Values of the Cuspidal Characters on Classes of GL(n, q)

5.4 Cuspidal Characters

5.4.2 Values of the Cuspidal Characters on Classes of GL(n, q)

The values of the cuspidal characters ofGL(n, q) on all the conjugacy classes ofGL(n, q) are easy to compute. We follow the description of Green [28]. Suppose thatf ∈Fq[t] with a rootα.By Lemma 5.2.9 we know that α^q, α^q2· · ·α^q∂f−1 are the other roots of f. Let θ be an arbitrary character of F^∗qⁿ.We defineθ(f) byθ(f) =

∂f−1

X

i=0

θ(α^qi).Now if d|n,we identifyθwithθ↓^F

∗ qn

F^∗_qd.Next we define the class functionχ_θ on g∈GL(n, q) by

χ_θ(g) =







θ(f)φ_l(σ)−1(q^∂f) if [g] is a primary class withσ ` _∂fⁿ, 0 if [g] is not a primary class,

(5.19) whereφl(σ)−1 is the function defined in (5.2).

Theorem 5.4.4. The class function χ_θ defined in (5.19)is a generalized character of GL(n, q) for any character θ of F^∗_qⁿ and if θk∈N D(F^∗_qⁿ),then (−1)ⁿ⁻¹χθ_k ∈Irr(GL(n, q)).

PROOF. See Fulton [22], Green [27] or Green [28].

Example 5.4.1. Consider the central elements g = αIn = (t−α,1, n,1ⁿ), α ∈ F^∗q of GL(n, q), which are self-classes. Since the characteristic polynomial of g is (t−α)ⁿ, it follows by Definition 5.2.2 that [g] is primary. If θk is any character of F^∗_qⁿ and F^∗_qⁿ =hθi,then

θ_k(f) =θ_k(t−α) =

1−1

X

i=0

θ_k(α^qi) =θ_k(α) =θ(α^k).

Also

φl(σ)−1(q^∂f) =φ_l(1ⁿ)−1(q) =

n−1

Y

i=1

(1−qⁱ) = (1−q)(1−q²)· · ·(1−qⁿ⁻¹).

Now ifθ_k ∈N D(F^∗qⁿ),then Theorem 5.4.4 asserts that (−1)ⁿ⁻¹χ_θk ∈Irr(GL(n, q)) and atg=αI_n, we have

(−1)ⁿ⁻¹χ_θk(αIn) = (−1)ⁿ⁻¹(1−q)(1−q²)· · ·(1−qⁿ⁻¹)θ(α^k)

= (q−1)(q²−1)· · ·(qⁿ⁻¹−1)θ(α^k).

In Example 5.4.1 if α = 1, that is the identity matrix of GL(n, q), then we have the following Corollary.

Corollary 5.4.5. The degree of a cuspidal character of GL(n, q) is(q−1)(q²−1)· · ·(qⁿ⁻¹−1).

The next theorem is of great importance for characters of GL(n, q). It shows that the cuspidal characters are the atoms from which any character ofGL(n, q) is build up.

Chapter 5 — The Character Table of GL(n, q)

Theorem 5.4.6. Every character ofGL(n, q) is either cuspidal or a constituent of anJ

−product of cuspidal characters.

PROOF. The proof is inductively onn.By definition, all characters ofGL(1, q) are cuspidal. Assume the result is true ∀1 ≤m ≤ n−1. Let χ ∈Irr(GL(n, q)). If χ is cuspidal, then there is nothing to prove. Let χ be a principal series character. Thenχ is a constituent of

k

O

i=1

χ_i↑^GL(n,q)_P

λ for some P_λ such that λ = (λ₁, λ₂,· · · , λ_t) ∈ P(n), t 6= 1 and χ_i ∈ Irr(GL(λ_i, q)). By hypothesis each χi is either cuspidal or a constituent of J

−product of some cuspidal characters. Hence χ is a constituent of anJ

−product of cuspidal characters.

Remark 5.4.3. By the above theorem (or by definition), we know that any χ_k∈Irr(GL(1, q)) = Irr(F^∗q) is a cuspidal character. Recall that characters ofGL(2, q) fall into four types, where char- acters of typeχ⁽⁴⁾ are the cuspidal characters ofGL(2, q).We also recall thatχ⁽³⁾_k,l =χ_kχ_l↑^GL(2,q)_{U T(2,q)}= χ_kχ_l,while the characters χ⁽¹⁾_k and χ⁽²⁾_k appeared as constituents of χ_k,k where χ_kχ_k↑^GL(2,q)_{U T(2,q)}= χ_kχ_k.This show that any χ⁽³⁾_k,l is anJ

-product of cuspidal characters, while any characterχ⁽¹⁾_k or χ⁽²⁾_k appears as a constituent of an J

-product of cuspidal characters. This confirms Theorem 5.4.6 forGL(2, q).

We illustrate the indexing and the values of the cuspidal characters of GL(2, q), GL(3, q) and GL(4, q) in the following examples.

Example 5.4.2. Consider GL(2, q) and let θ_k be a character of F^∗_q2. We determine the non- decomposable characters of F^∗_q2. The norm map N2,1 : F^∗_q2 −→ F^∗_q is given by N2,1(r) = r.r^q = r^q+1, r ∈ F^∗_q2. Now θ_k has two conjugate characters, namely θ_k itself and θ_k = θ^q_k, where θ_k^q is given byθ_k^q(r) =θk(r^q).Ifq+ 1|k,we can see that θ_k^q =θk.Thereforeθk∈N D(F^∗_q2) if and only if k∈K ={1,2,· · · , q²−1} − {q+ 1,2(q+ 1),· · ·,(q−1)(q+ 1)}.Thus |K|=q²−q.It easily to see that for any k∈K, we haveθ_kq ∈K and θ_kq =θ_k.Therefore in indexing the cuspidal characters of GL(2, q), whenever we choose k∈K, we take off kq from K. Thus we get a set of ^q2₂^−q such k to index the cuspidal characters χ_k of GL(2, q).

Next we calculate the values of the cuspidal charactersχ_θk on classes of GL(2, q).Let g₁, g₂, g₃, g₄ be elements in classes of types T⁽¹⁾, T⁽²⁾, T⁽³⁾ T⁽⁴⁾ respectively. By Example 5.4.1 we have χ_θk(g1) = (−1)²⁻¹(1−q)θ_k(α) = (q−1)θ_k(α) = (q−1)αb^k. The characteristic polynomial of g2

is f(t) = (t−α) for some α ∈ F^∗q and the associated partition to [g₂] is λ = (2) ` 2. Thus by (5.19), we have θk(f) = θk(t−α) =θk(α) =αb^k.Also φl(λ)−1(q^∂f) =φ1−1(q) = 1.Now Theorem 5.4.4 asserts that χ_θk(g₂) = (−1)²⁻¹θ_k(α) = −αb^k.The characteristic polynomial of g₃ splits into two distinct linear factors. Immediately, χθk(g3) = 0. The last case where g4 has characteristic polynomialf(t) =t²+at+b∈Fq[t],which is irreducible, that is f has eigenvaluesr andr^q where

T C T

r∈Fq²\Fq.Here we haveθ_k(f) =

∂f−1

X

i=0

θ_k(r^qi) =θ_k(r) +θ_k(r^q).Also φ_l(λ)−1(q^∂f) =φ1−1(q²) = 1.

By Theorem 5.4.4 we have

χ_θk(g₄) = (−1)²⁻¹(θ_k(r) +θ_k(r^q)) =−(θ_k(r) +θ_k(r^q)) =−(rb^k+br^kq).

This completes the cuspidal characters ofGL(2, q).

Sometimes we may writeχ_k in place ofχ_θk.

Example 5.4.3. The case n = 3 is very similar to the case n= 2 since 2 and 3 are both prime numbers. Considerk∈ {1,2,· · ·, q³−1} such thatq²+q+ 1-k.If we choose suchk, we exclude kq and kq² from the set{1,2,· · · , q³−1}.Note that if q²+q+ 1-k,it does not also divide kq or kq².We get ^q3₃^−q cuspidal characters ofGL(3, q),which their values are given by

χ_k(g) =











































(q−1)²(q+ 1)αb^k ifg is of type T⁽¹⁾,

−(q−1)αb^k ifg is of type T⁽²⁾, αb^k ifg is of type T⁽³⁾, 0 ifg is of type T⁽⁴⁾, 0 ifg is of type T⁽⁵⁾, 0 ifg is of type T⁽⁶⁾, 0 ifg is of type T⁽⁷⁾, sb^k+bs^kq+bs^kq2 ifg is of type T⁽⁸⁾.

(5.20)

Example 5.4.4. We calculate the cuspidal characters of GL(4, q).Firstly we determine the non- decomposable characters of F^∗_q4. Assume that k ∈ {1,2,· · ·, q⁴−1} and let θk be a character of F^∗_q4. We consider the norm maps N_4,1 : F^∗_q4 −→ F^∗q and N_4,2 : F^∗_q4 −→ F^∗_q2, which are given by N_4,1(r) =r^q

4−1

q−1 =r^q3+q2+q+1 and N_4,2(r) = r

q4−1

q2−1 =r^q2+1, for all r ∈ F^∗_q4.Now θ_k ∈ N D(F^∗_q4) if and only if q³+q²+q+ 1-k and q²+ 1 -k.Note that q³+q²+q+ 1 = (q²+ 1) +q(q² + 1) = (q+ 1)(q² + 1). This is reduced to say that θ_k ∈ N D(F^∗_q4) if and only if q²+ 1 - k. Equivalently θk ∈ N D(F^∗_q4) ⇐⇒ k ∈ {1,2,· · · , q⁴ −1} \ {q² + 1,2(q²+ 1),· · ·,(q² −1)(q² + 1)}. This gives (q⁴−1)−(q²−1) =q⁴−q² non-decomposable characters ofF^∗_q4. Now each orbit of the action of Γ = Γ(F_q⁴ :Fq) onN D(F^∗_q4) consists of four conjugate characters namely, θkΓ ={θ_k, θ^q_k, θ_k^q2, θ^q_k³}.

To parameterize a cuspidal character of GL(4, q), we choose from eachθ_k^Γ a representative char- acter θk since χθk = χ_θ^q

k = χ

θ^q_k² = χ

θ^q_k³. Therefore we have ¹₄(q⁴ −q²) cuspidal characters χθk of GL(4, q).Note thatI₄(q) = ¹₄(q⁴−q²).

Chapter 5 — The Character Table of GL(n, q)

To evaluate the cuspidal characters on classes of GL(4, q) we use Theorem 5.4.4 and similar steps used in calculating the values of the cuspidal characters of GL(2, q). For example consider [g] of type T⁽¹⁸⁾, which is given by the data ((t² +at+b),2,2,(1,1)). Let r and r^q be the roots of f(t) = t² +at+b. Then θk(f) = θk(r) +θk(r^q) and φl(λ)−1(q^∂f) = φl((1,1))−1(q²) = (1−q²).

Therefore we have

χ_k =χ_θk = (−1)⁴⁻¹(1−q²)(θ_k(r) +θ_k(r^q)) = (q²−1)(br^k+rb^kq).

Similarly we can calculate the values of the cuspidal characters on all other primary classes of GL(4, q).Let ∆ ={6,7,8,9,10,11,12,13,14,15,16,17,18,21}.In Table 5.4 we have skipped giving relevant information for classes of types T⁽ⁱ⁾ for i ∈ ∆, since all classes of these types are not primary and by Theorem 5.4.4 the values of the cuspidal characters on these classes are zero. The values of the cuspidal characters on classes of GL(4, q) are given in Table 5.4.

Remark 5.4.4. Note that the values of the cuspidal characters ofGL(2, q) given by Example 5.4.2, where we used the non-decomposable characters of F^∗_q2,the same as the values of χ⁽⁴⁾_k =πk given in Table 4.2, whereπ_k is written as a combination in terms of some characters ofGL(2, q).

T C T

Table5.4:ValuesofthecuspidalcharactersofGL(4,q) TypeofRootsoftheAssociated classescharacteristicpartitionλwithθk(f)φl(λ)−1(q∂f )(−1)n−1 χθk polynomialofgtheclass T(1) α,α∈F∗ q14 θk(α)(q−1)(q2 −1)(q3 −1)(q−1)(q2 −1)(q3 −1)θk(α) T(2) α,α∈F∗ q(2,1,1)θk(α)(q−1)(q2 −1)−(q−1)(q2 −1)θk(α) T(3) α,α∈F∗ q(2,2)θk(α)−(q−1)(q−1)θk(α) T(4) α,α∈F∗ q(3,1)θk(α)−(q−1)(q−1)θk(α) T(5) α,α∈F∗ q(4)θk(α)−1θk(α) T(19) r,rq ,r∈Fq2\Fq(1,1)θk(r)+θk(rq )−(q2 −1)(q2 −1)(θk(r)+θk(rq )) T(20) r,rq ,r∈Fq2\Fq(2)θk(r)+θk(rq )1θk(r)+θk(rq ) T(22) s,sq ,sq2 ,(2)θk(s)+θk(sq )1θk(s)+θk(sq ) sq3 ,s∈Fq4\Fq2θk(sq2 )+θk(sq3 )+θk(sq2 )+θk(sq3 ) T(i) ,i∈∆0

Chapter 5 — The Character Table of GL(n, q)