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  "body": "\n## Example\n\nLet $G = \\mathrm{SL}_2(\\R)$, the [[group|Q0GBCB9RA4]] of real $2 \\times 2$ matrices of determinant $1$.\nIt is a connected simple Lie group that is not compact.\nSo each nontrivial [[irreducible|FX3NEFYFFN]] [[unitary representation|74BPYJNZTM]] of $G$ is infinite-dimensional.\nIndeed, a nontrivial unitary representation on $\\C^n$ has a differential that embeds $\\mathfrak{sl}_2(\\R)$ into $\\mathfrak{u}(n)$.\nEach subalgebra of $\\mathfrak{u}(n)$ has an inner product for which every $\\operatorname{ad} X$ is skew-symmetric.\nThe algebra $\\mathfrak{sl}_2(\\R)$ has none, since $\\operatorname{ad} X$ has real nonzero eigenvalues for $X = \\operatorname{diag}(1, -1)$.\n\nPrincipal series.\nFix $\\varepsilon \\in \\set{0, 1}$ and $\\nu \\in i\\R$.\nLet $H_{\\varepsilon, \\nu}$ be the space of functions $f$ on $\\R^2 \\setminus \\set{0}$ with $f(\\lambda x) = \\sgn(\\lambda)^{\\varepsilon} \\abs{\\lambda}^{-1-\\nu} f(x)$ for $\\lambda \\ne 0$.\nIts norm is the norm of $L^2$ on the unit circle.\nThe action is $(\\pi(g) f)(x) = f(g^{-1} x)$.\nHere $\\abs{f}^2$ is homogeneous of degree $-2$.\nFor such functions the integral over the circle is unchanged by linear maps of determinant $1$, so $\\pi$ is unitary.\n\nDiscrete series.\nFor an integer $k \\ge 2$, let $D_k^+$ be the space of holomorphic functions $f$ on the upper half plane $\\set{z = x + iy : y \u003e 0}$ with $\\int \\abs{f(z)}^2 y^{k-2} \\, dx \\, dy \u003c \\infty$.\nFor $g^{-1} = \\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix}$, the action is $(\\pi(g) f)(z) = (cz + d)^{-k} f\\big(\\frac{az + b}{cz + d}\\big)$.\nThis is an irreducible unitary representation whose matrix coefficients are square integrable on $G$.\n\n## Remarks\n\nBargmann classified the irreducible unitary representations of $G$ in 1947.\nUp to equivalence they are the following.\n\n1. The trivial representation.\n2. The irreducible principal series.\n3. The complementary series, a family with a real parameter in an interval.\n4. The discrete series $D_k^{\\pm}$ for $k \\ge 2$.\n5. The two limits of discrete series $D_1^{\\pm}$, the two pieces of the reducible principal series with $\\varepsilon = 1$ and $\\nu = 0$.\n\nHere $D_k^-$ is the complex conjugate of $D_k^+$, realized on antiholomorphic functions.\n\nThe Casimir element of $\\mathfrak{sl}_2(\\R)$ acts on each irreducible representation by a scalar, and this scalar is a main invariant in the classification.\nThese representations govern the spectral theory of the Laplacian on hyperbolic surfaces $\\Gamma \\backslash \\mathbb{H}$, and holomorphic cusp forms of weight $k$ give copies of the discrete series $D_k^{\\pm}$ in $L^2(\\Gamma \\backslash G)$.\n\n## Explanations\n\n### L1 human\n\nThe group $\\mathrm{SL}_2(\\R)$ moves the points of the hyperbolic plane.\nUnlike a compact group, it has no nontrivial length-preserving action on a finite-dimensional space.\nSo its interesting representations live on spaces of functions, such as functions on the circle or holomorphic functions on the upper half plane.\nBargmann found all of them, and they come in a few families.\n",
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