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  "body": "\n## Definition\n\nLet $\\mathfrak{g}$ be a complex semisimple Lie algebra.\n*Weyl's unitary trick* is the method of studying the finite-dimensional complex representations of $\\mathfrak{g}$ through the following steps:\n\n1. choose a compact real form $\\mathfrak{k}$ of $\\mathfrak{g}$, that is, a real Lie subalgebra with $\\mathfrak{g} = \\mathfrak{k} \\oplus i\\mathfrak{k}$ whose Killing form is negative definite;\n2. restrict from $\\mathfrak{g}$ to $\\mathfrak{k}$, a bijection from complex representations of $\\mathfrak{g}$ to representations of $\\mathfrak{k}$ on complex spaces that keeps invariant subspaces;\n3. pass to the simply connected Lie group $K$ with Lie algebra $\\mathfrak{k}$, a [[compact Lie group|9TFC6ADC1C]] to whose [[representations|WYGRXCG2RN]] those of $\\mathfrak{k}$ integrate;\n4. prove the desired property for representations of the compact group $K$, for example by averaging over its Haar measure, and transfer it back to $\\mathfrak{g}$.\n\n## Remarks\n\nExample: for $\\mathfrak{g} = \\mathfrak{sl}(2, \\C)$, one takes $\\mathfrak{k} = \\mathfrak{su}(2)$ and $K = SU(2)$.\nComplete reducibility of representations of $SU(2)$, proved by averaging an inner product, gives complete reducibility of finite-dimensional representations of $\\mathfrak{sl}(2, \\C)$.\nNon-example: the real form $\\mathfrak{sl}(2, \\R)$ of $\\mathfrak{sl}(2,\\C)$ cannot replace $\\mathfrak{k}$ in step 3, because every connected Lie group with Lie algebra $\\mathfrak{sl}(2,\\R)$, such as $SL(2,\\R)$, is not compact.\nThe same method applies to a connected complex semisimple Lie group $G$ with a compact real form $K$, using holomorphic representations of $G$.\nIn step 3, the integrated representation of $K$ has the same invariant subspaces as the representation of $\\mathfrak{k}$.\nThe compactness of $K$ in step 3 is a theorem of Weyl.\nIt says that a connected Lie group whose Lie algebra has a negative definite Killing form is compact, and so is its universal cover.\n",
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