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  "body": "\n## Definition\n\nLet $p$ be a prime, let $q$ be a power of $p$, and let $\\overline{\\F}_q$ be an [[algebraic closure|0X6AB5P3G1]] of the [[finite field|0ZXZ82FTX2]] $\\F_q$.\nLet $\\mathbf{G}$ be a connected reductive group over $\\overline{\\F}_q$ with a Frobenius endomorphism $F$ for an $\\F_q$-structure.\nLet $G = \\mathbf{G}^F$ be the finite group of fixed points of $F$.\nLet $\\mathbf{T}$ be an $F$-stable maximal torus of $\\mathbf{G}$, and let $T = \\mathbf{T}^F$.\nLet $\\mathbf{B}$ be a Borel subgroup containing $\\mathbf{T}$, not necessarily $F$-stable, with unipotent radical $\\mathbf{U}$.\nThe Deligne–Lusztig variety is\n$$\\tilde{X} = \\set{g \\in \\mathbf{G} : g^{-1} F(g) \\in F(\\mathbf{U})}.$$\nThe group $G$ acts on $\\tilde{X}$ by left multiplication and $T = \\mathbf{T}^F$ acts by right multiplication.\nLet $\\ell \\ne p$ be a prime and let $\\theta \\colon T \\to \\overline{\\Q}_\\ell^\\times$ be a character.\nLet $H^i_c(\\tilde{X})_\\theta$ be the subspace of the $\\ell$-adic cohomology with compact support $H^i_c(\\tilde{X}, \\overline{\\Q}_\\ell)$ on which $T$ acts through $\\theta$.\nThe *Deligne–Lusztig virtual character* is the function on $G$\n$$R_{\\mathbf{T}}^\\theta(g) = \\sum_{i \\ge 0} (-1)^i \\tr\\big(g, H^i_c(\\tilde{X})_\\theta\\big).$$\n*Deligne–Lusztig theory* is the study of the representations of $G$ through the virtual characters $R_{\\mathbf{T}}^\\theta$.\n\n## Remarks\n\nThe virtual character $R_{\\mathbf{T}}^\\theta$ does not depend on the choice of $\\mathbf{B}$.\nEvery irreducible [[character|RMWMYBY8TC]] of $G$ occurs with nonzero multiplicity in some $R_{\\mathbf{T}}^\\theta$.\nIts degree is $R_{\\mathbf{T}}^\\theta(1) = \\pm [G : T]_{p'}$, the part of the index prime to $p$, up to a sign.\nExample: let $\\mathbf{G} = GL_2$ with $F$ raising matrix entries to the $q$-th power, let $\\mathbf{T}$ be the diagonal torus and $\\mathbf{B}$ the upper triangular matrices, both $F$-stable.\nThen $R_{\\mathbf{T}}^\\theta$ is induced from the upper triangular subgroup $B = \\mathbf{B}^F$ of $GL_2(\\F_q)$, by the character of $B$ that equals $\\theta$ on $T$ and is $1$ on $\\mathbf{U}^F$.\nFor $\\theta = 1$ it has degree $q + 1 = [G : B]$, and it is the sum of the trivial and the Steinberg character.\nSecond example: for $\\mathbf{G} = GL_2$ there is also an $F$-stable maximal torus $\\mathbf{T}$ with $T \\cong \\F_{q^2}^\\times$, contained in no $F$-stable Borel subgroup.\nFor a character $\\theta$ of $\\F_{q^2}^\\times$ with $\\theta \\ne \\theta^q$, the function $-R_{\\mathbf{T}}^\\theta$ is an irreducible cuspidal character of degree $q - 1$, which is not induced from any proper parabolic subgroup.\nThis shows why the cohomological construction is needed.\n",
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