3 Induced representations 3.5 Mackey’s formula — nonexaminable4 Linear Lie groups and their Lie algebras

3.6 Example: GL₂(F_p) — nonexaminable

Let $\mathbb{F}_{p}$ be the field of integers modulo $p$ , with $p$ a prime. We will construct some of the irreducible representation of $G=GL_{2}(\mathbb{F}_{p})$ by induction from the subgroup of upper triangular matrices. We will use Mackey theory to show that these representations are irreducible.

3.6.1 The group GL₂(F_p)

Proposition 3.26.

The order of the group $GL_{2}(\mathbb{F}_{p})$ is

p(p+1)(p-1)^{2}.

We let $B\subset GL_{2}(\mathbb{F}_{p})$ be the subgroup of upper triangular invertible matrices:

B=\left\{\begin{pmatrix}*&*\\ 0&*\end{pmatrix}\in GL_{2}(\mathbb{F}_{p})\right\}.

The group $B$ is sometimes called a Borel subgroup.

Proposition 3.27.

The index of $B$ in $G$ is $p+1$ . A set of left coset representatives for $G/B$ is

\left\{\begin{pmatrix}0&1\\ 1&0\end{pmatrix}\right\}\cup\left\{\begin{pmatrix}1&0\\ i&1\end{pmatrix}:0\leq i\leq p-1\right\}.

We write $w=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$ . We will want to know the double cosets of $B$ in $G$ , and the result is

Proposition 3.28.

We have

B\backslash G/B=B\cup BwB.

For later use, we compute

w\begin{pmatrix}x&z\\ 0&y\end{pmatrix}w^{-1}=\begin{pmatrix}y&0\\ z&x\end{pmatrix}

so that

B_{w}=B\cap{}^{w}B=\left\{\begin{pmatrix}x&0\\ 0&y\end{pmatrix}:x,y\in\mathbb{F}_{p}^{\times}\right\}.

3.6.2 Characters of B and their inductions.

Let $\chi_{1}$ and $\chi_{2}$ be two characters of $\mathbb{F}_{p}^{\times}$ . Then there is a character

\chi_{1}\cdot\chi_{2}:B\to\mathbb{C}^{\times}

defined by

(\chi_{1}\cdot\chi_{2})\left(\begin{pmatrix}x&z\\ 0&y\end{pmatrix}\right)=\chi_{1}(x)\chi_{2}(y).

Define

\pi(\chi_{1},\chi_{2})=\operatorname{Ind}_{B}^{G}(\chi_{1}\cdot\chi_{2}).

If $\chi:\mathbb{F}_{p}^{\times}\to\mathbb{C}^{\times}$ is a character, then we get a character

\chi\circ\det:G\to\mathbb{C}^{\times}

taking $g$ to $\chi(\det(g))$ .

Lemma 3.29.

If $\chi_{1},\chi_{2},\chi:\mathbb{F}_{p}^{\times}\to\mathbb{C}^{\times}$ are characters, then

\pi(\chi_{1}\chi,\chi_{2}\chi)\cong(\chi\circ\det)\otimes\pi(\chi_{1},\chi_{2}).

Lemma 3.30.

On $B_{w}$ , we have

{}^{w}(\chi_{1}\cdot\chi_{2})=\chi_{2}\cdot\chi_{1}.

Theorem 3.31.

If $\chi_{1},\chi_{2}:\mathbb{F}_{p}^{\times}\to\mathbb{C}^{\times}$ are characters, then:

1.

If $\chi_{1}\neq\chi_{2}$ , then

$\pi(\chi_{1},\chi_{2})$

is an irreducible representation of dimension $p+1$ .
2.

There is an irreducible representation $\sigma$ of dimension $p$ such that, for any $\chi:\mathbb{F}_{p}^{\times}\to\mathbb{C}^{\times}$ ,

$\pi(\chi,\chi)\cong(\chi\circ\det)\otimes\left(\mathbbm{1}\oplus\sigma\right).$
3.

There is an isomorphism

$\pi(\chi_{1},\chi_{2})\cong\pi(\chi_{1}^{\prime},\chi_{2}^{\prime})$

if and only if $(\chi_{1},\chi_{2})=(\chi_{1},\chi_{2})$ or $(\chi_{2},\chi_{1})$ .
4.

For $\chi\neq\chi^{\prime}$ , the representations $(\chi\circ\det)\otimes\sigma$ and $(\chi^{\prime}\circ\det)\otimes\sigma$ are not isomorphic.

One can prove the theorem using Mackey theory, but it is also possible to avoid that and directly compute the inner products of the induced characters.

The representations $\pi(\chi_{1},\chi_{2})$ , $\chi\circ\det$ , and $\chi\circ\det\otimes\sigma$ are not all of the irreducible representations of $GL_{2}(\mathbb{F}_{p})$ . There is another family of $p-1$ -dimensional representations called the cuspidal representations, which are more difficult to construct: see [1] Lecture 5, or [3] Chapter 28.

3.6.3 Computing the induced character

Finally, we can compute the character of $\pi(\chi_{1},\chi_{2})$ .

We will need to know the conjugacy classes of $G$ and of $B$ . The characteristic polynomial almost determines the conjugacy classes; the result is as follows. Recall that the size of the conjugacy class of $g\in G$ is $\frac{|G|}{|C_{G}(g)|}$ where $C_{G}(g)$ is the centralizer of $g$ in $G$ .

Proposition 3.32.

Every element of $G$ is conjugate to exactly one of the following elements:

1.

$\begin{pmatrix}x&0\\ 0&x\end{pmatrix},x\in\mathbb{F}_{p}^{\times}$
2.

$\begin{pmatrix}x&1\\ 0&x\end{pmatrix},x\in\mathbb{F}_{p}^{\times}$
3.

$\begin{pmatrix}x&0\\ 0&y\end{pmatrix},x,y\in\mathbb{F}_{p}^{\times},x\neq y$
4.

$\begin{pmatrix}0&-d\\ 1&t\end{pmatrix},t\in\mathbb{F}_{p},d\in\mathbb{F}_{p}^{\times}$ such that $X^{2}-tX+d=0$ has no solutions in $\mathbb{F}_{p}$ .

For each type of element $G$ , the order of the centralizer subgroup is:

1.

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

$p(p-1)$
3.

$(p-1)^{2}$
4.

$(p-1)(p+1)$ .

Proof.

We sketch the proof. Let $X^{2}-tX+d$ be the characteristic polynomial of an element $g\in G$ . If it has distinct roots in $\mathbb{F}_{p}$ , then $g$ is diagonalizable and we are in case 3. If it has no roots in $\mathbb{F}_{p}$ , then choosing a basis $v, g v$ (for some $V\in\mathbb{F}_{p}^{2}$ ) puts us in case 4. If the roots are identical, both equal to $x$ , there are two possibilities: either the matrix is a scalar matrix, giving case 1, or $v,(g-xI)v$ is a basis for some $v\in\mathbb{F}_{p}^{2}$ and then we are in case 2.

In each case, the centralizer can be computed explicitly and checked to have the order claimed (the most difficult case is the fourth). ∎

Now let $\chi:\mathbb{F}_{p}^{\times}\to\mathbb{C}^{\times}$ be a character. We compute the character of $\pi(\chi,\mathbbm{1})$ ; after twisting by a character of the form $\chi^{\prime}\circ\det$ this gives the general case. In the case $\chi=\mathbbm{1}$ , we also record the character of $\sigma$ .

Theorem 3.33.

The characters are as follows (where $x\neq y$ , and $X^{2}-tX+d$ has no solutions in $\mathbb{F}_{p}$ ):

\begin{array}[]{cc|ccc}&\begin{pmatrix}x&0\\ 0&x\end{pmatrix}&\begin{pmatrix}x&1\\ 0&x\end{pmatrix}&\begin{pmatrix}x&0\\ 0&y\end{pmatrix}&\begin{pmatrix}0&-d\\ 1&t\end{pmatrix}\\ &1&(p-1)(p+1)&p(p+1)&p(p-1)\\ \hline\cr\pi(\chi,\mathbbm{1})&(p+1)\chi(x)&\chi(x)&\chi(x)+\chi(y)&0\\ \sigma&p&0&1&-1\\ \end{array}

3.6 Example: GL2(Fp) — nonexaminable

3.6.1 The group GL2(Fp)