Tensor products of irreducible representations of the group G = GL(3,q)

0.16 MB·English

by  L. Aburto-Hageman

TENSOR PRODUCTS OF IRREDUCIBLE REPRESENTATIONS
OF THE GROUP GL(3,F )
q
9 LUISAABURTO-HAGEMAN,JOSE´ PANTOJA,ANDJORGESOTO-ANDRADE
0
0
2 Abstract. Wedescribethetensorproductsoftwoirreduciblelinearcomplex
n representations of the group G = GL(3,Fq) in terms of induced representa-
tions by linear characters of maximal torii and also in terms of classical and
a
J generalized Gelfand-Graev representations. Our results include MacDonald’s
conjectures for G and at the same time they are extensions to G of finite
2
counterparts to classical results on tensor products of holomorphic and anti-
holomorphic representations of the group SL(2,R). Moreover they provide
]
T an easy way to decompose these tensor products, with the help of Frobenius
R reciprocity. Wealsostatesomeconjectures forthegeneralcaseofGL(n,Fq).
.
h
t
a
m 1. Introduction
[
This paper studies the tensor product of two irreducible linear complex repre-
1 sentations of the group G = GL(3,F ), generalizing previous work [1] of the first
q
v two named authors on GL(2,F ) and SL(2,F ).
q q
2
Our main idea is to express tensor products of irreducible representations in
9
terms of induced representations. Indeed, this paper adds to the experimental evi-
2
0 dence which strongly suggests that, for classicalgroups, tensor products of generic
. irreduciblerepresentationsareessentiallyinducedrepresentationsbysuitablelinear
1
characters from either of the involved torii. Here “essentially” means “up to lower
0
9 dimensional correcting terms”.
0 General results of this type are of interest for several reasons.
v: First, for finite classical groups, tensor products of irreducible representations
i realized as induced representations of this kind, have appeared already in the con-
X
text of the famous MacDonald’s conjectures, which state, in the case of GL(n,F ),
q
r thatthe tensorproductofthe canonicalSteinberg representationStwithageneric
a
irreducible representation R (θ), associated to a character θ, in general position,
T
ofa maximaltorus T ofGL(n,F ), equalsthe representationinduced byθ fromT
q
to GL(n,F ). These conjectures were proved much later by Deligne and Lusztig
q
as a corollary of their construction in [4].
Second,itwaspointedouttousbyA.Guichardet,thatremarkablyenoughthe-
oretical physicists, like Rideau [6], have been interested on their own in describing
tensor products of irreducible unitary representations of a same series for classical
real Lie groups like SL(2,R), proving that the tensor product of holomorphic and
antiholomorphic discrete series representations of SL(2,R) is given by a suitable
ThefirstandsecondauthorswerepartiallysupportedbyPontificiaUniversidadCato´licadeVal-
para´ıso. ThesecondandthirdauthorswerepartiallysupportedbyFONDECYTGrants1040444,
1070246andbyPICSCNRS1413
Mathematics Subject Classification(2000): 20C33,20C15.
1
2 LUISAABURTO-HAGEMAN,JOSE´ PANTOJA,ANDJORGESOTO-ANDRADE
induced representation from the corresponding torus. Rideau’s results were ex-
tended to principal series representations as well by Guichardet, and we realized
that analogous results would hold for GL(2,F ) if adequate correcting terms were
q
introduced. So in fact, Rideau’s results and MacDonald’s conjectures dwell under
the same roof. A complete description of tensor products of irreducible represen-
tations of GL(2,F ) and SL(2,F ) in terms of induced representations, was then
q q
given in [1].
Third, tensor products of irreducible representations may be decomposed quite
easily,via Frobenius reciprocity,once youhave describedthem interms ofinduced
representations (see section 4 below).
Fourth,therealizationofthetensorproductoftwoirreduciblerepresentationsas
aninduced representationfroma linear characteralsoallows us to guess andglean
interesting relations between special functions of various sorts. Indeed, such an
induced representation may be looked upon as a ”twisted” natural representation,
for which spherical functions may be calculated, in the multiplicity-free case, as in
[8]. Forinstance,recentwork[5]onthe relationshipbetweenclassicalKloosterman
sums and the so calledLegendre sums andSoto-Andrade sums for G=PGL(2,q),
may be understood as a consequence of the fact that
St⊗St=(Ind1)⊕St
T ↑G
where T denotes the anisotropic torus of G, if you recall that this induced
representation is just the natural multiplicity-free representation of G associated
to its homographic action on the (double cover of) finite Poincar´e’s upper half
plane,whosesphericalfunctions aregivenby the lasttwoaforementionedsums [8].
AlsoGel’fand-Graev representationsappearin this way(see subsection3.2below).
Fifth, there is, on the other hand, a non obvious but close connections between
tensorproductsandGelfandmodels forthe classicalgroups. RecallthataGelfand
modelforagroupGisanyrepresentationof G whichdecomposeswithmultiplicity
one as the sum of all the irreducible representations of G.
Intriguing experimental evidence suggests that quite often Gelfand models or
”quasi-models” may be obtained as tensor products of Steinberg representations.
ThefirstcaseisG=PGL(2,q),wherethe tensorsquareofthe Steinbergrepresen-
tation affords a quasi-model for G, where only the sign representation is missing.
Analogous results seem to hold for PGL(n,q). It is indeed very interesting to re-
alize a Gelfand model or quasi-model as an induced representation from a linear
character,specially from the unit character. In the latter case we say that we have
a geometrical Gelfand model or quasi-model.
Sixth, although the problem of decomposing tensor products in the p-adic case
was treated in [2, 3, 9], apparently tensor products of two different series of rep-
resentations have not been studied, and the description of this tensor products as
induced representation by linear characters from torii has not been worked out.
So, in this paper we describe tensor products of two irreducible representations
of the group G = GL(3,F ) as induced representations by linear characters from
q
torii of G, up to adequate lower dimensional correcting terms. We notice that the
situation is more complex than in dimension 2, and subtler in some cases (see for
examplecase10oftheorem1below). Weconcentratehereon”generic”irreducible
representations, since the ”non generic” or ”degenerate” lower dimensional irre-
ducible representationsmay be obtainedfromlimiting casesof the generic ones,as
TENSOR PRODUCTS OF IRREDUCIBLE REPRESENTATIONS OF THE GROUP GL(3,Fq) 3
in lemma 2. We show then how to get formulasfor the non genericrepresentations
from the generic ones.
Dependingontheseriesofrepresentationstowhichanirreduciblerepresentation
belongs, a torus is associated in a natural way, and it is the one we use in the
corresponding formula. Thus in the cases where we deal with the tensor product
of two irreducible representations belonging to two different series, we give two
descriptions, one in terms of each torus.
However, another approach is also possible in this case: to express our tensor
product in terms of the intersection of the two different torii (the center Z(G) of
G) times the upper unipotent subgroup N. More specifically, let α be a character
of k× =Z(G)=Z and let ψ be a non trivial character of k× seen as character on
N by
1 x z
 0 1 y 7−→ψ(x+y).
0 0 1
 
TheclassicalGelfand-Graevrepresentationof GistherepresentationIndG (αψ).
ZN
We call generalized Gelfand-Graev representation of G, any representation of the
formIndG (αψ), where N is the subgroupof N defined by the condition x=0.
ZN1 1
Proposition1 belowdescribes the tensor productofa discreteseriesrepresentation
and a principal series representation of G as the sum of these two Gelfand-Graev
representations,classical and generalized.
Asimilarresultshouldbe obtainedinthe generalcaseofdimensionn. We state
this as a conjecture, after proposition 1 below.
The paper is organized as follows: section 2 gives the preliminaries, mainly the
character table of the group G; section 3 is devoted to our main results.
2. Preliminaries
2.1. Notations. Let F be the finite field of q = pn elements, p prime. Then the
q
quadratic and cubic extensions of Fq are Fq2 and Fq3 respectively. We identify
z ∈ Fq3 with the Fq-automorphism of Fq3 given by x 7−→ zx, and we identify
w ∈ Fq2 with the Fq-automorphism of Fq2 given by x 7−→ wx. We also denote by
z the matrix of the above automorphism with respect to the basis 1,σ,σ2 , σ a
generator of Fq3 over Fq. Similarly, for w in Fq2, w also denotes the(cid:8)matrix(cid:9)of the
automorphismx7−→wx with respect to the basis {1,τ}, τ a generatorof Fq2 over
F .
q
TheabovedefinesamonomorphismofF× intoGwhoseimageistheanisotropic
q3
torus T .
a
We denote by T the isotropic torus of G; thus T is the image of F××F××F×
i i q q q
in G. Similarly we denote by T the intermediate torus
m
x 0
T = | a∈F×, x∈F× .
m (cid:26)(cid:18) 0 a (cid:19) q q2(cid:27)
Finally, we set L equal to the intermediate Levi subgroup GL(2,F )×F×.
q q
We adopt the convention that η denotes a representation of G as well as its
character.
2.2. The conjugacy classes of G= GL(3,F ). The elements
q
4 LUISAABURTO-HAGEMAN,JOSE´ PANTOJA,ANDJORGESOTO-ANDRADE
a 0 0 a 1 0 a 1 0
Ta = 0 a 0 ; Ta = 0 a 0 ; Ta = 0 a 1 
1 11
0 0 a 0 0 a 0 0 a
     
a 0 0 a 1 0 a 0 0
Tab = 0 a 0 ; Tab = 0 a 0 ; Tabc = 0 b 0 
1
0 0 b 0 0 b 0 0 c
     
(where a 6= b, a 6= c, b 6= c) are representatives of different conjugacy classes in G.
In addition, if we set
Tκa = κ 0 ; Tz = z (where a ∈ F×, κ ∈ F× and z ∈ F×), then the
(cid:18) 0 a (cid:19) q q2 q3
setof3by3matrices Ta,Ta,Ta,Tab,Tab,Tabc,Tκa,Tz isa full set of represen-
1 11 1
tatives of the conjugac(cid:8)y classes of G. (cid:9)
Moreover the following holds.
Lemma 1. We have
(1) For T =Ta,Ta,Tab, we have X ∈G|XTX−1∈T =φ and also
1 11 1 i
X ∈G|XTκaX−1 ∈T =(cid:8) X ∈G|XTzX−1 ∈(cid:9)T =φ.
i i
(2) X(cid:8)∈G|XTabX−1 ∈T =(cid:9) (cid:8) (cid:9)
i
(cid:8) (cid:9)
r s 0 k 0 0 0 0 1
=  t u 0 , 0 r s  0 1 0 ,
 0 0 k 0 t u 1 0 0
    
r 0 s 1 0 0
 r s
 0t k0 0u  00 01 10 |k ∈F×q ,(cid:18) t u (cid:19)∈GL(2,Fq)
  
0 0 k k 0 0 0 0 1 
We note that  r s 0 = 0 r s  0 1 0  and
t u 0 0 t u 1 0 0
    
r s 0 r 0 s 1 0 0
 0 0 k = 0 k 0  0 0 1 
t u 0 t 0 u 0 1 0
    
(3) X ∈G|XTabcX−1 ∈T =T ×S
i i 3
(4) (cid:8)For T =Ta,Ta,Tab,Tab(cid:9)we get X ∈G|XTX−1∈T =φ; moreover
1 11 1 a
X ∈G|XTabcX−1 ∈T =(cid:8)φ and X ∈G|XTκaX(cid:9)−1 ∈T =φ
a a
(5) X(cid:8)∈G|XTzX−1 ∈T =T(cid:9) ×Γ (cid:8) (cid:9)
a a 3
(cid:8)where Γ3 is the Galois gr(cid:9)oup of the cubic extension generated by Frobenius
automorphism, acting naturally on T .
a
(6) For T =Ta,Ta,Tab, we get X ∈G|XTX−1∈T =φ, and also
1 11 1 m
X ∈G|XTzX−1 ∈T (cid:8)=φ (cid:9)
m
(7) X(cid:8)∈G|XTabX−1 ∈T (cid:9)=L
m
(8) (cid:8)X ∈G|XTκaX−1 ∈T (cid:9) =T × Γ2 0
m m (cid:26)(cid:18) 0 1 (cid:19)(cid:27)
(cid:8) (cid:9)
where Γ is (isomorphic to) the Galois group of the quadratic extension
2
of the field F , acting through Frobenius automorphism of the quadratic
q
extension.
2.3. The characters of G= GL(3,F ). The followingtable givesthe irreducible
q
characters of the group G. This result leans on the work of Steinberg [7].
TENSOR PRODUCTS OF IRREDUCIBLE REPRESENTATIONS OF THE GROUP GL(3,Fq) 5
Character Table of GL(3,F )
q
Elementary divisors of Represen- χ1 χq2+q χq3
conjugacy classes tatives α α α
x−a,x−a,x−a Ta α3(a) q2+q α3(a) q3α3(a)
1,x−a,(x−a)2 Ta α3(a) (cid:0) qα3(cid:1)(a) 0
1
1,1,(x−a)3 Ta α3(a) 0 0
11
1,x−a,(x−a)(x−b) Tab α a2b (q+1)α a2b qα a2b
1,1,(x−a)2(x−b) Tab α(cid:0)a2b(cid:1) α a2b(cid:0) (cid:1) (cid:0)0 (cid:1)
1
1,1,(x−a)(x−b)(x−c) Tabc α(cid:0)(abc(cid:1)) 2α(cid:0)(abc(cid:1)) α(abc)
1,1,(x−a)(x−κ)(x−κq) Tκa α(aκκq) 0 −α(aκκq)
1,1,(x−z)(x−zq) x−zq2 Tz α zzqzq2 −α zzqzq2 α zzqzq2
a,b,c∈F×,a6=b6=(cid:16)c6=a (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
q
κ ∈Fq2 −Fq,z ∈Fq3 −Fq
Number of characters q−1 q−1 q−1
Series Principal series Principal series Principal series
Represen- χq2+q+1 χq(q2+q+1)
tatives α,β α,β
Ta q2+q+1 αβ2 (a) q q2+q+1 αβ2 (a)
Ta (cid:0) (q+1) α(cid:1)β(cid:0)2 (a(cid:1)) (cid:0) q αβ2(cid:1)(cid:0)(a) (cid:1)
1
Ta αβ(cid:0)2 (a)(cid:1) (cid:0) 0(cid:1)
11
Tab (q+1)α(a)(cid:0)β(a(cid:1)b)+β2(a)α(b) (q+1)α(a)β(ab)+qβ2(a)α(b)
Tab α(a)β(ab)+β2(a)α(b) α(a)β(ab)
1
Tabc α(a)β(bc) α(a)β(bc)
(aP,b,c) (aP,b,c)
Tκa α(a)β(κκq) −α(a)β(κκq)
Tz 0 0
Number of characters (q−1)(q−2) (q−1)(q−2)
Series Principal series Principal Series
Represen- χ(q+1)(q2+q+1) χq3−1 χ(q−1)(q2−1)
tatives α,β,γ α,λ ϕ
Ta (q+1) q2+q+1 (αβγ)(a) q3−1 α(a)λ(a) (q−1) q2−1 ϕ(a)
Ta (2(cid:0)q+1)(αβγ(cid:1))(a) (cid:0) −α((cid:1)a)λ(a) −(q(cid:0)−1)ϕ(cid:1)(a)
1
Ta (αβγ)(a) −α(a)λ(a) ϕ(a)
11
Tab (q+1) (αβ)(a)γ(b) (q−1)α(b)λ(a) 0
(αP,β,γ)
Tab (αβ)(a)γ(b) −λ(a)α(b) 0
1
(αP,β,γ)
Tabc α(a)β(b)γ(c) 0 0
(αP,β,γ)
Tκa 0 −α(a)(λ+λq)(κ) 0
Tz 0 0 ϕ+ϕq +ϕq2 (z)
(cid:16) (cid:17)
∧ ∧
λ∈ F× ϕ∈ F×
q2 q3
(cid:16) (cid:17) (cid:16) (cid:17)
λ6=λq ϕ6=ϕq
Number of
1(q−1)(q−2)(q−3) 1q(q−1)2 1q q2−1
characters 6 3 3
(cid:0) (cid:1)
Principal series Intermediate series Discrete series
6 LUISAABURTO-HAGEMAN,JOSE´ PANTOJA,ANDJORGESOTO-ANDRADE
Thefollowinglemmadescribesthedecompositionofthecharacterobtainedfrom
an irreducible character in the ”degenerate” case, i.e., when two or more of the
(distinct) parameters are now taken to be equal.
Lemma 2. We have
(1) χ(q−1)(q2−1) =χ1 −χq2+q+χq3
α◦N3 α α α
(2) χq3−1 =χq3+q2+q−χq2+q+1
α,β◦N2 α,β α,β
(3) χq3−1 =χq3 −χ1
α,α◦N2 α α
(4) χq(q2+q+1) =χq3 +χq2+q
α,α α α
(5) χq2+q+1 =χ1 +χq2+q
α,α α α
(6) χ(q+1)(q2+q+1) =χq2+q+1+χq(q2+q+1)
α,β,β α,β α,β
(7) χ(q+1)(q2+q+1) =χ1 +2χq2+q+χq3
α,α,α α α α
Proof. Follows from the character table of G. (cid:3)
3. Tensor Products of Irreducible Characters of the Group G.
3.1. Description in terms of induced representations from torii. In what
follows,wedenotebyαanextensiontoF× (ortoF×,whicheveristhe case)ofthe
q3 q2
characterα∈F× . Onthe other hand, for charactersλ andµof F× (or ofF×), λ
q e q3 q2 |
denotes the restriction to F× of the character λ and µq stands for µ◦F, where F
c q
is the Frobenius F×-automorphism of F× (or F×), given by F(x)=xq.
q q3 q2
Theorem 1. With notations as above, we have:
(1) i. χq3−1⊗χ(q+1)(q2+q+1) =IndG (γδλ,αβ)+χq3−1 +
α,λ β,γ,δ Tm αβ,γfδλ
+ χq3−1 +χq3−1 ⊗χq2+q
(cid:16) αγ,βfδλ αδ,βfγλ(cid:17) 1
ii. χq3−1 ⊗χ(q+1)(q2+q+1) =IndG (αλβ,αλγ,αλδ)−χ(q+1)(q2+q+1)
αλ,(α◦N2)λ2 β,γ,δ Ti αλβ,αλγ,αλδ
(2) χq3−1⊗χq3−1 =IndG (λµ,αβ)−χq3−1
α,λ β,µ Tm αβ,λµq
(3) i. χq3−1⊗χ(q−1)(q2−1) =IndG αλϕ −χ(q−1)(q2−1)
α,λ ϕ Ta(cid:16) | (cid:17) αeλe|ϕ
ii. χqα3,λ−1⊗χ(ϕq−1)(q2−1) =IndGTm(λe,eαϕ)+χqα3−ϕ,1λ⊗(cid:16)χ11−χq12+q(cid:17)
(4) χq3 ⊗χ(q+1)(q2+q+1) =IndG (αδ,βδ,γδ)
δ α,β,γ Ti
(5) χq3 ⊗χ(q−1)(q2−1) =IndG (α◦N )ϕ
α ϕ Ta 3
(6) χq3 ⊗χq3−1 =IndG ((α◦N )λ,αβ)
α β,λ Tm 2
(7) i. χq3 ⊗χq3 =IndG (αβ◦N ,αβ)+χq3
α β Tm 2 αβ
ii. χq3 ⊗χq3 =IndG (αβ,αβ,αβ)−χq3 ⊗ 2χq2+q+χ1
α β Ti αβ (cid:16) 1 1(cid:17)
iii. χq3 ⊗χq3 =IndG αβ◦N +χq3 ⊗ χq2+q−χ1
α β Ta 3 αβ (cid:16) 1 1(cid:17)
(8) i. χ(q−1)(q2−1)⊗χ(q+1)(q2+q+1) =IndG ϕ αβγ +χ(q−1)(q2−1)⊗
ϕ α,β,γ Ta (cid:16) (cid:17) ϕ(αgβγ)
2χq2+q+χ1 g
1 1
(cid:16) (cid:17)
TENSOR PRODUCTS OF IRREDUCIBLE REPRESENTATIONS OF THE GROUP GL(3,Fq) 7
ii. χ(q−1)(q2−1)⊗χ(q+1)(q2+q+1) =IndG (α,β,ϕγ)+χ(q+1)(q2+q+1)
ϕ α,β,γ Ti α,β,ϕγ
⊗ χ1−χq2+q
1 1
(cid:16) (cid:17)
(9) χ(q−1)(q2−1)⊗χ(q−1)(q2−1) =
ϕ ψ
IndG ϕψ+χ(q−1)(q2−1)+χ(q−1)(q2−1)−χ(q−1)(q2−1)⊗ χq2+q+χ1
Ta ϕψq ϕψq2 ϕψ (cid:16) 1 1(cid:17)
(q+1)(q2+q+1) (q+1)(q2+q+1)
(10) χ ⊗χ =
α,β,γ δ,ε,η
IndG (αδ,βε,γη)+ χ(q+1)(q2+q+1)+
Ti αδ,βε,γη
(δεη)∈PS3−{I}
χ(q+1)(q2+q+1)+χ(q+1)(q2+q+1) ⊗ χq2+q−2χ1
(cid:18) αε,βη,γδ αη,βδ,γε (cid:19) (cid:16) 1 1(cid:17)
Proof. The proof consists in computing the relevant characters on the different
conjugacyclassesofG(Seesection2). Specifically,thecomputationsrelyonlemma
1 and the character table of G.
As an example of the above, we present a proof of 8.i.
a) Inthe caseofthe conjugacyclassesgivenby Ta,Ta,Ta,Tab,Tab,Tabc and
1 11 1
Tκa, the result follows directly from both the character table and lemma
1.
b) We have
χ(q−1)(q2−1)⊗χ(q+1)(q2+q+1) (Tz)= ϕ+ϕq +ϕq2 (z) 0=0
(cid:18) ϕ α,β,γ (cid:19) h(cid:16) (cid:17) i
Also, using the character table and part 7 of Lemma 1, we have
IndG ϕ αβγ +χ(q−1)(q2−1)⊗ 2χq2+q+χ1 (Tz)=
(cid:18) Ta (cid:16) (cid:17) ϕ(αgβγ) (cid:16) 1 1(cid:17)(cid:19)
1 g q
ϕ αβγ XTzX−1 −ϕ αβγ (z)− ϕ αβγ (z)−
|T |
a XTzXXP∈−G1∈Ta (cid:16)g(cid:17)(cid:0) (cid:1) (cid:16)g(cid:17) (cid:16) (cid:16)g(cid:17)(cid:17)
q2
ϕ αβγ (z)
(cid:16) (cid:16) (cid:17)(cid:17)
Now, by part 5 of lemma 1, the above expression becomes
g
1 q
ϕ αβγ XTzX−1 −ϕ αβγ (z)− ϕ αβγ (z)−
|T |
a X∈TPa×Γ3 (cid:16) (cid:17)(cid:0) (cid:1) (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
ϕ αβγ q2(z)g= |Ta| ϕ αβγ (z)+gϕ αβγ (zq)+gϕ αβγ (zq2)
|T |
(cid:16) (cid:16) (cid:17)(cid:17) a (cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17)
g q g gq2 g
−ϕ αβγ (z)− ϕ αβγ (z)− ϕ αβγ (z)=0.
(cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
g g g (cid:3)
Finally, using lemma 2, the next corollary address the case of irreducible repre-
sentations that arisefrom limit cases of previous formulaswhere now some param-
eters coincide.
Corollary 1. With notations as above, we have
(1) χq3 ⊗ χq2+q+1+χq(q2+q+1) =IndG (αβ,αγ,αγ)
α (cid:20) β,γ β,γ (cid:21) Ti
(2) χq3 ⊗χq2+q =2χq3 +IndG (αβ◦N ,αβ)−IndG (α◦N )(β◦N )
α β αβ Tm 2 Ta 3 3
8 LUISAABURTO-HAGEMAN,JOSE´ PANTOJA,ANDJORGESOTO-ANDRADE
Proof. The first formula follows from4 of theorem 1 above,taking two parameters
to be equal. To prove the second, it is enough to use parts 7 and 5 of theorem 1
and to reduce terms applying part 1 of Lemma 2. (cid:3)
3.2. DescriptionintermsofGelfand-Graevrepresentations. Thenextpropo-
sitiondescribesthetensorproductofadiscreteandaprincipalseriesrepresentation
of G as the sum of the classicaland a generalized Gelfand-Graev representation of
G.
WedenotebyN thestandardunipotentsubgroupofG,andbyN thesubgroup
2
of N whose (1,2) entry is 0, i.e.,
1 x z 1 0 z
N = 0 1 y |x,y,z ∈k, N1 = 0 1 y |y,z ∈k.
 0 0 1   0 0 1 
   
For a non trivial character ϕ of k+, we also denote by ϕ the character on N
   
1 x z
defined by ϕ( 0 1 y )=ϕ(x+y).
0 0 1
 
Proposition 1. With notations as above, we have
χ(q−1)(q2−1)⊗χ(q+1)(q2+q+1) =IndG (ψβγδ)ϕ+IndG (ψβγδ)ϕ
ψ β,γ,δ ZN ZN1
Proof. We compute first the involved characters on the conjugacy class of Ta :
1
χ(q−1)(q2−1)⊗χ(q+1)(q2+q+1) (Ta)=−(q−1)(2q+1)(ψβγδ)(a).
(cid:20) ψ β,γ,δ (cid:21) 1
p q r
Let D1 = X |XT1aX−1 ∈ZN1 = l m n ∈G, and
(cid:8) (cid:9)  0 t 0 
 
p q r  
let D2 = 0 m n  |s6=0,p6=0,ms−tn6=0, we see that,
 0 t s 
 
 p q r
D = X |XT1aX−1 ∈ZN =D1∪D2 = l m n ∈G|ls=0,
(cid:8) (cid:9)  0 t s 
so  
 
IndG (ψβγδ)ϕ+IndG (ψβγδ)ϕ (Ta)=
ZN ZN1 1
1(cid:2) 1 (cid:3)
(ψβγδ)ϕ XTaX−1 + (ψβγδ)ϕ XTaX−1 =
|ZN| 1 |ZN | 1
XP∈D (cid:0) (cid:1) 1 XP∈D1 (cid:0) (cid:1)
1 1 a−1l
= + (ψβγδ)(a)ϕ +
(cid:20)|ZN| |ZN |(cid:21) (cid:18) t (cid:19)
1 t∈F×q;Pq,m∈Fq
pn−lr6=0
1 a−1ps
+ (ψβγδ)(a)ϕ =
|ZN| (cid:18)ms−tn(cid:19)
s,p∈F×qP;r,q∈Fq
ms−tn6=0
q+1 1
= −q3(q−1)2 + −q4(q−1)2 (ψβγδ)(a)
(cid:20)q3(q−1) q3(q−1) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1)
=−(q−1)(2q+1)(ψβγδ)(a).
The computations on the conjugacy class of Ta is similar to the above one. In
11
this case the sum corresponding to ZN has no support.
1
The remaining cases are straightforward. (cid:3)
TENSOR PRODUCTS OF IRREDUCIBLE REPRESENTATIONS OF THE GROUP GL(3,Fq) 9
The above proposition suggests that for G = GL(n,F ) the tensor product of
q
a cuspidal (discrete) and a principal series representation may be expressed as the
directsumof (n−2)(n−1)+1packetsofgeneralizedGelfand-Graevrepresentationsof
2
the same dimension, the first one consisting simply of the classical Gelfand-Graev
representation and the last one consisting only of the generalized Gelfand-Graev
representation induced from Z times the upper unipotent subgroup with all non
zero upper diagonal entries in the last column.
To describe our general conjecture more precisely we introduce some notations:
LetG=GL(n,F )andZ be the centerofG.Letαbe anycharacterofZ andϕ
q
any non trivial characterof F+. We extend as usual ϕ to a character,still denoted
q
by ϕ, of the standard upper unipotent subgroup N, or any of its subgroups, by
ϕ(u )=ϕ(u +u +···+u ) for u=(u )∈N.
ij 12 23 (n−1)n ij
For any subgroup N′ of N, we put
Γ (αϕ)=IndG (αϕ).
N′ ZN′
Let n˜ =1+2+···+(n−2)= (n−2)(n−1).
2
We define the family of numbers c (n) for 0≤j ≤n˜ by
j
(q+1)(q2+q+1)···+(qn−2+qn−1+···+1)= c (n)qj
j
0≤Xj≤n˜
We willcallGelfand-Graev interpolating family, anyfamily {N } ofsets of
i 0≤j≤n˜
subgroups of the standard upper unipotent subgroup N of G such that, for all i:
a. |N |=c (n)
i i
b. each subgroup N′ ∈ N is defined by the vanishing of i upper unipotent
i
entries, so that [N :N′]=qi.
Notice that N consists only of N and N contains only one subgroup, of order
0 n˜
q(n−1).
Conjecture 1. Fix a non trivial character ϕ of F+. There exists a Gelfand-Graev
q
interpolating family {N } such that
i 0≤j≤n˜
χ(q−1)(q2−1)···(qn−1−1)⊗χ(q+1)(q2+q+1)..(qn−1+qn−2+...+q+1) =
ψ β1,...,βn
= [ Γ ((ψβ ...β )ϕ)]
N′ 1 n
0≤Mj≤n˜ NM′∈Ni
4. Application: Clebsch-Gordan coefficients for the tensor product
of two cuspidal representations of G.
Finally, we notice that the decomposition of tensor products of irreducible rep-
resentations in irreducible constituents can be easily computed, once you have de-
scribed these tensor products in terms of induced representations.
Forexample,considerthecaseof thetensorproduct χ(q−1)(q2−1)⊗χ(q−1)(q2−1)
ϕ ψ
of two cuspidal (discrete series) representations of G.
If hΩ,Θi stands for the usual inner product of two complex valued functions
G
on G, i.e.,
1
hΩ,Θi = Ω(t)Θ(t),
G |G|
tX∈G
10 LUISAABURTO-HAGEMAN,JOSE´ PANTOJA,ANDJORGESOTO-ANDRADE
where Θ(t) is the conjugate of the complex number Θ(t), then we have that the
multiplicities of the irreducible representations of G in this tensor product are
given as follows:
(1) χ(q−1)(q2−1)⊗χ(q−1)(q2−1),χ1 =1 if and only if ϕψ =α◦N
ϕ ψ α 3
D E
(2) χ(q−1)(q2−1)⊗χ(q−1)(q2−1),χ(q+1)(q2+q+1) =hϕψ,αβγi
(cid:28) ϕ ψ α,β,γ (cid:29) k×
G
(3) χ(q−1)(q2−1)⊗χ(q−1)(q2−1),χ(q−1)(q2−1) =
ϕ ψ Λ
D E
hϕψ,Λi +hϕψ,Λqi + ϕψ,Λq2 +(q−3)hϕψ,Λi +
K× K× k×
D EK×
χ(q−1)(q2−1),χ(q−1)(q2−1) + χ(q−1)(q2−1),χ(q−1)(q2−1) =
ϕψq Λ ϕψq2 Λ
D E D E
hϕψ,Λi +hϕψ,Λqi + ϕψ,Λq2 +hϕψq,Λi +hϕψq,Λqi + ϕψq,Λq2
K× K× K× K×
D EK× D EK×
+ ϕψq2,Λ + ϕψq2,Λq + ϕψq2,Λq2 +(q−3)hϕψ,Λi
k×
D EK× D EK× D EK×
References
[1] Aburto, L., Pantoja, J.: Tensor Products of Irreducible Representations of the Groups
GL(2,k)andSL(2,k),kaFiniteField,Comm.inAlgebra,28,2507-2514(2000)
[2] Asmuth,C.,Repka,J.: TensorProductsforSl(2,k)II,Supercuspidalrepresentations,Pacific
J.Math.,97,1-18(1981)
[3] Asmuth,C.,Repka,J.: TensorProductsforSl(2,k)I,Complementaryseriesandthespecial
representation,PacificJ.Math.97,271-282(1981)
[4] Deligne,P.,Lusztig,G.: Representationsofreductivegroupsoverfinitefields,Ann.ofMath.
103,103-161(1976)
[5] Kable,A.: Legendresums,Soto-AndradesumsandKloostermansums,PacificJ.Math,206,
129-157(2002)
[6] Rideau, G.: Sur lar´eduction duproduit tensoriel des repr´esentations de las´eriediscrete de
SL(2,RAnn.Inst.H.Poincar´e,SectionA(N.S.)4,67-76(1966)
[7] Steinberg, R.: Therepresentations ofGL(3,q), GL(4,q), PGL(3,q)andPGL(4,q), Canadian
J.ofMath., 3,225-235(1951)
[8] Soto-Andrade,J.,Vargas,J.: ThetwistedsphericalfunctionsonfinitePoincare’supperhalf
plane(anycharacteristic),J.Algebra,248,724-746(2002)
[9] Tsuchikawa,M.: ThePlanchereltransformonSL2anditsapplicationstothedecomposition
oftensorproductsofirreduciblerepresentations,J.MathKyotoUniv.,22,369-433(1982/83)
Instituto de Matema´tica, Pontificia Universidad Cato´lica de Valpara´ıso, Casilla
4059,Valpara´ıso,Chile
E-mail address: [email protected]
Instituto de Matema´tica, Pontificia Universidad Cato´lica de Valpara´ıso, Casilla
4059,Valpara´ıso,Chile
E-mail address: [email protected]
Departamentode Matema´ticas, Facultad de Ciencias, Universidad de Chile, Casilla
653,Santiago,Chile
E-mail address: [email protected]