Deligne-Lusztig theory

Deligne-Lusztig theory constructs virtual characters of finite groups of Lie type from compactly supported étale cohomology of varieties associated with Frobenius and flags. Rational maximal tori and their finite-group characters provide induction data, while the geometry explains character orthogonality and cuspidality.

For a rational maximal torus $T$, choose a Borel subgroup containing it with unipotent radical $U$. The alternating compactly supported étale cohomology of ${\mathcal{L}}_F^{-1}(U)$, in the appropriate $T^F$-isotypic component, defines the virtual character of $G^F$ $R_T^G(\theta )$. It is independent of the chosen Borel. When the Borel subgroup is rational, this is ordinary induction from its rational points after inflation of a group representation from $T^F$.

For a rational Levi subgroup $M$ of a rational parabolic subgroup and a rational maximal torus $T$, the torus case of the induction-restriction formula isIn particular it vanishes if there is no rational conjugate of $T$ in $M$. If $T\subset M$, transitivity gives $R_T^G=R_M^G{R}_T^M$. Together with adjunction these identities show that an irreducible character up to sign arising from $T$ is cuspidal exactly when $T$ is elliptic.

For a rational Borel subgroup $B=TU$, the principal series is ${\mathrm{Ind}}_{B^F}^{G^F}\tilde{\theta }$, with $\tilde{\theta }$ inflated from a linear character of $T^F$. The Mackey restriction formula and rational Bruhat decomposition give its self-inner-product as the size of the Weyl stabilizer of $\theta$. It is irreducible precisely when that stabilizer is trivial.

For $N=q-1$, write a diagonal-torus character as $\theta (\mathrm{diag}(a,b,(ab{)}^{-1}))=\chi (a{)}^r\chi (b{)}^s$, with $\chi$ a generator of the character group of ${\mathbb{F}}_q^{\times }$. Its Weyl stabilizer is trivial exactly when $r,s,r-s$ are all nonzero modulo $N$ and, if $3\mid N$, the unordered pair $\{r,s\}$ is not $\{N/3,2N/3\}$. The first conditions exclude transpositions; the exceptional pair excludes cyclic permutation of three characters modulo a common twist.

For a connected reductive algebraic group and a Weyl group element $w$, this locally closed subset of the flag variety consists of flags in relative position $w$ to their Frobenius image. For rational $B$ it is ${\mathcal{L}}_F^{-1}(B\dot{w}B)/B$, using the twisted right action identity of the Lang map. Its dimension is the length of $w$.

Choose rational $B=TU$, a representative $\dot{w}$, and $x$ with $x^{-1}F(x)=\dot{w}$. Set $T^{\prime }=xT{x}^{-1}$ and $U^{\prime }=F(xU{x}^{-1})$. Then $T^{\prime }$ is rational, andThe map sends $h$ to $hxB$. Quotienting only by the finite group $T^{\prime F}$ leaves fibres isomorphic to the positive-dimensional unipotent intersection when that intersection is nontrivial. For $w=1$ in ${\mathrm{SL}}_2$, this is an affine-line fibre over each rational flag, so the missing quotient cannot be ignored in a variety isomorphism.