Deligne-Lusztig characters

$ \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Irr}{Irr} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\SO}{SO} \DeclareMathOperator{\Sp}{Sp} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\GU}{GU} \DeclareMathOperator{\PSL}{PSL} \DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\Cent}{Cent} \DeclareMathOperator{\inn}{inn} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\E}{E} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\diag}{diag} \newcommand{\Ql}{\overline{\mathbb{Q}_{\ell}}} \newcommand{\gmod}[1]{#1\text{-}\mbf{mod}} \newcommand{\Fq}[1]{\mathbb{F}_q\text{-rank}(#1)} \newcommand{\mbf}[1]{\mathbf{#1}} \newcommand{\ss}{\mathrm{ss}} $ A Deligne–Lusztig Character is a virtual character of a finite reductive group, (here we take virtual character to mean a $\mathbb{Z}$-linear combination of irreducible characters). Such a group arises in the following way. Let $\mbf{G}$ denote a connected reductive algebraic group defined over an algebraic closure $\mathbb{K} = \overline{\mathbb{F}_p}$ of the finite field $\mathbb{F}_p$ of prime characteristic $p > 0$. Furthermore let $F$ be a generalised Frobenius endomorphism of $\mbf{G}$ then we say the fixed point group $G = G = \{g \in \mbf{G} \mid F(g) = g\}$ is a finite reductive group. Note that a generalised Frobenius endomorphism of $\mbf{G}$ is an endomorphism $F$ such that $F^n$ is a Frobenius endomorphism for some $n \in \mathbb{N}$. Let us take $n$ to be minimal such that $F^n$ is a Frobenius endomorphism endowing $\mbf{G}$ with an $\mathbb{F}_{p^r}$-rational structure. Throughout we will denote by $q$ the value $p^{r/n}$, which is in general a rational number but is an integer when $F$ is a Frobenius endomorphism. Note that more generally we say a finite group is a finite group of Lie type if it can be obtained as a central quotient of a finite reductive group.

Example: We can take $\mbf{G}$ to be the special linear group $\SL_n(\mathbb{K})$ and $F$ to be the Frobenius endomorphism given by $F(x_{ij}) = (x_{ij}^q)$ where $q = p^a$ for some $a \in \mathbb{N}$. The resulting finite reductive group $G$ is the finite special linear group $\SL_n(q) = \SL_n(\mathbb{F}_q)$ defined over the finite field of order $q$. The quotient group $\PSL_n(q) = \SL_n(q)/Z(\SL_n(q))$, where $Z(\SL_n(q))$ is the centre of $\SL_n(q)$, is a finite group of Lie type called the projective special linear group. This group is a finite simple group except in the following cases:

• $n = 2$ and $q = 2$ in which case we have $\PSL_2(2)$ is isomorphic to the symmetric group $\mathfrak{S}_3$,
• $n = 2$ and $q = 3$ in which case we have $\PSL_2(3)$ is isomorphic to the alternating group $\mathfrak{A}_4$.

Warning: The group $\PSL_n(q)$ cannot in general be realised as the fixed point group of a connected reductive algebraic group under a generalised Frobenius endomorphism. This is due to the fact that the centre of $\SL_n(q)$ is disconnected, (see [DM91, Example 3.14]).

Basic Construction

The Deligne–Lusztig Bimodule

The key idea of Deligne and Lusztig is to construct for each $F$-stable Levi subgroup $\mbf{L} \subset \mbf{P}$ of a parabolic subgroup $\mbf{P} \subseteq \mbf{G}$ a family of bimodules satisfying certain nice properties. Note that by $F$-stable we mean $F(\mbf{L}) = \mbf{L}$, (we do not make this assumption on $\mbf{P}$). They do this in the following way. Let $\mbf{U}$ be the unipotent radical of $\mbf{P}$ so that we have a semidirect product decomposition $\mbf{P} = \mbf{U} \rtimes \mbf{L}$, (note in particular we have $\mbf{L}$ is contained in the normaliser of $\mbf{U}$ in $\mbf{G}$). We denote by $\mathscr{L} : \mbf{G} \to \mbf{G}$ the Lang-Steinberg map which is defined by $\mathscr{L}(g) = g^{-1}F(g)$ for all $g \in \mbf{G}$. The preimage $\mathscr{L}^{-1}(\mbf{U}) = \{g \in \mbf{G} \mid g^{-1}F(g) \in \mbf{U}\}$ of $\mbf{U}$ under $\mathscr{L}$ is an affine algebraic set as it is a closed subset of $\mbf{G}$. As such this is an affine variety and the direct product $\mbf{G}^F \times \mbf{L}^F$ acts on $\mathscr{L}^{-1}(\mbf{U})$ on the left as a finite group of automorphisms under the action $(g,l)\cdot x = gxl^{-1}$. This follows from the fact that for $g \in \mbf{G}^F$, $l \in \mbf{L}^F$ and for all $x \in \mathscr{L}^{-1}(\mbf{U})$ we have \begin{align*} \mathscr{L}(gx) &= x^{-1}g^{-1}F(g)F(x) = x^{-1}F(x) \in \mbf{U},\\ \mathscr{L}(xl) &= l^{-1}x^{-1}F(x)F(l) = l^{-1}x^{-1}F(x)l \in l^{-1}\mbf{U}l = \mbf{U}. \end{align*}

What we would now like to do is obtain a vector space from the variety $\mathscr{L}^{-1}(\mbf{U})$ which inherits an action of the direct product $\mbf{G}^F \times \mbf{L}^F$, we do this by passing to cohomology. Let $\Ql$ denote an algebraic closure of the field of $\ell$-adic numbers where $\ell$ is a prime number distinct from $p$. For $i \in \mathbb{Z}$ we can construct the $\ell$-adic cohomology group with compact support, which we denote by $H_c^i(\mathscr{L}^{-1}(\mbf{U}),\overline{\mathbb{Q}}_{\ell})$. It is a non-obvious result that each element $(g,l) \in \mbf{G}^F \times \mbf{L}^F$ induces a linear endomorphism of the cohomology group [DL76, Proposition 10.2]. In particular the cohomology group inherits a bimodule structure for the group algebras $\Ql \mbf{G}^F$, (acting on the left), and $\Ql\mbf{L}^F$, (acting on the right). We call the family of bimodules $H_c^{\bullet}(\mathscr{L}^{-1}(\mbf{U}),\overline{\mathbb{Q}}_{\ell})$ the Deligne–Lusztig bimodules associated to $\mbf{L}$ and $\mbf{P}$.

Notation: Let $H$ and $K$ be finite groups then we will sloppily write $H$-module, module-$H$ and $H$-module-$K$ to mean respectively left $\Ql H$-module, right $\Ql H$-module and $(\Ql H,\Ql K)$-bimodule.

Deligne–Lusztig Induction and Restriction

of Modules

The Deligne–Lusztig bimodule allows one to construct from each $\mbf{L}^F$-module a new $\mbf{G}^F$-module in the following way. Assume $M$ is an $\mbf{L}^F$-module then we define a virtual left $\mbf{G}^F$-module \begin{equation*} R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}}(M) = \sum_{i \in \mathbb{Z}} (-1)^iH_c^i(\mathscr{L}^{-1}(\mbf{U}),\overline{\mathbb{Q}}_{\ell}) \otimes_{\Ql\mbf{L}^F} M. \end{equation*} We call the map $R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}}$ a Deligne–Lusztig induction map associated to $\mbf{L}$ and $\mbf{P}$. Note that formally we define a virtual module to be a $\mathbb{Z}$-linear combination of simple modules. One can also obtain a map in the other direction. We take $\mathscr{L}^{-1}(\mbf{U})^{\vee}$ to be the variety $\mathscr{L}^{-1}(\mbf{U})$ endowed with the right action of $\mbf{L}^F \times \mbf{G}^F$ given by $x\cdot(l,g) = g^{-1}xl$. We now obtain that the cohomology groups $H_c^i(\mathscr{L}^{-1}(\mbf{U})^{\vee},\overline{\mathbb{Q}}_{\ell})$ are $\mbf{L}^F$-module-$\mbf{G}^F$ bimodules and for any $\mbf{G}^F$-module $N$ we define a virtual left $\mbf{L}^F$-module \begin{equation*} {}^*R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}}(N) = \sum_{i \in \mathbb{Z}} (-1)^iH_c^i(\mathscr{L}^{-1}(\mbf{U})^{\vee},\overline{\mathbb{Q}}_{\ell}) \otimes_{\Ql G} N. \end{equation*} We call the map ${}^*R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}}$ a Deligne–Lusztig restriction map.

of Characters

Notation: If $H$ is any finite group we denote by $\Cent(H)$ the $\Ql$-vector space of all class functions on $H$, (i.e. all functions $f : H \to \Ql$ such that $f(ghg^{-1}) = f(h)$ for all $g$, $h \in H$). Denote by $\Irr(H) \subset \Cent(H)$ the set of all irreducible characters of $H$ obtained from irreducible $\Ql$-representations of $H$. We then denote by $\mathbb{Z}\Irr(H) \subset \Cent(H)$ the $\mathbb{Z}$-span of $\Irr(H)$ in $\Cent(H)$, (note that this can be identified with the Grothendieck group of the category of $\Ql H$-modules).

One can now formulate the above process in terms of characters, indeed it is possible to express the virtual character of the virtual modules $R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}}(M)$ and ${}^*R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}}(N)$ in terms of the characters of $M$ and $N$. Before getting on to the specific expression of these virtual characters we will need the following notion. Let $\mbf{X}$ be any affine variety and $H$ a finite group which acts on $\mbf{X}$ as a group of automorphisms. We define \begin{equation*} \mathcal{L}(h,\mbf{X}) = \sum_{i\geqslant0}(-1)^i\tr(h,H_c^i(\mbf{X},\overline{\mathbb{Q}}_{\ell})) \end{equation*} for any $h \in H$ to be the Lefschetz number of $h$ acting on $\mbf{X}$. With this in place we may now describe the virtual characters as follows, (see [DM91, Proposition 11.2]), \begin{align*} R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}}(\chi_M)(g) &= \frac{1}{|\mbf{L}^F|}\sum_{l \in \mbf{L}^F} \chi_M(l^{-1})\mathcal{L}((g,l),\mathscr{L}^{-1}(\mbf{U}))\\ {}^*R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}}(\chi_N)(l) &= \frac{1}{|G|}\sum_{g \in \mbf{g}^F} \chi_N(g^{-1})\mathcal{L}((l,g),\mathscr{L}^{-1}(\mbf{U})^{\vee}) \end{align*} where here $\chi_M$ and $\chi_N$ denote the characters of the modules in the previous section. Extending by linearity we obtain $\mathbb{Z}$-linear maps \begin{align*} R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}} &: \mathbb{Z}\Irr(\mbf{L}^F) \to \mathbb{Z}\Irr(\mbf{G}^F)\\ {}^*R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}} &: \mathbb{Z}\Irr(\mbf{G}^F) \to \mathbb{Z}\Irr(\mbf{L}^F). \end{align*} which are also referred to as Deligne–Lusztig induction and restriction.

The Case of a Torus

Assume now that $\mbf{L} = \mbf{T}$ is a maximal torus of $\mbf{G}$ and $\mbf{P} = \mbf{B}$ is a Borel subgroup of $\mbf{G}$ containing $\mbf{T}$. Assume $\theta \in \Irr(\mbf{T}^F)$ is an irreducible character of the finite abelian group $\mbf{T}^F$ then we call the corresponding virtual character $R_{\mbf{T} \subset \mbf{B}}^{\mbf{G}}(\theta)$ a Deligne–Lusztig character of $G$. We will see below that $R_{\mbf{T} \subset \mbf{B}}^{\mbf{G}}(\theta)$ depends only on $\mbf{T}$ and $\theta$ and not on the choice of Borel subgroup containing $\mbf{T}$, therefore this character is often denoted simply by $R_{\mbf{T}}^{\mbf{G}}(\theta)$.

Remark: The definition of $R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}}$ for the case when $\mbf{L}$ is a maximal torus and $\mbf{P}$ is a Borel subgroup was given in [DL76, §1.20]. The construction was later generalised in [Lus76, §1] to the case of an arbitrary Levi and parabolic subgroup.

The Mackey Formula

Notation: Assume $H$ is a finite group and $K \leqslant H$ is a subgroup of $H$. Given $g \in H$ we denote by ${}^gK$ the conjugate subgroup $gKg^{-1} = \{gkg^{-1} \mid k \in K\}$ and by $(\inn g)_K : \Cent(K) \to \Cent({}^gK)$ the map given by $(\inn g)_K(\psi)(k) = \psi(g^{-1}kg)$ for all $k \in {}^gK$. We will also denote the character $(\inn g)_K(\psi)$ by ${}^g\psi$.

In the case of induction and restriction maps for characters of finite groups one has the Mackey formula which describes the composition of these two maps in terms of similar maps of smaller subgroups. Let us also assume now that $\mbf{M} \subset \mbf{Q}$ is an $F$-stable Levi subgroup of a parabolic subgroup $\mbf{Q} \subseteq \mbf{G}$. We denote by $\mathcal{S}(\mbf{L},\mbf{M})$ the set of all $g \in \mbf{G}$ such that $\mbf{L}\cap{}^g\mbf{M}$ contains a maximal torus of $\mbf{G}$. We may the consider the double cosets $\mbf{L}^F\backslash\mathcal{S}(\mbf{L},\mbf{M})^F/\mbf{M}^F = \{\mbf{L}^Fx\mbf{M}^F \mid x \in \mathcal{S}(\mbf{L},\mbf{M})^F\}$ and we denote by $\mathcal{X}(\mbf{L},\mbf{M})^F \subseteq \mathcal{S}(\mbf{L},\mbf{M})^F$ a minimal set of coset representatives such that $\mbf{L}^F\backslash\mathcal{X}(\mbf{L},\mbf{M})^F/\mbf{M}^F = \mbf{L}^F\backslash\mathcal{S}(\mbf{L},\mbf{M})^F/\mbf{M}^F$. The following equality \begin{equation*} {}^*R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}} \circ R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}} = \sum_{g \in \mathcal{X}(\mbf{L},\mbf{M})^F} R_{\mbf{L}\cap{}^g\mbf{M} \subset \mbf{L} \cap {}^g\mbf{Q}}^{\mbf{L}}\circ {}^*R_{\mbf{L}\cap{}^g\mbf{M} \subset \mbf{P} \cap {}^g\mbf{M}}^{{}^g\mbf{M}}\circ(\inn g)_{\mbf{M}^F} \end{equation*} is called the Mackey formula for Deligne–Lusztig induction and restriction. It was first stated by Deligne and published in [LS79, Lemma 2.5]. It is known to hold if one of the following holds:

• $\mbf{P}$ and $\mbf{Q}$ are $F$-stable, (this is due to Deligne - see [LS79, Theorem 2.5]).
• $\mbf{L}$ or $\mbf{M}$ is a maximal torus, (this is due to Deligne and Lusztig - see [DL83, Theorem 7]).
• $F$ is a Frobenius endomorphism and $q > 2$.
• $F$ is a Frobenius endomorphism and $\mbf{G}$ does not contain an $F$-stable quasi-simple component of type ${}^2\E_6$, $\E_7$ or $\E_8$.

The latter two cases are obtained through a detailed case by case analysis, which is recent work of Bonnafé and Michel [BM11].

Remark: It is conjectured that the Mackey formula holds without any restriction.

Consequences of the Mackey Formula

Independence of the Parabolic

When the Mackey formula holds then it is known that $R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}}$ is independent of the parabolic subgroup containing $\mbf{L}$, (see for instance the argument in [DM91, Proposition 6.1]). In particular when $\mbf{L} = \mbf{T}$ is a torus we have $R_{\mbf{T} \subset \mbf{B}}^{\mbf{G}}$ is independent of the Borel subgroup containing $\mbf{T}$ so we may write this simply as $R_{\mbf{T}}^{\mbf{G}}$.

The Inner Product Formula

Notation: We assume fixed once and for all an involutary automorphism $\overline{\mathbb{Q}}_{\ell} \to \overline{\mathbb{Q}}_{\ell}$, denoted $x \mapsto \overline{x}$, such that $\overline{\omega} = \omega^{-1}$ for every root of unity $\omega \in \overline{\mathbb{Q}}_{\ell}^{\times}$. Assume $H$ is a finite group then vector space $\Cent(H)$ is an inner product space with respect to the product $\langle-,-\rangle_H : \Cent(H) \times \Cent(H) \to \overline{\mathbb{Q}}_{\ell}$ given by \begin{equation*} \langle f, f' \rangle_H = \frac{1}{|H|}\sum_{h \in H}f(h)\overline{f'(h)}, \end{equation*} for all $f,f' \in \Cent(H)$. Note that $\Irr(H)$ then forms an orthonormal basis of $\Cent(H)$ with respect to this product. We will often write $\langle - , - \rangle$ without the subscript if there is no confusion over the underlying finite group.

Assume now that $\mbf{T}$ and $\mbf{T}'$ are two $F$-stable maximal tori of $\mbf{G}$ then the inner product of $R_{\mbf{T}_1}^{\mbf{G}}(\theta)$ and $R_{\mbf{T}_2}^{\mbf{G}}(\theta')$ for any $\theta \in \Irr(\mbf{T}^F)$ and $\theta' \in \Irr(\mbf{T}'^F)$ is given by \begin{equation*} \langle R_{\mbf{T}}^{\mbf{G}}(\theta), R_{\mbf{T}'}^{\mbf{G}}(\theta') \rangle = \frac{1}{|\mbf{T}^F|} |\{n \in G \mid {}^n\mbf{T} = \mbf{T}'\text{ and }{}^n\theta = \theta'\}|. \end{equation*} This is obtained simply by rewriting the Mackey formula and using the fact that Frobenius reciprocity holds for Deligne–Lusztig induction and restriction. In other words given any $F$-stable Levi subgroup $\mbf{L} \subset \mbf{P}$ of a parabolic subgroup $\mbf{P} \subseteq \mbf{G}$ we have \begin{equation*} \langle R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}}(\varphi), \chi \rangle_{\mbf{G}^F} = \langle \varphi, {}^*R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}}(\chi)\rangle_{\mbf{L}^F} \end{equation*} for all $\varphi \in \Cent(\mbf{L}^F)$ and $\chi \in \Cent(\mbf{G}^F)$.

Remark: The inner product formula was first proved by Deligne and Lusztig in [DL76, Theorem 6.8]. From this they also obtained that the the map $R_{\mbf{T} \subset \mbf{B}}^{\mbf{G}}$ was independent of $\mbf{B}$.

Deligne–Lusztig Characters

Parameterisation

Notation: Let $\nabla(\mbf{G},F)$ denote the set of all pairs $(\mbf{T},\theta)$ such that $\mbf{T}$ is an $F$-stable maximal torus of $\mbf{G}$ and $\theta \in \Irr(\mbf{T}^F)$. We say two pairs $(\mbf{T}',\theta')$, $(\mbf{T},\theta) \in \nabla(\mbf{G},F)$ are rationally conjugate if there exists an element $x \in G$ such that $\mbf{T}' = {}^x\mbf{T}$, (where ${}^x\mbf{T}$), and $\theta' = {}^x\theta$. This defines an equivalence relation $\sim_{G}$ on $\nabla(\mbf{G},F)$ and we denote the set of all equivalence classes by $\nabla(\mbf{G},F)/G$.

Proposition: We claim that the Deligne–Lusztig characters are parameterised by the equivalence classes $\nabla(\mbf{G},F)/G$. In other words we have $R_{\mbf{T}}^{\mbf{G}}(\theta) = R_{\mbf{T}'}^{\mbf{G}}(\theta')$ if and only if $(\mbf{T},\theta) \sim_{G} (\mbf{T}',\theta')$.

$(\Leftarrow)$: Let $x \in G$ and consider the rational conjugate $({}^x\mbf{T},{}^x\theta) \in \nabla(\mbf{G},F)$ of the pair $(\mbf{T},\theta) \in \nabla(\mbf{G},F)$. Let $\mbf{B}$ be a Borel subgroup of $\mbf{G}$ containing $\mbf{T}$ with unipotent radical $\mbf{U}$ then ${}^x\mbf{B}$ is a Borel subgroup of $\mbf{G}$ containing ${}^x\mbf{T}$ with unipotent radical ${}^x\mbf{U}$, (here we have used the fact that all Borel subgroups are conjugate [Hum75, §21.3 - Theorem]). Computing the corresponding Deligne–Lusztig character at an arbitrary element $g \in G$ we have \begin{align*} R_{{}^x\mbf{T} \subset {}^x\mbf{B}}^{\mbf{G}}({}^x\theta)(g) &= \frac{1}{|({}^x\mbf{T})^F|}\sum_{t \in ({}^x\mbf{T})^F} {}^x\theta(t^{-1})\mathcal{L}((g,t),\mathscr{L}^{-1}({}^x\mbf{U})),\\ &= \frac{1}{|\mbf{T}^F|}\sum_{t \in \mbf{T}^F}{}^x\theta({}^xt^{-1})\mathcal{L}((g,{}^xt),\mathscr{L}^{-1}({}^x\mbf{U})),\\ &= \frac{1}{|\mbf{T}^F|}\sum_{t \in \mbf{T}^F}\theta(t^{-1})\mathcal{L}((g,t),\mathscr{L}^{-1}(\mbf{U})), \end{align*} which is $R_{\mbf{T} \subset \mbf{B}}^{\mbf{G}}(\theta)(g)$. Note that the second equality holds because ${}^x(\mbf{T}^F) = ({}^x\mbf{T})^F$ and conjugate subgroups have the same order. The map $\varphi : \mathscr{L}^{-1}(\mbf{U}) \to \mathscr{L}^{-1}({}^x\mbf{U})$ given by $\varphi(h) = hx^{-1}$ is a bijective morphism of varieties satisfying $\varphi((g,t) \cdot h) = (g,{}^xt)\cdot \varphi(h)$ for all $(g,t) \in \mbf{G}^F \times \mbf{T}^F$, hence the third equality holds by a property of $\ell$-adic cohomology, (see [Car93, Property 7.1.5]). As the Deligne–Lusztig characters don't depend upon the choice of Borel subgroup we have the desired result.

$(\Rightarrow)$: If $R_{\mbf{T}}^{\mbf{G}}(\theta) = R_{\mbf{T}'}^{\mbf{G}}(\theta')$ then the inner product $\langle R_{\mbf{T}}^{\mbf{G}}(\theta), R_{\mbf{T}'}^{\mbf{G}}(\theta') \rangle_{G}$ is non-zero but by the inner product formula we must have $(\mbf{T},\theta) \sim_{G} (\mbf{T}',\theta')$ as required.

Conjugacy Classes of Maximal Tori

Notation: Let $H$ be a group and $\varphi : H \to H$ a homomorphism. We say two elements $x$, $y \in H$ are $\varphi$-conjugate if there exists $h \in H$ such that $x = h^{-1}y\varphi(h)$. This forms an equivalence relation on $H$ and we call the equivalence classes the $\varphi$-conjugacy classes of $H$. We denote the set of all $\varphi$-conjugacy classes by $H^1(\varphi,H)$.

Let $\mathfrak{T}$ denote the set of all $F$-stable maximal tori of $\mbf{G}$. As all maximal tori in $\mbf{G}$ are conjugate we have $\mbf{G}$ acts transitively on the set $\mathfrak{T}$ by conjugation. However the finite group $G$ also acts on $\mathfrak{T}$ by conjugation and we would like to understand how $\mathfrak{T}$ splits into $G$-orbits. In particular this would give us a greater understanding of the set $\nabla(\mbf{G},F)/G$.

Let us fix now an element $\mbf{T}_0 \in \mathfrak{T}$ then a maximal torus $\mbf{T}$ of $\mbf{G}$ is contained in $\mathfrak{T}$ if and only if there exists $g \in \mbf{G}$ such that $\mbf{T} = {}^g\mbf{T}_0$ and $\mathscr{L}(g) \in N_{\mbf{G}}(\mbf{T}_0)$, (here $N_{\mbf{G}}(\mbf{T}_0)$ denotes the normaliser of $\mbf{T}_0$ in $\mbf{G}$). The element $\mathscr{L}(g)$ naturally projects onto an element of the Weyl group $W(\mbf{T}_0) = N_{\mbf{G}}(\mbf{T}_0)/\mbf{T}_0$ of $\mbf{G}$, defined with respect to $\mbf{T}_0$, hence we have a map $\mathfrak{T} \to W(\mbf{T}_0)$. Note that as $\mbf{T}_0$ is $F$-stable we will obtain an induced action of $F$ on $W(\mbf{T}_0)$, which we again denote by $F$. We now have the following result describing the $G$-conjugacy classes

Lemma (see [Car95, Proposition 3.3.3] or [Gec03, §4.3.7]): The map $\mathfrak{T} \to W(\mbf{T}_0)$ determines a bijection $\mathfrak{T}/G \to H^1(F,W(\mbf{T}_0))$ where $\mathfrak{T}/G$ denotes the set of $G$-conjugacy classes of $F$-stable maximal tori of $\mbf{G}$.

Let $w \in W(\mbf{T}_0)$ be an element of the Weyl group and $\dot{w} \in N_{\mbf{G}}(\mbf{T}_0)$ be a representative in the normaliser. We denote by $\mbf{T}_w$ an $F$-stable maximal torus of $\mbf{G}$ such that $\mbf{T}_w = g\mbf{T}_0g^{-1}$ for some $g \in \mbf{G}$ satisfying $\mathscr{L}(g) = \dot{w}$. We say $\mbf{T}_w$ is an $F$-stable maximal torus obtained from $\mbf{T}_0$ by twisting with $w$. The corresponding fixed point group under $F$ is given by \begin{align*} \mbf{T}_w^F &= \{{}^gt \mid t \in \mbf{T}_0\text{ and } F({}^gt) = {}^gt\},\\ &= \{{}^gt \mid t \in \mbf{T}_0\text{ and } F(t) = t^{\dot{w}}\} \end{align*} because $F(g) = g\dot{w}$. Hence, using the morphism $t \mapsto g^{-1}tg$, we may identify $\mbf{T}_w^F$ with the subgroup of $\mbf{T}_0$ consisting of all elements satisfying $F(t) = t^{\dot{w}}$. Note that this description of $\mbf{T}_w^F$ does not depend upon the choice of representative $\dot{w}$ as $\mbf{T}_0$ is abelian.

Degree

Recall that if $H$ is a finite group and $\chi \in \Cent(H)$ is a class function then we call $\chi(1)$ the degree of $\chi$, (where here $1$ denotes the identity element in $H$). Before we can describe the degree of a Deligne–Lusztig character must first introduce the notion of $\mathbb{F}_q$-rank. For this we need the following notions:

• Assume that $\mbf{S}$ is an $F$-stable torus of $\mbf{G}$ then $F$ acts on the character group $X(\mbf{S}) = \Hom(\mbf{S},\mathbb{K}^{\times})$ of $\mbf{S}$ by $F(\chi) = \chi\circ F$. We say $\mbf{S}$ is a split torus if there exists a power $r = p^b$ of $p$ such that $F(\chi) = r\chi$ for all $\chi \in X(\mbf{S})$.
• Assume $\mbf{T}$ is an $F$-stable maximal torus of $\mbf{G}$ then $\mbf{T}$ contains a maximal split subtorus $\mbf{S} \leqslant \mbf{T}$, (see [DM91, Proposition 8.2]). We define the $\mathbb{F}_q$-rank of $\mbf{T}$ to be the dimension of $\mbf{S}$ which we denote by $\Fq{\mbf{T}}$.
• We say a torus $\mbf{T}$ of $\mbf{G}$ is maximally-split if it is an $F$-stable maximal torus contained in an $F$-stable Borel subgroup of $\mbf{G}$.
• We define the $\mathbb{F}_q$-rank of $\mbf{G}$ to be the $\mathbb{F}_q$-rank of a maximally-split torus of $\mbf{G}$, we denote this by $\Fq{\mbf{G}}$.

Let $\mathfrak{T}$ denote the set of all $F$-stable maximal tori of $\mbf{G}$ then the subset of $\mathfrak{T}$ consisting of all maximally-split tori forms a single orbit under the conjugation action of $G$, (see for instance [Gec03, Example 4.3.3]). It is an easy consequence of this that the $\mathbb{F}_q$-rank of $\mbf{G}$ does not depend upon the choice of maximally-split torus used to define it.

With the notion of $\mathbb{F}_q$-rank in place we can now state the following result concerning the degree of a Deligne–Lusztig character. Assume $\mbf{H}$ is a connected reductive algebraic group defined over $\mathbb{K}$ equipped with a generalised Frobenius endomorphism $F'$. We denote by $e_{\mbf{H}}$ the sign $(-1)^{\Fq{\mbf{H}}}$ where $\Fq(\mbf{H})$ is the $\mathbf{F}_q$-rank of $\mbf{H}$ with respect to the Frobenius endomorphism $F'$. For any $(\mbf{T},\theta) \in \nabla(\mbf{G},F)$ we have \begin{equation*} R_{\mbf{T}}^{\mbf{G}}(\theta)(1) = e_{\mbf{G}}e_{\mbf{T}}[\mbf{G}^F:\mbf{T}^F]_{p'} \end{equation*} where $[\mbf{G}^F:\mbf{T}^F]_{p'}$ denotes the largest divisor of the index $[\mbf{G}^F:\mbf{T}^F] = |\mbf{G}^F|/|\mbf{T}^F|$ which is coprime to $p$. In particular the degree of the Deligne–Lusztig character is independent of $\theta$.

Remark: The degree $R_{\mbf{T}}^{\mbf{G}}(\theta)(1)$ is a monic polynomial in $q$ in the following sense. For each $F$-stable maximal torus $\mbf{T}$ of $\mbf{G}$ there exists a monic polynomial $P_{\mbf{T}}(t) \in \mathbb{Z}[t]$ such that $R_{\mbf{T}}^{\mbf{G}}(\theta)(1) = P_{\mbf{T}}(q)$. This statement follows from the fact that the orders of finite reductive groups are polynomials in $q$, (see for instance [Gec03, Corollary 4.2.5] and [Car93, §2.9]).

Examples of $\mathbb{F}_q$-rank

For this subsection only we assume that $\mbf{G}$ is the general linear group $\GL_n(\mathbb{K})$. If $r = p^b$ is a power of $p$, (with $b \in \mathbb{N}$), we define $F_r : \mbf{G} \to \mbf{G}$ to be the Frobenius endomorphism given by $F_r(x_{ij}) = (x_{ij}^r)$. We denote by $\mbf{T}_0$ the subgroup of all diagonal matrices and by $\mbf{B}_0$ the subgroup of all upper triangular matrices. The group $\mbf{B}_0$ is then a Borel subgroup of $\mbf{G}$ which contains the maximal torus $\mbf{T}_0$. A set of coset representatives for $\mbf{T}_0$ in $N_{\mbf{G}}(\mbf{T}_0)$ is given by the set of permutation matrices in $\mbf{G}$, (every row and column of such a matrix contains only one non-zero entry which is equal to 1). In particular the Weyl group $W = N_{\mbf{G}}(\mbf{T}_0)/\mbf{T}_0$ is isomorphic to the symmetric group $\mathfrak{S}_n$.

Let us denote by $\dot{w}_0 \in \mbf{G}$ the matrix given by \begin{equation*} \sum_{i = 1}^n E_{i,n-(i-1)} \end{equation*} where $E_{i,j} \in \mbf{G}$ is an elementary matrix, (every entry of such a matrix is 0 except the entry in the $i$th row and $j$th column which is 1). The element $\dot{w}_0$ is a representative in $N_{\mbf{G}}(\mbf{T}_0)$ for the longest word in $W$. We then define an automorphism $\tau : \mbf{G} \to \mbf{G}$ such that $\tau(x) = \dot{w}_0x^{-T}\dot{w}_0^{-1}$ where here $x^{-T}$ denotes the inverse transpose of the matrix. Both of the maps $F_r$ and $F_r\circ\tau$ are Frobenius endomorphism giving rise to, (in general), non-isomorphic finite reductive groups. In particular the case $F = F_r$ gives rise to the finite general linear group $\GL_n(\mathbb{F}_q)$, (here $q = r$), and the case $F = F_r \circ \tau$ gives rise to the finite unitary group $\GU_n(\mathbb{F}_q)$, (here $q = r^2$).

Under both choices for $F$ the subgroups $\mbf{T}_0$ and $\mbf{B}_0$ are $F$-stable hence $\mbf{T}_0$ is a maximally split torus of $\mbf{G}$. Note that the transpose map takes upper triangular matrices to lower triangular matrices but it's easy to verify that conjugation by $\dot{w}_0$ takes lower triangular matrices to upper triangular matrices hence $\mbf{B}_0$ is stable under $\tau$. If $F = F_r$ then $\mbf{T}_0$ is a split torus because for all $t \in \mbf{T}_0$ we have $F(t) = t^r$ hence $F(\chi)(t) = \chi(F(t)) = \chi(t^r) = (r\chi)(t)$ for all $\chi \in X(\mbf{T}_0)$. The $\mathbb{F}_q$-rank of $\mbf{T}_0$ is then just the dimension of $\mbf{T}_0$ which is $n$, hence $e_{\mbf{T}_0} = (-1)^n$.

Assume now $F = F_r \circ \tau$ then $\mbf{T}_0$ is in general not a split torus. Indeed let $t \in \mbf{T}_0$ be the diagonal matrix $\diag(t_1,\dots,t_n)$ for $t_i \in \mathbb{K}^{\times}$ then we have \begin{equation*} F(t) = \diag(t_n^{-r},\dots,t_1^{-r}). \end{equation*} We can describe a maximum split subtorus $\mbf{S} \leqslant \mbf{T}_0$ as the subgroup \begin{equation*} \mbf{S} = \{ \diag(t_1,\dots,t_n) \mid t_i = t_{n-(i-1)}^{-1}\text{ for all }1\leqslant i\leqslant n \}. \end{equation*} Let $n = 2k + \varepsilon$ where $k \in \mathbb{N}$ and $\varepsilon \in \{0,1\}$ then the dimension of $\mbf{S}$ is $k + \varepsilon$ so the $\mathbb{F}_q$-rank of $\mbf{T}_0$ is $k + \varepsilon$ and $e_{\mbf{T}_0} = (-1)^{k + \varepsilon}$.

Irreducibility

Assume $\mbf{T}$ is an $F$-stable maximal torus of $\mbf{G}$ then we denote the Weyl group of $\mbf{G}$ with respect to $\mbf{T}$ by $W(\mbf{T})$. As $\mbf{T}$ is $F$-stable we will obtain an induced action of $F$ on $W(\mbf{T})$ and as $\mbf{T}$ is connected we have $W(\mbf{T})^F = N_{\mbf{G}}(\mbf{T})^F/\mbf{T}^F$ by the Lang-Steinberg theorem, (see [DM91, Corollary 3.13]). Given an element $w \in W(\mbf{T})^F$ we denote by $\dot{w} \in N_{\mbf{G}}(\mbf{T})^F$ a representative in the normaliser. We then have an action of the group $W(\mbf{T})^F$ on $\Irr(\mbf{T}^F)$ given by $\theta \mapsto {}^w\theta$ where ${}^w\theta(t) = \theta(\dot{w}^{-1}t\dot{w})$, (note that this doesn't depend on the choice of representative as $\mbf{T}^F$ acts trivially).

Let $(\mbf{T},\theta)$ be a pair in $\nabla(\mbf{G},F)$ then we say the character $\theta$ is in general position if the stabiliser of $\theta$ under the action of $W(\mbf{T})^F$ is trivial. The reason for this definition is that we then have $\langle R_{\mbf{T}}^{\mbf{G}}(\theta), R_{\mbf{T}}^{\mbf{G}}(\theta)\rangle = 1$ if and only if $\theta$ is in general position. The statement $\langle R_{\mbf{T}}^{\mbf{G}}(\theta), R_{\mbf{T}}^{\mbf{G}}(\theta)\rangle = 1$ can be translated into saying that $\pm R_{\mbf{T}}^{\mbf{G}}(\theta)$ is an irreducible character, (note that the Deligne–Lusztig character is a virtual character). In fact as we know the degree of the Deligne–Lusztig character we can precisely compute the required sign to obtain the following theorem.

Theorem: Assume $(\mbf{T},\theta) \in \nabla(\mbf{G},F)$ is such that $\theta$ is in general position then $e_{\mbf{G}}e_{\mbf{T}}R_{\mbf{T}}^{\mbf{G}}(\theta)$ is an irreducible character of $G$.

Uniform Class Functions

Let $\Cent(G)^0$ denote the subspace of $\Cent(G)$ obtained as the $\overline{\mathbb{Q}_{\ell}}$-span of the set $\{R_{\mbf{T}}^{\mbf{G}}(\theta) \mid (\mbf{T},\theta) \in \nabla(\mbf{G},\theta)\}$ of all Deligne–Lusztig characters. We call the elements of $\Cent(G)^0$ uniform class functions. It is known that in general this is a proper subspace of $\Cent(G)$, indeed one already encounters in $\SL_2(\mathbb{F}_q)$ four irreducible characters which are not uniform, (see [DM91, §15.9] or [Bon11]). Although $\Cent(G)^0$ can be a proper subspace many important class functions are uniform. The following class functions are known to be uniform:

1. The characteristic function $f_s^{G} \in \Cent(G)$ of a semisimple element $s \in G$, (see [DM91, Proposition 12.20] or [DL76, Proposition 7.5]). Here we take characteristic function to mean the normalised characteristic function defined by\begin{equation*} f_s^{G}(g) = \begin{cases} |C_{G}(s)| &\text{if there exists }h \in G\text{ such that }hgh^{-1} = s,\\ 0 &\text{otherwise}. \end{cases}

\end{equation*} Warning: The characteristic function of an arbitrary element of $G$ is not necessarily uniform, (for instance those of unipotent elements).

1. The character of the regular representation of $G$, (this is just the above statement with $s=1$).
2. The trivial character of $G$, (see [DM91, Proposition 12.13] or [DL76, Proposition 7.14.1]).
3. The Steinberg character of $G$, (this follows from the definition of the Steinberg character and the statement for the trivial character - see [DL76, Proposition 7.14.2]).

The fact that these class functions are uniform has some important consequences. Firstly as every irreducible character occurs with non-zero multiplicity in the character of the regular representation we have every irreducible character of $G$ occurs with non-zero multiplicity in some Deligne–Lusztig character. This allows one to give a rough partitioning of the irreducible characters of $G$ based on which Deligne–Lusztig characters they occur in, (this is also discussed below in the section on Lusztig series).

Note that the following remark is made explicit in [DL76, Proposition 7.14.1]. Assume $\chi \in \Cent(G)$ is any class function then given any semisimple element $s \in G$ we have from the definition of the inner product that $\chi(s) = \langle \chi, f_s^{G}\rangle_{G}$. The decomposition of $f_s^{G}$ as a sum of Deligne–Lusztig characters is explicit. Specifically we have \begin{equation*} f_s^{G} = \frac{e_{C_{\mbf{G}}(s)^{\circ}}}{|{C_{\mbf{G}}(s)^{\circ}}^F|_p}\sum_{\substack{\mbf{T} \in \mathfrak{T} \\ s \in \mbf{T}}} e_{\mbf{T}} \sum_{\theta \in \Irr(\mbf{T}^F)} \theta(s)^{-1} R_{\mbf{T}}^{\mbf{G}}(\theta) \end{equation*} where here $\mathfrak{T}$ denotes the set of all $F$-stable maximal tori of $\mbf{G}$ and $C_{\mbf{G}}(s)^{\circ}$ denotes the connected component of the centraliser of $s$, (see [Car93, Proposition 7.5.5]). Hence once the multiplicities of the irreducible characters in the Deligne–Lusztig characters are known we can determine the value of every irreducible character on semisimple elements. In fact Lusztig has solved the latter problem, (see the section Jordan decomposition of characters).

Lusztig Series

The Dual Group

Let $\mbf{T}_0$ be a maximal torus of $\mbf{G}$ and let us denote by $\Phi \subset X(\mbf{T}_0)$ the roots of $\mbf{G}$ with respect to $\mbf{T}_0$. We denote by $Y(\mbf{T}_0) = \Hom(\mbf{K}^{\times},\mbf{T}_0)$ the cocharacter group of $\mbf{T}_0$ and by $\Phi^{\vee} \subset Y(\mbf{T}_0)$ the coroots of $\mbf{G}$ with respect to $\mbf{T}_0$, (note that $\Phi^{\vee}$ is the root system dual to $\Phi$). We say a connected reductive algebraic group $\mbf{G}^{\star}$ is a dual group for $\mbf{G}$ if there exists a maximal torus $\mbf{T}_0^{\star}$ of $\mbf{G}^{\star}$ and an isomorphism $\varphi : X(\mbf{T}_0) \to Y(\mbf{T}_0^{\star})$ mapping $\Phi$ to $\Phi^{\vee}$. By the classification of connected reductive algebraic groups the group $\mbf{G}^{\star}$ exists and is unique up to isomorphism.

Let us fix a generalised Frobenius endomorphism $F^{\star} : \mbf{G}^{\star} \to \mbf{G}^{\star}$ then we assume that $\mbf{T}_0^{\star}$, (resp. $\mbf{T}_0$), is maximally split with respect to $F^{\star}$, (resp. $F$). We say the triples $(\mbf{G},\mbf{T}_0,F)$ and $(\mbf{G}^{\star},\mbf{T}_0^{\star},F^{\star})$ are in duality if our isomorphism $\varphi : X(\mbf{T}_0) \to Y(\mbf{T}_0^{\star})$ can be chosen such that $\varphi\circ F = F^{\star}\circ \varphi$. We say the groups $G^{\star} = {\mbf{G}^{\star}}^{F^{\star}}$ and $G$ are finite reductive groups in duality, (note that $G^{\star}$ is uniquely determined up to isomorphism).

Example (see [Car95, §4.4]): The dual of a simply connected group is adjoint and vice versa. For instance the following pairs of groups are dual:

• the special linear group $\SL_n(\mathbb{K})$ and the projective general linear group $\PGL_n(\mathbb{K})$. The finite groups $\SL_n(\mathbb{F}_q)$ and $\PGL_n(\mathbb{F}_q)$ are finite reductive groups in duality.
• the symplectic group $\Sp_{2n}(\mathbb{K})$ and the odd rank special orthogonal group $\SO_{2n+1}(\mathbb{K})$. The finite groups $\Sp_{2n}(\mathbb{F}_q)$ and $\SO_{2n+1}(\mathbb{F}_q)$ are finite reductive groups in duality.
• the general linear groups $\GL_n(\mathbf{K})$ and $\GL_n(\mathbf{F}_q)$ are in duality with themselves.

Partitioning Irreducible Characters

Notation: Let $\nabla^{\star}(\mbf{G},F)$ denote the set of all pairs $(\mbf{T}^{\star},s)$ such that $\mbf{T}^{\star}$ is an $F^{\star}$-stable maximal torus of $\mbf{G}^{\star}$ and $s \in T^{\star}$. We say two pairs $({\mbf{T}^{\star}}',s')$, $(\mbf{T}^{\star},s) \in \nabla^{\star}(\mbf{G},F)$ are rationally conjugate if there exists an element $x \in G^{\star}$ such that ${\mbf{T}^{\star}}' = {}^x\mbf{T}^{\star}$ and $s' = {}^xs$. This defines an equivalence relation $\sim_{G^{\star}}$ on $\nabla^{\star}(\mbf{G},F)$ and we denote the set of all equivalence classes by $\nabla^{\star}(\mbf{G},F)/G^{\star}$.

There is a strong relationship between the equivalence classes $\nabla^{\star}(\mbf{G},F)/G^{\star}$ and $\nabla(\mbf{G},F)/G$, namely we have the following result.

Proposition (see [DM91, Proposition 13.13]): We have a bijective correspondence between the sets of equivalence classes $\nabla(\mbf{G},F)/G \to \nabla^{\star}(\mbf{G},F)/G^{\star}$ such that $(\mbf{T},1)\mapsto (\mbf{T}^{\star},1)$, where $1$ denotes the trivial character of $\mbf{T}^F$ or the identity in $G^{\star}$.

By the parameterisation of Deligne–Lusztig characters given above we may unambiguously denote any Deligne–Lusztig character $R_{\mbf{T}}^{\mbf{G}}(\theta)$ by $R_{\mbf{T}^{\star}}^{\mbf{G}}(s)$ where the equivalence class containing $(\mbf{T}^{\star},s) \in \nabla^{\star}(\mbf{G},F)$ is in bijective correspondence with the equivalence class containing $(\mbf{T},\theta) \in \nabla(\mbf{G},F)$. With this in mind we can now state one of the most fundamental results concerning Deligne–Lusztig characters.

Theorem (see [Bon11, Théorème 11.8]): Suppose $({\mbf{T}^{\star}}',s')$ and $(\mbf{T}^{\star},s)$ are not rationally conjugate then $R_{{\mbf{T}^{\star}}'}^{\mbf{G}}(s')$ and $R_{\mbf{T}^{\star}}^{\mbf{G}}(s)$ have no irreducible constituents in common.

In particular we may now describe a partitioning of the irreducible characters $\Irr(G)$ of $G$ as follows. Let $s \in G^{\star}$ be a semisimple element then we denote by $[s]$ the $G^{\star}$-conjugacy class containing $s$. We define the Lusztig series of $G$ associated to $[s]$ to be the set \begin{equation*} \mathcal{E}(G,s) = \mathcal{E}(G,[s]) = \{\chi \in \Irr(G) \mid \langle \chi,R_{\mbf{T}^{\star}}^{\mbf{G}}(s)\rangle \neq 0\text{ for some }(\mbf{T}^{\star},s)\in\nabla^{\star}(\mbf{G},F)\}. \end{equation*} As every irreducible character occurs in some Deligne–Lusztig character the above theorem then says that we have a partition \begin{equation*} \Irr(G) = \bigsqcup_{[s]} \mathcal{E}(G,s), \end{equation*} where $[s]$ runs over all $G^{\star}$-conjugacy classes of semisimple elements.

Semisimple and Regular Characters

Assume now that $s \in G^{\star}$ is a semisimple element then the connected component of the centraliser $C_{\mbf{G}^{\star}}(s)^{\circ}$ is again a connected reductive algebraic group, (see [Car93, Theorem 3.5.4]). We will denote by $W^{\star}(s)^{\circ}$ the Weyl group of $C_{\mbf{G}^{\star}}(s)^{\circ}$ with respect to some choice of $F^{\star}$-stable maximal torus $\mbf{T}_s^{\star}$ containing $s$, (note that $\mbf{T}_s^{\star}$ is also an $F^{\star}$-stable maximal torus of $\mbf{G}^{\star}$). Assume $f \in \Cent(W^{\star}(s)^{\circ})$ is a class function of the Weyl group then we define a corresponding class function \begin{equation*} R_f^{\mbf{G}}(s) = \frac{1}{|\mbf{W}(s)^{\circ}|}\sum_{w \in \mbf{W}(s)^{\circ}} f(w)R_{\mbf{T}_w^{\star}}^{\mbf{G}}(s) \in \Cent(G), \end{equation*} where here we denote by $\mbf{T}_w^{\star}$ the $F^{\star}$-stable maximal torus obtained from $\mbf{T}_s^{\star}$ by twisting with $w$.

We say an irreducible character $\chi \in \Irr(G)$ is regular if it occurs with non-zero multiplicity in $R_{\sgn}^{\mbf{G}}(s)$ for some $s \in G^{\star}$. Furthermore we say $\chi$ is semisimple if it occurs with non-zero multiplicity in $R_{1}^{\mbf{G}}(s)$ for some $s \in G^{\star}$. Here we denote by 1 and $\sgn$ the trivial and sign characters of $W^{\star}(s)^{\circ}$. We have the following important result due to Deligne and Lusztig.

Theorem (see [DL76, Theorem 10.7]): Assume the centre $Z(\mbf{G})$ of $\mbf{G}$ is connected then for each $s \in G^{\star}$ we have $e_{\mbf{G}}e_{C_{\mbf{G}^{\star}}(s)^{\circ}}R_1^{\mbf{G}}(s)$ and $e_{\mbf{G}}e_{C_{\mbf{G}^{\star}}(s)^{\circ}}R_{\sgn}^{\mbf{G}}(s)$ are irreducible characters of $G$. In particular each Lusztig series contains a unique semisimple and a unique regular character, (note that these characters may coincide).

For the above theorem one may also see the formulation in [Bon06, Théorème 15.10] and the remarks there. Note that when $s$ is the identity of $G^{\star}$ we have $R_1^{\mbf{G}}(1)$ is the trivial character of $G$ and $R_{\sgn}^{\mbf{G}}(1)$ is the Steinberg character of $G$.

Remark: We have given here a somewhat unconventional definition of semisimple and regular characters. Indeed one usually defines an irreducible character to be a regular character if it occurs with non-zero multiplicity in a Gelfand–Graev character of $G$. Similarly one usually defines an irreducible character to be a semisimple character if it occurs with non-zero multiplicity in the Alvis–Curtis dual of a Gelfand–Graev character of $G$. We have avoided this formalism here to keep things simpler. Our definition coincides with this one due to the decomposition of the Gelfand–Graev character, see again [DL76, Theorem 10.7] for the connected centre case and [DM, Theorem 14.49] for the general case.

Jordan Decomposition of Characters

Definition: An irreducible character $\chi \in \Irr(G)$ is called unipotent if it occurs in the Lusztig series $\mathcal{E}(G,1)$.

We assume for this subsection that $Z(\mbf{G})$ is connected. Under this assumption we have that the centraliser of every semisimple element of $\mbf{G}^{\star}$ is connected, (see [DM91, Remarks 13.15(ii)]). In particular for each semisimple element $s$ we may consider the series of unipotent characters of the centraliser $C_{\mbf{G}^{\star}}(s)^{F^{\star}}$. Note that all that we have discussed above has only been for a connected algebraic group, hence the assumption that $C_{\mbf{G}^{\star}}(s)$ is connected is very important for the following result.

Theorem (see [Lus84a, Main Theorem 4.23]). Assume $Z(\mbf{G})$ is connected then there exists a bijection \begin{equation*} \Psi_s : \mathcal{E}(G,s) \to \mathcal{E}(C_{\mbf{G}^{\star}}(s)^{F^{\star}},1) \end{equation*} with the following properties:

• If $\chi \in \mathcal{E}(G,s)$ and $\mbf{T}^{\star}$ is an $F^{\star}$-stable maximal torus of $\mbf{G}^{\star}$ containing $s$ then

\begin{equation*} \langle \chi, R_{\mbf{T}^{\star}}^{\mbf{G}}(s) \rangle_G = e_{\mbf{G}}e_{C_{\mbf{G}^{\star}}(s)}\langle \Psi_s(\chi), R_{\mbf{T}^{\star}}^{C_{\mbf{G}^{\star}}(s)}(1) \rangle_{C_{\mbf{G}^{\star}}(s)^{F^{\star}}} \end{equation*}

• Let us denote by $\chi_{\ss} \in \mathcal{E}(G,s)$ the unique semisimple character in the Lusztig series. For any irreducible character $\chi \in \mathcal{E}(G,s)$ we have the degree of $\chi$ is given by

\begin{equation*} \chi(1) = \chi_{\ss}(1)\Psi_s(\chi)(1). \end{equation*} Note: The first bullet point implies the second, (see [DM91, Remark 13.24]).

We call the bijection in the above theorem the Jordan decomposition of characters. The Jordan decomposition reduces the problem of studying the degrees of irreducible characters and the multiplicities of irreducible characters in Deligne–Lusztig characters to the unipotent case. In [Lus84a] Lusztig determines the degrees and multiplicities for a unipotent character of an arbitrary finite reductive group. Indeed as unipotent characters are insensitive to the centre of $\mbf{G}$ one can reduce this problem to the case where $\mbf{G}$ is an adjoint simple group, (see [DM91, Proposition 13.20]). For information on the degrees and multiplicities of unipotent characters in simple groups see the appendices in [Car95, §§13.5-13.9].

Remark: If the centraliser $C_{\mbf{G}^{\star}}(s)$ is contained in a Levi subgroup of $\mbf{G}^{\star}$ then the bijection $\Psi_s$ in the above theorem can be realised using a Deligne–Lusztig induction map $R_{\mbf{L} \subset \mbf{P}}^{\mbf{G}}$, (see for instance [DM91, pg. 113 to 118]). In particular the main work in [Lus84a] is to deal with those series $\mathcal{E}(G,s)$ for which $C_{\mbf{G}^{\star}}(s)$ is not contained in a Levi subgroup, (such semisimple elements are called isolated elements).

Character Sheaves

The algebraic group $\mbf{G}$ considered so far has been defined over an algebraically closed field of prime characteristic, so that one has a Frobenius endomorphism. However, Lusztig has used ideas on intersection cohomology to develop a "geometric theory of characters" which works equally well for reductive groups over arbitrary algebraically closed fields. This geometric theory of characters uses the theory of perverse sheaves, developed in [BBD82]. The irreducible perverse sheaves are the intersection cohomology complexes. A character sheaf is a particular type of irreducible perverse sheaf. In the case of a reductive group over the algebraic closure of a finite field, a character sheaf fixed by the Frobenius mapping has a characteristic function which is a class function on the finite group $G$. These characteristic functions form an orthonormal basis of $\Cent(G)$. They are not, in general, the irreducible characters of $G$ but are, roughly speaking, Fourier transforms of the irreducible characters. Thus, there is a close connection between the irreducible characters of $G$ and the $F$-stable irreducible character sheaves. However, since the theory of character sheaves is valid in the wider context of reductive groups over arbitrary algebraically closed fields, the ideas of the Deligne–Lusztig theory have an analogue in arbitrary connected reductive groups. An exposition of the theory of character sheaves can be found in [Lus85], while [Lus84b], [Lus87] provide useful introductory material.

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