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  "body": "\n## Definition\n\nLet $G$ be a [[real reductive Lie group|42ZCE14AD2]] with [[maximal compact subgroup|T2EB8R9HEV]] $K$.\nLet $P$ be a minimal [[parabolic subgroup|Z2FEBW5MQM]] of $G$ with [[Langlands decomposition|BM3TWJ0D4P]] $P = MAN$, and let $\\mathfrak a$ be the Lie algebra of $A$.\nLet $\\rho \\in \\mathfrak a^*$ be half the sum of the roots of $\\mathfrak a$ in the Lie algebra of $N$, counted with multiplicity.\nFor $a \\in A$ and $\\lambda \\in \\mathfrak a^*_\\C$, write $a^\\lambda = e^{\\lambda(\\log a)}$.\nLet $\\sigma$ be an [[irreducible|FX3NEFYFFN]] unitary representation of $M$ on $V_\\sigma$, and let $\\nu \\in \\mathfrak a^*_\\C$.\nThe *principal series representation* $I(\\sigma, \\nu) = \\mathrm{Ind}_P^G(\\sigma \\otimes e^\\nu \\otimes 1)$ acts on the space of measurable functions $f \\colon G \\to V_\\sigma$ such that\n$$f(x m a n) = a^{-(\\nu + \\rho)} \\sigma(m)^{-1} f(x) \\qquad (x \\in G, \\ m \\in M, \\ a \\in A, \\ n \\in N)$$\nand $\\int_K \\norm{f(k)}^2 \\, dk \u003c \\infty$, by left translation $(I(\\sigma, \\nu)(g) f)(x) = f(g^{-1} x)$.\n\n## Remarks\n\nThe shift by $\\rho$ makes $I(\\sigma, \\nu)$ unitary when $\\nu \\in i \\mathfrak a^*$; these are the unitary principal series.\nEvery function in the space is determined by its restriction to $K$, because $G = KP$ by the Iwasawa decomposition.\n\nExample: for $G = \\mathrm{SL}_2(\\R)$, let $P$ be the upper triangular matrices, so $M = \\set{\\pm 1}$, $A$ is the positive diagonal matrices and $N$ the unipotent upper triangular ones.\nHere $\\rho(\\log \\operatorname{diag}(t, t^{-1})) = \\log t$, so $\\operatorname{diag}(t, t^{-1})^\\rho = t$.\nThe restriction to $K = \\mathrm{SO}(2)$ identifies the space with the functions on the circle that are even or odd according to $\\sigma(-1) = \\pm 1$.\nSo each character $e^{in\\theta}$ of $K$ with $(-1)^n = \\sigma(-1)$ occurs once, and $I(\\sigma, \\nu)$ is admissible.\n\nNon-example: inducing the trivial representation from the compact subgroup $K$ instead of from $P$ gives the representation of $G$ on $L^2(G/K)$.\nIt is not a principal series representation: $K$ is not parabolic, and the representation is not even admissible.\n\nSome authors call representations induced from any parabolic subgroup, with discrete series or tempered data on $M$, generalised principal series.\n",
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