\n\n## Explanations\n\n### L1 human\n\nFor a finite group, the character of a representation records the trace of each group element and tells the representation apart from all others.\nFor a group like $\\mathrm{SL}_2(\\R)$ acting on an infinite-dimensional space, the trace of a single element is usually infinite.\nThe trick is to smear the group element out with a smooth bump function before taking the trace.\nThe result is no longer a function on the group but a \"generalised function\", and it still remembers the representation.\n",
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