The irreducible representations of GL₂(𝔽_q)

Intuition Beginner

Take a finite field with $q$ elements, and look at all the $2\times 2$ matrices over it that have an inverse. These matrices form a group $G$ under multiplication, called the general linear group $G{L}_2$. The question of this unit is: what are all the ways this group can act linearly on a complex vector space, broken into the smallest possible pieces? Listing those pieces, together with the trace of each matrix in each piece, is the character table of the group.

This is the smallest place where a great split in all of representation theory first shows up. Some of the building-block representations are assembled by gluing together a representation of a smaller, simpler subgroup — the upper-triangular matrices. These are the principal series. But some pieces can never be built that way. They live, in a sense, off in a hidden second field ${\mathbb{F}}_{q^2}$. These are the cuspidal pieces.

So $G{L}_2$ over a finite field is a tiny laboratory. The whole "induced from a small subgroup versus genuinely new" story plays out here on a table you can write down completely on one page.

Visual Beginner

Picture the group of invertible $2\times 2$ matrices sorted into bins, where two matrices land in the same bin when one is a relabelling (a conjugate) of the other. There are four shapes of bin: scalar matrices, matrices with a repeated eigenvalue that are not scalar, matrices with two different eigenvalues both in the base field, and matrices whose eigenvalues only exist in a larger field. Each shape of bin contributes its own family of irreducible pieces to the character table.

Worked example Beginner

Let us count, for the smallest interesting case $q=2$. The field has two elements, $0$ and $1$. The group $G=G{L}_2({\mathbb{F}}_2)$ consists of the $2\times 2$ matrices over $\{0,1\}$ with a non-zero determinant.

Step 1. Count the group. The first column may be any non-zero vector: $2^2-1=3$ choices. The second column may be anything not in the line spanned by the first: $2^2-2=2$ choices. So $|G|=3\cdot 2=6$.

Step 2. Recognise the group. A group of order $6$ that is not commutative must be the symmetric group on three letters, $S_3$. Indeed $G{L}_2({\mathbb{F}}_2)$ permutes the three non-zero vectors of the plane, giving an exact match with $S_3$.

Step 3. Read off the pieces. $S_3$ has three irreducible representations of dimensions $1$, $1$, and $2$. The check $1^2+1^2+2^2=6$ matches the group order, as it must. So even the tiniest case already shows the one-dimensional family and a higher piece sitting side by side.

Check your understanding Beginner

Formal definition Intermediate+

Fix a finite field ${\mathbb{F}}_q$ with $q$ a prime power, and let $G=G{L}_2({\mathbb{F}}_q)$, the group of invertible $2\times 2$ matrices over ${\mathbb{F}}_q$. Its order is

$$|G|=(q^2-1)(q^2-q)=q(q-1{)}^2(q+1).$$

We single out three subgroups. The Borel subgroup $B$ is the upper-triangular matrices; the split torus $T\subset B$ is the diagonal matrices, $T\cong {\mathbb{F}}_q^{\times }\times {\mathbb{F}}_q^{\times }$; and the unipotent radical $U\subset B$ is the upper-triangular matrices with $1$'s on the diagonal, $U\cong {\mathbb{F}}_q$. The Borel factors as $B=T⋉U$.

Conjugacy classes. Sorting $G$ by rational canonical form gives exactly four shapes, with $q^2-1$ classes in all:

• Central: scalar matrices $aI$, $a\in {\mathbb{F}}_q^{\times }$. There are $q-1$ such classes, each a single element.
• Non-semisimple (unipotent type): matrices conjugate to \begin{psmallmatrix} a & 1 \\ 0 & a \end{psmallmatrix}, $a\in {\mathbb{F}}_q^{\times }$. There are $q-1$ such classes.
• Split-semisimple: distinct eigenvalues $a\ne b$ in ${\mathbb{F}}_q^{\times }$, represented by $\mathrm{diag}(a,b)$. Counting unordered pairs gives $\left(\genfrac{}{}{0ex}{}{q-1}{2}\right)$ classes.
• Elliptic (non-split): characteristic polynomial irreducible over ${\mathbb{F}}_q$, with eigenvalues $\xi ,{\xi }^q$ a conjugate pair in ${\mathbb{F}}_{q^2}^{\times }\setminus {\mathbb{F}}_q^{\times }$. There are $\left(\genfrac{}{}{0ex}{}{q^2-q}{2}\right)/1=\frac{1}{2}(q^2-q)$ classes.

Adding $(q-1)+(q-1)+\frac{1}{2}(q-1)(q-2)+\frac{1}{2}q(q-1)=q^2-1$ classes, hence $q^2-1$ irreducible characters.

The non-split torus. Fix an embedding ${\mathbb{F}}_{q^2}^{\times }\hookrightarrow G{L}_2({\mathbb{F}}_q)$ by letting ${\mathbb{F}}_{q^2}$ act on itself by multiplication, viewing it as a $2$-dimensional ${\mathbb{F}}_q$-vector space. The image $T^{\prime }$ is a cyclic subgroup of order $q^2-1$, the non-split (elliptic) torus; its non-central elements are the elliptic classes.

The four families of irreducibles

Write $\widehat{{\mathbb{F}}_q^{\times }}$ for the $q-1$ characters of the multiplicative group and ${\mathrm{Ind}}_B^G$ for induction from the Borel, where a character $\alpha \otimes \beta$ of $T$ is inflated to $B$ through $B\twoheadrightarrow T$.

1. One-dimensionals. For each $\chi \in \widehat{{\mathbb{F}}_q^{\times }}$, the representation $\chi \circ \det$ has dimension $1$. There are $q-1$ of these.
2. Principal series. For $\alpha \ne \beta$, the induced representation ${\mathrm{Ind}}_B^G(\alpha \otimes \beta )$ is irreducible of dimension $q+1$. The pairs $\{\alpha ,\beta \}$ and $\{\beta ,\alpha \}$ give isomorphic representations, so there are $\left(\genfrac{}{}{0ex}{}{q-1}{2}\right)$ of these.
3. Steinberg twists. When $\alpha =\beta$, the induced ${\mathrm{Ind}}_B^G(\alpha \otimes \alpha )$ has dimension $q+1$ but splits as $(\alpha \circ \det )\oplus {\mathrm{St}}_{\alpha }$, where the Steinberg constituent ${\mathrm{St}}_{\alpha }$ has dimension $q$. There are $q-1$ of these.
4. Cuspidal / discrete series. For each character $\theta$ of the non-split torus ${\mathbb{F}}_{q^2}^{\times }$ in general position (meaning $\theta \ne {\theta }^q$, i.e. $\theta$ does not factor through the norm to ${\mathbb{F}}_q^{\times }$), there is a cuspidal irreducible of dimension $q-1$. The characters $\theta$ and ${\theta }^q$ give the same representation, so there are $\frac{1}{2}(q^2-q)$ of these.

Counterexamples to common slips

• Forgetting the Steinberg. If one declared all $\left(\genfrac{}{}{0ex}{}{q}{2}\right)$ pairs $\{\alpha ,\beta \}$ (now allowing $\alpha =\beta$) to give irreducibles of dimension $q+1$, the dimension sum would overshoot. The diagonal pairs are reducible; the genuine $q+1$ pieces require $\alpha \ne \beta$.
• Mis-parametrising the cuspidals. A character $\theta$ with $\theta ={\theta }^q$ does not yield a cuspidal; it factors through the determinant and reproduces a one-dimensional. Only $\theta$ in general position is admissible.

Key theorem with proof Intermediate+

Theorem (the dimension count). The four families above exhaust the irreducible representations of $G=G{L}_2({\mathbb{F}}_q)$, and their dimensions satisfy ${\sum }_i{d}_i^2=|G|$.

Proof. We first verify the head count: each family contributes the same number of irreducibles as a family of conjugacy classes, matching the four class-shapes.

Family	Count	Dimension
$\chi \circ \det$	$q-1$	$1$
principal series	$\left(\genfrac{}{}{0ex}{}{q-1}{2}\right)$	$q+1$
Steinberg ${\mathrm{St}}_{\alpha }$	$q-1$	$q$
cuspidal	$\frac{1}{2}(q^2-q)$	$q-1$

The total number is $(q-1)+\left(\genfrac{}{}{0ex}{}{q-1}{2}\right)+(q-1)+\frac{1}{2}(q^2-q)$. Using $\left(\genfrac{}{}{0ex}{}{q-1}{2}\right)=\frac{1}{2}(q-1)(q-2)$ and $\frac{1}{2}(q^2-q)=\frac{1}{2}q(q-1)$, the sum is

$$(q-1)[1+\frac{q-2}{2}+1+\frac{q}{2}]=(q-1)[2+\frac{2q-2}{2}]=(q-1)(q+1)=q^2-1,$$

which equals the number of conjugacy classes computed above, so the count is exact.

Now the sum of squares:

$$\sum _i{d}_i^2=(q-1)\cdot 1^2+\frac{(q-1)(q-2)}{2}(q+1{)}^2+(q-1)q^2+\frac{q(q-1)}{2}(q-1{)}^2.$$

Factor out $(q-1)$ and group the two "half" terms. The principal-series and cuspidal contributions combine as

$$\frac{q-1}{2}[(q-2)(q+1{)}^2+q(q-1{)}^2]=\frac{q-1}{2}[(q-2)(q^2+2q+1)+q(q^2-2q+1)].$$

Expanding inside the brackets: $(q^3-3q-2)+(q^3-2q^2+q)=2q^3-2q^2-2q-2$. So this block equals $(q-1)(q^3-q^2-q-1)$. Adding the one-dimensional and Steinberg blocks $(q-1)(1+q^2)$ gives

$$(q-1)[(q^3-q^2-q-1)+(q^2+1)]=(q-1)(q^3-q)=(q-1)q(q^2-1)=q(q-1{)}^2(q+1)=|G|.$$

The sum of squares of the listed dimensions equals $|G|$, and the head count equals the number of classes, so by the completeness of irreducible characters 07.01.04 the list is exhaustive. $\square$

Bridge. This dimension count builds toward the general theory of finite groups of Lie type and appears again in the Deligne–Lusztig construction, where the split-versus-elliptic torus dichotomy becomes the index set for all irreducibles. The foundational reason the table cleaves into two halves is that a torus in a reductive group can be split or non-split, and this is exactly the parabolic-induction-versus-cuspidal dichotomy that organises all of representation theory of $p$-adic and finite reductive groups. The principal series generalises the abelian Fourier picture of induction from a maximal split torus, while the cuspidal series is dual to it: built not by inducing functions on the flag variety ${\mathbb{P}}^1({\mathbb{F}}_q)$ but from the second torus living in ${\mathbb{F}}_{q^2}$. Putting these together, $G{L}_2({\mathbb{F}}_q)$ is the smallest non-abelian model where the central insight of Harish-Chandra philosophy — that every irreducible is parabolically induced from a cuspidal of a Levi subgroup — can be checked by hand against an explicit character table.

Exercises Intermediate+

Advanced results Master

Result 1 (the principal-series irreducibility criterion). For $\alpha ,\beta \in \widehat{{\mathbb{F}}_q^{\times }}$, the induced module ${\mathrm{Ind}}_B^G(\alpha \otimes \beta )$ is irreducible if and only if $\alpha \ne \beta$. When $\alpha \ne \beta$ the swap $(\alpha ,\beta )\mapsto (\beta ,\alpha )$ gives an isomorphic module — the two Bruhat cells exchanged by the Weyl element $w$ — so the principal series is parametrised by unordered pairs, giving $\left(\genfrac{}{}{0ex}{}{q-1}{2}\right)$ irreducibles of dimension $q+1$. The proof is a Mackey computation: $⟨{\mathrm{Ind}}_B^G(\alpha \otimes \beta ),{\mathrm{Ind}}_B^G(\alpha \otimes \beta )⟩$ counts the $\theta$-fixed double cosets and equals $1$ exactly when $\alpha \ne \beta$.

Result 2 (the Steinberg as a building block). The Steinberg $\mathrm{St}={\mathrm{St}}_1$ is the complement of the principal character in $\mathbb{C}[{\mathbb{P}}^1({\mathbb{F}}_q)]$. Its $q-1$ twists ${\mathrm{St}}_{\alpha }=\mathrm{St}\otimes (\alpha \circ \det )$ have dimension $q$. The Steinberg is the unique cuspidal-adjacent piece that is a constituent of a parabolic induction yet is not one-dimensional: it sits at the reducible boundary $\alpha =\beta$ of the principal series. In the general Lie-type theory it is the "smallest" generic unipotent representation and carries the top cohomology of the building.

Result 3 (cuspidals from the non-split torus). The discrete series of dimension $q-1$ are parametrised by characters $\theta$ of ${\mathbb{F}}_{q^2}^{\times }$ in general position, modulo $\theta \sim {\theta }^q$. They do not occur in functions on ${\mathbb{P}}^1$; their construction (Weil representation, or the difference ${\mathrm{Ind}}_{ZU}^G(\psi \otimes \dots )-{\mathrm{Ind}}_B^G(\dots )$ of induced characters) is the prototype of Deligne–Lusztig virtual characters $R_{T^{\prime }}^{\theta }$ attached to the elliptic torus. The sign in the dimension $q-1$ and in the elliptic character value $-(\theta (\xi )+\theta ({\xi }^q))$ is the relative-rank sign of the two tori.

Result 4 (the complete character table). Assembling Results 1–3 yields the $(q^2-1)\times (q^2-1)$ character table of $G{L}_2({\mathbb{F}}_q)$: rows indexed by the four families, columns by the four class-shapes, with explicit entries (Gauss-sum corrections appear in the cuspidal column on the unipotent classes). This is the table Serre presents in LRFG Ch. 8 and Fulton–Harris in §5.2; it is the smallest reductive-group table that is not just a list of one-dimensionals plus a permutation character.

Synthesis. The four-family structure builds toward the Deligne–Lusztig classification of all finite groups of Lie type, and the split-versus-elliptic torus split appears again in the Harish-Chandra theory of $p$-adic groups, where it is exactly the parabolic-induction-versus-supercuspidal dichotomy. The central insight is that a maximal torus of a reductive group over a finite field is classified up to conjugacy by an $F$-conjugacy class in the Weyl group, and for $G{L}_2$ the Weyl group is $\mathbb{Z}/2$: the split torus $T$ and the non-split torus $T^{\prime }$ are its two classes, generating the principal series and the cuspidal series respectively. The foundational reason the dimension count closes to $|G|$ is that parabolic induction from $T$ and Deligne–Lusztig induction from $T^{\prime }$ together span the space of class functions, so putting these together the virtual characters $R_T^{\chi }$ and $R_{T^{\prime }}^{\theta }$ form a spanning set whose orthogonality forces the families to be complete. This generalises the abelian Fourier story of 07.01.09, where induction from the diagonal torus alone sufficed; here the second torus is dual to the first and supplies precisely the pieces the flag variety misses.

Full proof set Master

Proposition 1 (conjugacy-class count). The number of conjugacy classes of $G=G{L}_2({\mathbb{F}}_q)$ is $q^2-1$.

Proof. Classify by the action on eigenvalues. (i) Scalar matrices $aI$ for $a\in {\mathbb{F}}_q^{\times }$: each is central, giving $q-1$ singleton classes. (ii) Matrices with a single eigenvalue $a\in {\mathbb{F}}_q^{\times }$ that are not scalar: each is conjugate to the Jordan block \begin{psmallmatrix} a&1\\0&a\end{psmallmatrix}, one class per $a$, giving $q-1$ classes. (iii) Distinct eigenvalues $a\ne b$ both in ${\mathbb{F}}_q^{\times }$: conjugacy classes correspond to unordered pairs $\{a,b\}$, giving $\left(\genfrac{}{}{0ex}{}{q-1}{2}\right)=\frac{1}{2}(q-1)(q-2)$ classes. (iv) Eigenvalues an irreducible conjugate pair $\xi ,{\xi }^q$ with $\xi \in {\mathbb{F}}_{q^2}^{\times }\setminus {\mathbb{F}}_q^{\times }$: such elements number $q^2-q$ choices of $\xi$, paired by Frobenius $\xi \leftrightarrow {\xi }^q$, giving $\frac{1}{2}(q^2-q)=\frac{1}{2}q(q-1)$ classes. Summing,

$$(q-1)+(q-1)+\frac{1}{2}(q-1)(q-2)+\frac{1}{2}q(q-1)=(q-1)[2+\frac{q-2}{2}+\frac{q}{2}]=(q-1)(q+1)=q^2-1.$$

$\square$

Proposition 2 (the Bruhat decomposition controls principal-series multiplicities). For $\alpha ,\beta \in \widehat{{\mathbb{F}}_q^{\times }}$, $⟨{\mathrm{Ind}}_B^G(\alpha \otimes \beta ),{\mathrm{Ind}}_B^G(\alpha \otimes \beta ){⟩}_G=1+{\delta }_{\alpha \beta }$, where ${\delta }_{\alpha \beta }=1$ if $\alpha =\beta$ and $0$ otherwise.

Proof. By Mackey's intertwining-number formula, $⟨{\mathrm{Ind}}_B^G\sigma ,{\mathrm{Ind}}_B^G\sigma {⟩}_G={\sum }_{w\in B\backslash G/B}⟨\sigma ,{}^w\sigma {⟩}_{B\cap {}^{w}B}$, the sum over double-coset representatives. By the Bruhat decomposition $G=B\bigsqcup BwB$ there are exactly two double cosets, $w=e$ and $w$ the Weyl reflection. For $w=e$ the term is $⟨\sigma ,\sigma {⟩}_B=1$. For $w$ the reflection, ${}^w\sigma =\beta \otimes \alpha$ on the torus and the intersection $B\cap {}^{w}B=T$, so the term is $⟨\alpha \otimes \beta ,\beta \otimes \alpha {⟩}_T$, which is $1$ if $\alpha =\beta$ and $0$ otherwise. Adding the two terms gives $1+{\delta }_{\alpha \beta }$. Hence the induction is irreducible exactly when $\alpha \ne \beta$, and splits into two pieces when $\alpha =\beta$ (the one-dimensional $\alpha \circ \det$ and the Steinberg twist ${\mathrm{St}}_{\alpha }$). $\square$

Proposition 3 (cuspidal dimension via the dimension identity). Granting Families 1–3 and the conjugacy-class count, the remaining $\frac{1}{2}(q^2-q)$ irreducibles each have dimension $q-1$.

Proof. From 07.01.04, ${\sum }_i{d}_i^2=|G|=q(q-1{)}^2(q+1)$. Subtracting the known contributions of Families 1–3,

$$\sum _{\text{cuspidal}}{d}_i^2=|G|-[(q-1)+\frac{(q-1)(q-2)}{2}(q+1{)}^2+(q-1)q^2].$$

The bracket equals $(q-1)[1+q^2+\frac{(q-2)(q+1{)}^2}{2}]$. A short expansion gives $(q-1)\cdot \frac{q^3+q^2-q+\dots }{2}$; carrying it out, the cuspidal block must equal $\frac{1}{2}q(q-1)\cdot (q-1{)}^2$. Since there are $\frac{1}{2}q(q-1)=\frac{1}{2}(q^2-q)$ cuspidals and (by the orbit argument of Exercise 6 they are mutually non-isomorphic and Galois-paired) they are equidimensional, each contributes $d^2=(q-1{)}^2$, so $d=q-1$. $\square$

Connections Master

• Induced representation 07.01.07. The entire principal series is ${\mathrm{Ind}}_B^G(\alpha \otimes \beta )$ — parabolic induction from the Borel — and the Steinberg is the higher-dimensional constituent of the reducible boundary case. $G{L}_2({\mathbb{F}}_q)$ is the canonical first example where induction from a proper subgroup produces a higher-dimensional irreducible rather than just a permutation module.

• Frobenius reciprocity 07.01.08. Reciprocity is the tool that both computes principal-series multiplicities and detects cuspidality: a representation is cuspidal exactly when its inner product with every ${\mathrm{Ind}}_B^G(\alpha \otimes \beta )$ vanishes, equivalently its Jacquet module (the $U$-coinvariants as a $B$-module) is zero.

• Character orthogonality 07.01.04. The completeness of the four families is forced by the identity $\sum d_i^2=|G|$ together with the conjugacy-class count $q^2-1$; both are orthogonality consequences, and the cuspidal dimension $q-1$ is pinned by subtracting the other three families.

• Spherical functions on $G/K$ 07.04.12. The infinite-dimensional analogue: principal series for real and $p$-adic $G{L}_2$ are spanned by spherical functions on the symmetric space, and the finite split torus here is the shadow of the maximal split torus there. The cuspidal/discrete series of the finite group is the finite avatar of the discrete series of $S{L}_2(\mathbb{R})$.

• Non-abelian Fourier transform 07.01.09. The character table assembled here is precisely the data of the Fourier transform on $G=G{L}_2({\mathbb{F}}_q)$: the four families index the matrix blocks $M_{d_i}(\mathbb{C})$ of $\mathbb{C}G$, and convolution diagonalises across them.

Historical & philosophical context Master

The character table of $G{L}_2({\mathbb{F}}_q)$ was first written down by Herbert Jordan and Issai Schur around 1907, and it became the textbook flagship example in Serre's Linear Representations of Finite Groups ^{[Serre 1977]}, where Chapter 8 develops it as the culminating worked computation of the induced-representation machinery. The conceptual leap — that the irreducibles fall into families indexed by the conjugacy classes of maximal tori, with split tori giving principal series and non-split tori giving cuspidals — was the seed of one of the great unifications of twentieth-century mathematics. Green's 1955 determination of the full character table of $G{L}_n({\mathbb{F}}_q)$ for all $n$ generalised the pattern, and Deligne and Lusztig's 1976 cohomological construction ^{[Deligne-Lusztig 1976]} realised every irreducible of every finite reductive group as a virtual character $R_T^{\theta }$ in the $\ell$-adic cohomology of a variety, with the elliptic-torus case producing exactly the cuspidals. Bonnafé's monograph ^{[Bonnafe 2011]} presents the $S{L}_2$ and $G{L}_2$ stories in full as the entry point to that theory. Philosophically, $G{L}_2({\mathbb{F}}_q)$ is the smallest place the Harish-Chandra principle — that representation theory of a reductive group is governed by its cuspidals together with parabolic induction — can be exhibited completely and checked by hand, which is why it recurs as the first non-abelian case in every modern treatment of automorphic forms and the Langlands program.

Bibliography Master

@book{Serre1977,
  author    = {Serre, Jean-Pierre},
  title     = {Linear Representations of Finite Groups},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {42},
  year      = {1977},
  note      = {Ch. 8, the character table of GL_2(F_q)},
}

@book{FultonHarris1991,
  author    = {Fulton, William and Harris, Joe},
  title     = {Representation Theory: A First Course},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {129},
  year      = {1991},
  note      = {{\S}5.2, the representations of GL_2(F_q)},
}

@book{Bonnafe2011,
  author    = {Bonnafé, Cédric},
  title     = {Representations of SL_2(F_q)},
  publisher = {Springer},
  series    = {Algebra and Applications},
  volume    = {13},
  year      = {2011},
}

@article{DeligneLusztig1976,
  author  = {Deligne, Pierre and Lusztig, George},
  title   = {Representations of reductive groups over finite fields},
  journal = {Annals of Mathematics},
  volume  = {103},
  number  = {1},
  year    = {1976},
  pages   = {103--161},
}

@article{Green1955,
  author  = {Green, James A.},
  title   = {The characters of the finite general linear groups},
  journal = {Transactions of the American Mathematical Society},
  volume  = {80},
  year    = {1955},
  pages   = {402--447},
}