Reductive group

A reductive algebraic group is a smooth connected affine algebraic group whose largest connected normal unipotent algebraic group is trivial.

A finite group of Lie type is obtained from rational points of a connected reductive algebraic group with a suitable Frobenius endomorphism of an algebraic group, often followed by central quotient or passage to a derived subgroup. Its complex representations combine finite-group character theory with the geometry of Borel subgroups, maximal algebraic tori, and the Weyl group.

For a rational parabolic subgroup $P=L⋉U$, use inflation of a group representation for a complex $L^F$-module across $P^F\to L^F$ and induce it to $G^F$. This is Harish-Chandra induction. Its adjoint, Harish-Chandra restriction, takes $U^F$-invariants. These are ordinary finite-group group representation functors and should not be confused with general Deligne-Lusztig induction from a nonrational Borel.

For rational parabolic subgroups $P=LU$ and $Q=MV$, define $\mathcal{S}(M,L)=\{x:M\cap {}^{x}L\text{ contains a maximal algebraic torus of }G\}$. Then ${}^{\ast }{R}_M^G{R}_L^G$ is the sum over $M^F\backslash \mathcal{S}(M,L{)}^F/L^F$ of $R_{M\cap {}^{x}L}^M{}^{\ast }{R}_{M\cap {}^{x}L}^{{}^{x}L}\mathrm{ad}x$, with the intersection parabolics understood. It follows from ordinary finite-group Mackey restriction formula, taking $V^F$-invariants, and the compatible-Levi intersection lemma for parabolics.

Restriction first views a $G^F$-module as a $P^F$-module, then takes the subspace fixed by the unipotent radical. The resulting $L^F$-module is adjoint to Harish-Chandra induction by ordinary Frobenius reciprocity. Over characteristic zero, averaging makes invariants exact and identifies them with coinvariants.

An irreducible complex $G^F$-module is cuspidal if its Harish-Chandra restriction is zero for every proper rational parabolic subgroup. By adjunction, it occurs in no Harish-Chandra induction from a proper rational Levi subgroup. This is a finite-group notion, distinct from a cuspidal automorphic representation.

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.

A Borel subgroup is a maximal closed connected solvable subgroup of an affine algebraic group.

A parabolic subgroup of a connected reductive group is a closed subgroup containing a Borel subgroup. Its quotient in the group is projective.

A Levi subgroup appears in a group version of a Levi decomposition. A Levi subgroup is a reductive complement to the unipotent radical of a parabolic subgroup.

For a connected reductive algebraic group $G$ and a Borel subgroup $B$, the quotient $G/B$ is its complete flag variety. Quotients $G/P$ by parabolic subgroups are partial flag varieties.

A root datum is a quadruple $(X^{\ast },\Phi ,X_{\ast },{\Phi }^{\vee })$ of dual lattices, roots, and coroots with the natural perfect pairing and reflection axioms. A reductive group with maximal torus $T$ has $X^{\ast }=X^{\ast }(T)$ and $X_{\ast }=X_{\ast }(T)$.

For a connected reductive algebraic group, a Borel subgroup $B$, and a maximal torus $T\subseteq B$, the Weyl group $W=N_G(T)/T$ indexes the double cosets:

The relative position of Borel subgroups $B_1,B_2$ is the Weyl group index of their simultaneous conjugacy orbit. After moving the first subgroup to a fixed $B$, the second has the form $gB{g}^{-1}$ and its index is the double coset $BgB$. This identifies the diagonal action on pairs of flags with the Bruhat decomposition.

This is the Bruhat decomposition associated with a BN-pair. The Bruhat decomposition is the disjoint unionassociated with a BN-pair.