IRREDUCIBLE CHARACTERS OF GSp4, AND DIMENSIONS OF SPACES OF FIXED VECTORS JEFFERY BREEDING II Abstract. In this paper, we compute the conjugacy classes and the list of irreducible characters of GSp4,, where is odd. We also determine precisely which irreducible characters are non-cuspidal and which are generic. These characters are then used to compute dimensions of certain subspaces of fixed vectors of smooth admissible non-supercuspidal representations of GSp4, F, where F is a non-archimedean local field of characteristic zero with residue field of order. Contents. Introduction 2. Definitions and notations 3 3. Conjugacy classes 6 4. Induced characters 2 4.. Generic characters 2 4.2. Borel 2 4.3. Siegel 22 4.4. Klingen 23 5. Irreducible characters 24 5.. Constructing characters of GSp4, 25 5.2. Irreducible non-cuspidal representations 30 6. Dimension formulas 34 7. Non-cuspidal character tables 40 References 48. Introduction The passage of cuspidal Siegel modular forms of degree n to cuspidal automorphic representations of GSp2n, A is a natural extension of the classical theory for GL2 described in the seminal work [0] by Jacuet and Langlands. As in the GL2- case, these cuspidal automorphic representations π can be written in terms of local components π v, where v is a place of Q. 99 Mathematics Subject Classification. F46, F70, F85. Key words and phrases. representation theory, finite groups, p-adic groups. The author would like to thank Ralf Schmidt and Alan Roche for their valuable notes and comments on this paper. The author would especially like to thank the referee for their detailed comments and careful reading of this paper.
2 JEFFERY BREEDING II In the degree 2 case, let Γ be a discrete subgroup of Sp4, R. Dimensions of the spaces S 2 k Γ can be computed using the Selberg trace formula. A general formula for these dimensions has been given by Hashimoto [9]. In terms of the associated representations, these dimensions tell us essentially how many possibilities there are for the local factors of the representation by giving restrictions on the dimensions of its subspace of Γ -fixed vectors. Let N N be suare-free. In the case Γ = ΓN, the principal congruence subgroup of level N, the dimensions of spaces of ΓN-fixed vectors can be computed in terms of representations of the finite group GSp4, p for primes p N. Let π be an irreducible automorphic representation of GSp4, A, where A is the adele ring of Q. Write π in terms of its local components π = p π p, where π p is a representation of GSp4, Q p. Suppose π is associated to a Siegel cuspidal eigenform f Sk 2ΓN. Then, for each p N, the representation π p has a finite dimensional space of Γp-fixed vectors. This dimension would make a contribution to the dimension of the space Sk 2ΓN. The computation of the precise contribution can be achieved by determining the GSp4, p analogue of the representation π p of GSp4, Q p for each finite prime p. The dimension of the GSp4, p analogue is the dimension of the subspace of Γp-fixed vectors. The conjugacy classes and the complete list Table of all of the irreducible characters of the finite group GSp4, as well as their cuspidality and genericity is determined in this paper. Our method relies on Srinivasan s irreducible character tables for Sp4,, [8], and does not lead to the computation of the complete character table. In particular, our method will not yield the character values of some of the irreducible characters on conjugacy classes that have non-suare multipliers. Cuspidality is determined by defining cuspidal representations on the standard Borel, Siegel parabolic, and Klingen parabolic subgroups and then inducing. The irreducible non-cuspidal representations are precisely the irreducible constituents of these induced representations. Criteria are determined for these induced characters to be irreducible. If an induced character is reducible, then its irreducible constituents are determined. We find that we did not need the complete character table to identify the cuspidal characters. After computing what we need from the representation theory of GSp4,, we give formulas for dimensions of subspaces of Γp-fixed vectors of certain smooth admissible non-supercuspidal representations of GSp4, F, where F is a non-archimedean local field of characteristic zero with ring of integers o and maximal ideal p such that o/p is a finite field with elements. The referee has informed us that Shinoda had independently obtained the conjugacy classes [7] and the complete character table [6] of GSp4,, which he calls CSp4,, using the modern approach of Deligne-Lusztig theory. Shinoda remarks at the end of the introduction in [6] that Reid had also obtained this character table in [2]. We were not aware of these results when doing the computations in this paper. We give a correspondence between our notations and Shinoda s notations for the conjugacy classes and the irreducible characters of GSp4, in section 3 and section 5, respectively.
IRREDUCIBLE CHARACTERS OF GSp4, 3 2. Definitions and notations Let G be a finite group of Lie type. A Borel subgroup of G is a maximal closed connected solvable subgroup of G. For any such group G, any two Borel subgroups of G are conjugate. A parabolic subgroup of G is a subgroup of G which contains a Borel subgroup of G. Let B be a Borel subgroup of G. Then B has the Levi decomposition B = T U, where T is a maximal torus in G such that the centralizer of T in G is contained in B and U is a connected unipotent group. Moreover, all maximal tori of B are conjugate in B. Let T B be a maximal torus and let W = NT /T be its Weyl group, where NT is the normalizer of T in G. Then G has the following double coset decomposition, which is called its Bruhat decomposition with respect to B: G = BwB. w W Let F denote a finite field with = p n elements, with p an odd prime. Let κ be a generator of F and let ζ = κ 2, θ = κ 2 +, η = θ, and γ = θ +. The 4 element η is the generator of the set of elements in F whose norm over F 2 is and ζ is the generator of the set of elements in F whose norm over F 4 2 is. Fix a monomorphism from F into C and denote the images of ζ, θ, η, and γ under 4 this monomorphism by ζ, θ, η, and γ, respectively. Define the sets R = {,..., 4 2 }, R 2 is a set of 4 2 distinct positive integers i such that θ i, θ i, θ i, and θ i are all distinct, T = {,..., 2 3}, T 2 = {,..., 2 }, and T 3 = {,..., }. The general symplectic group of degree two over F, denoted GSp4,, is the set of all g GL4, such that t gjg = λgj, where J = and λg is an element of F which depends on g. We will call λg the multiplier of g. The set of all g GSp4, such that λg = is the subgroup Sp4,. Let Z = { I 4 : } denote the center of GSp4,. For any g G, there exists a uniue λg and g Sp4, such that g = λg λg The order of Sp4,, as computed by Wall in [20], is 4 4 2. So the order of GSp4, is 4 4 2. The standard Borel subgroup B of GSp4, is the set of all of the upper triangular matrices, { } B = GSp4,. Since GSp4, is a finite group of Lie type, any other Borel subgroup of GSp4, is conjugate to B. Using the Levi decomposition, every element g B can be g.
4 JEFFERY BREEDING II written uniuely as g = a b λ µ y cb x µ, ca λ with a, b, c F and x, y, λ, µ F. The first factor in is an element of the split torus of GSp4,. The subgroup of B of elements which have on every entry on the main diagonal is the unipotent radical of the Borel and will be denoted by N GSp4. The order of B is 4 3. The multiplier of the element g in is c. The standard Siegel parabolic subgroup P of GSp4, is { } P = GSp4,. Any other Siegel parabolic subgroup of GSp4, is conjugate to P. Using the Levi decomposition, every element p P can be written uniuely as µ y 2 p = x µ, a b c d λa/ λb/ λc/ λd/ with = ad bc F, λ F and x, y, µ F. The first factor in 2 is contained in a Levi subgroup of GSp4, and the second factor is in the unipotent radical of P. The order of P is 4 2 2. The multiplier of the element p in 2 is λ. We also define A = t A for any A GL2,. Then the Levi subgroup of P is the set of all matrices of the form A λ A where A GL2, and λ F. The standard Klingen parabolic subgroup Q of GSp4, is { } Q = GSp4,. Any other Klingen parabolic subgroup of GSp4, is conjugate to Q. Using the Levi decomposition, every element g Q can be written uniuely as 3 g =, t a b c d t λ µ y µ λ with = ad bc F, t F, and y, λ, µ F. The first factor in 3 is contained in a Levi subgroup of GSp4, and the second factor is in the unipotent radical of Q. The order of Q is 4 2 2. The multiplier of the element g in 3 is. We remind the reader of the following results in the representation theory of finite groups. The reader may wish to consult [5] for more details. Let G be a finite group and let χ and χ 2 be characters of G. The inner product of χ and χ 2 is χ, χ 2 = χ gχ 2 g. G This inner product is eual to g G dim Homρ, ρ 2,
IRREDUCIBLE CHARACTERS OF GSp4, 5 where ρ i is a representation with character χ i for i =, 2. Let H be a subgroup of G and let π, V be a representation of H. The induced representation of π, V on G, denoted by Ind G HV or by Ind G Hπ, is the space of functions f : G V satisfying fhg = πhfg, for h H, g G, with group action by right translation. If π has character χ, we denote the character of Ind G Hπ by χ G H or by IndG Hχ and call it the induced character of π. Let C be a conjugacy class of G. Then the conjugacy class C either has an empty intersection with H or it splits into finitely many distinct conjugacy classes of the subgroup H, say C = D... D r. One may compute the values of the induced character χ G H on the conjugacy class C using the following formula from [7]: χ G HC = G H r i= D i C χd i. We also recall some results from the representation theory of GL2,. These will be useful in our computations. More details can be found in [3] and in [7]. Let B GL2 be the standard Borel subgroup of GL2,, i.e., B GL2 is the subgroup of all upper triangular matrices, B GL2 = { GL2, }. By the Levi decomposition, every element in B GL2 can be written uniuely as y y 2 a, where the first factor is an element of the split torus of GL2,, denoted by T GL2. The subgroup of B GL2 of matrices which have on every entry on the main diagonal is the unipotent radical of B GL2 and will be denoted by N GL2. Let χ, χ 2 be characters of F. Define a representation χ of B GL2 by χ y x y 2 = χ y χ 2 y 2. Denote the representation of GL2, induced from χ by χ χ 2. The irreducible representations χ χ 2 are called representations of the principal series. The following two results can be found in [3]. Theorem 2.. Let χ, χ 2, µ and µ 2 be characters of F. Then χ χ 2 is an irreducible representation of degree + of GL2, unless χ = χ 2, in which case it is the direct sum of two irreducible representations having degrees and. We have χ χ 2 = µ µ 2 if and only if either χ = µ and χ 2 = µ 2 or 2 χ = µ 2 and χ 2 = µ. Let F denote the trivial character of F. Then, F F is not irreducible. It decomposes into the sum of two irreducible representations: F F = GL2 + St GL2, where GL2 is the trivial representation of GL2, and St GL2 is a - dimensional representation of GL2, called its Steinberg representation. Then, for any character χ of F, the irreducible representation of dimension contained in χ χ is the character g χdetg. The other irreducible constituent is the -dimensional representation which can be obtained by taking the tensor of χ with St GL2.
6 JEFFERY BREEDING II Proposition 2.2. Every irreducible cuspidal representation of GL2, has dimension. Every cuspidal representation is a direct sum of irreducible cuspidal representations. Let α, β be characters of F. Let φ be a character of F such that φ φ and 2 X φ = Xφ is an irreducible cuspidal representation of GL2,. Define Mx := x x and let d = x + y γ, where γ = γ /2 is a fixed element t F 2 such that t 2 = γ. The complete character table of GL2,, taken from [7], is the table below, no matter what choice for γ was made. Class diagx, x Mx diagx, y, x y d x y yγ x Class order 2 2 + 2 α GL2 αx 2 αx 2 αxy αd + αst GL2 αx 2 0 αxy αd + α β, α β + αxβx αxβx αxβy + αyβx 0 X φ φx φx 0 φd + φd 3. Conjugacy classes Our first step in the determination of the irreducible characters of GSp4, is the computation of its conjugacy classes. This list of classes can then be used to compute the conjugacy classes of the standard Borel, Siegel parabolic, and Klingen parabolic subgroups. One does this by determining which conjugacy classes have a non-empty intersection with the subgroup, determining how each class splits, and computing the order of the centralizer of each conjugacy class of the subgroup. The main tool we used for computing the conjugacy classes of GSp4, is a paper of Wall, [20]. This paper, incidentally, was also used by Srinivasan in [8] to determine the conjugacy classes of the symplectic group Sp4,. Unfortunately, Wall s results cannot be immediately used in our case. We can, however, use these results to find the classes of SO5,. This is particularly useful because SO5, is isomorphic to PGSp4, := GSp4, /Z. The reader can consult Appendix A.7 in [3] for a description of an isomorphism ρ 5 : PGSp4, F SO5, F, where F is a field of characteristic not eual to 2. The computation of the classes and their orders of this special orthogonal group are then used to determine the classes of GSp4,. One does this by computing the corresponding class of PGSp4, under an isomorphism and then pulling the class back to GSp4,. Consider the natural projection map from GSp4, to PGSp4, given by GSp4, PGSp4,, g g. Let g, h GSp4,. If g = xhx for some x GSp4,, then it is clear that λg = λh. Moreover, under the projection map, g = xhx = x h x. So if two elements are conjugate in GSp4,, they must be conjugate in PGSp4,. The list of class representatives in PGSp4,, when pulled back to GSp4,, hit class representatives of all the conjugacy classes of GSp4,. Suppose now that two elements g, h GSp4, are conjugate in PGSp4,, i.e. g = x h x, for some x PGSp4,. Then, for some γ i F, g = γ i I 4 xhx.
IRREDUCIBLE CHARACTERS OF GSp4, 7 Taking multipliers on both sides, we have λg = γ 2i λh. So the multiplier of g is a suare if and only if the multiplier of h is a suare. Write g and h as g = γ ig γ ig z jg g and h = γ i h γ i h z jh h, with g, h Sp4, F, i g, i h {0, }, and j g, j h T 3. So the multipliers of g and h respectively are λg = γ ig+2jg and λh = γ i h+2j h. If g and h are conjugate, then i g = i h. So γ 2jg = γ 2j h, i.e., γ jg 2 = γ j h 2. So λh = ±λg. It is possible that an element g of GSp4, is conjugate to g. For example, =. The centralizers are sometimes affected when we pull back our representatives in PGSp4, to GSp4,. There are two types of pullbacks. The first type consists of elements g such that g x gx for any x GSp4,. The second type consists of elements g such that g = x gx for some x GSp4,. Type. Let g GSp4, be of the first type, i.e., g is not conjugate to g. Let h C PGSp4, g. Then g = h g h. When pulled back to GSp4,, g = z 0 hgh, with z 0 = ±I. We have z 0 I since g is not conjugate to g. So z 0 = I, g = hgh, and h C GSp4, g. We get a short exact seuence Z C GSp4, g C PGSp4, g. Therefore #C GSp4, g = #C PGSp4, g. Type 2. Let g GSp4, be of the second type, i.e., g is conjugate to g. Define the set S g = {h GSp4, F : hgh = g}. Fix s 0 S g. The set S g is not a group, but there is a bijection of sets S g C GSp4, g, given by the map h s 0 h. Given h C PGSp4, g, either h C GSp4, g or h S g. The set S g C GSp4, g maps onto C PGSp4, g via the projection map. Moreover, C GSp4, g := S g C GSp4, g is a group with respect to matrix multiplication and the projection map is a group homomorphism. C GSp4, g is a subgroup of C GSp4, g of index 2. We get a short exact seuence Then Also, Z C GSp4, g C PGSp4,g. #C GSp4, g = #C PGSp4,g. 2 #C GSp4, g = #C GSp4, g. So #C GSp4, g = #C PGSp4, g. 2 Thus, given a class representative g PGSp4,, we pull it back to GSp4,. Then, we determine if g is of Type or Type 2. If it is of Type, then there are distinct conjugacy classes zg, for z Z, each of order #GSp4, / #C PGSp4, g. If the pullback is of Type 2, then there are /2 distinct conjugacy classes z i g, with i T 2. We list the conjugacy classes in the following table. The Type conjugacy classes are A k, A 2 k, A 3 k, A 32 k, A 5 k, B 3 k, C i, k, C 3 i, k, C 4 i, k, C 5 i, k,
8 JEFFERY BREEDING II C 6 i, j, k, D i, k, D 2 i, k, D 4 i, j, k, D 5 i, k, D 6 i, k, D 7 i, j, k, D 8 i, k, and D 9 i, k. The Type 2 conjugacy classes are B k, B 2 k, B 2 k, B 22 k, B 4 k, B 42 k, B 43 k, B 44 k, B 5 k, B 52 k, C 2 i, k, C 22 i, k, D 3 i, k, and D 32 i, k. Fix a, b F such that a 2 + b 2 γ is a suare and let c = γ +. The conjugacy classes of our groups can now be given. The reader should note that there are 2 + 2 + 4 conjugacy classes of GSp4,. The reader may notice that some of the class representatives given in the tables below are in a form with entries not in F, but in F 2 or F 4. These are given as representatives of the conjugacy class in GSp4, F that is stable under the action of the Frobenius, which raises each entry in the matrix to the -th power. Notation Table : Conjugacy classes of GSp4, Class representative Order of centralizer A k, I 4 #GSp4, A 2k, A 3k, A 32k, A 5k, + 4 2 2 3 2 2 3 2 2 B k, diag,,, 2 2 2 B 2k, + + 2 4 B 2k, diag,,, 2 2 B 22k, B 3k, B 4k, B 42k, + + + 2 2 2 2 2 2 2 2
Notation B 43k, B 44k, B 5k, B 52k, IRREDUCIBLE CHARACTERS OF GSp4, 9 Table Continued Class representative a 2 γk + a 2 γk+ b+ + c + 2+ + + + Order of centralizer 2 2 2 2 2 2 C i, k, i T, diag,, +i, +i 2 2 C 2i, k, diag,, +i, +i 3 i T, C 22i, k, i T, C 3i, k, i T, C 4i, k, i T, + +i +i+ +i +i +i i +i 2 2 2 C 5i, k, i T, diag+i,,, i 2 2 C 6i, j, k, i, j T, i < j, diag, +i, +j, +i+j 3 D i, k, i R 2, diag, θ i, θ i, +i 2 D 2i, k, diag θ i, θ i, θ i, θ i 2 2 i T 2, D 3i, k, i, θ i θ i θ i θ i 2 D 32i, k, i, diagγ /2 η i, γ /2 η i, γ /2 η i, γ /2 η i 2 + D 4i, j, k, i T 2, j T, diag θ i, θ i, +j θ i, +j θ i 2 D 5i, k, i T 2, θ i θ i θ i θ i θ i θ i 2 D 6i, k, i T 2, diag, η i, η i, 2 2
0 JEFFERY BREEDING II Notation Table Continued Class representative Order of centralizer D 7i, j, k, i, j T 2, i < j, diag η i, η j, η j, η i 2 + D 8i, k, i T 2, η i η i 2 D 9i, k, i R, diag ζ i, ζ i, ζ 2i, ζ 3i 2 + As noted before, the conjugacy classes of GSp4, were previously determined by Shinoda [7] using a different approach. We give a correspondence between our different notations. The classes D 32 i, k and D 7 i, j, k turn out to be different members of the class type L 0 from [6]. Table 2: Notations for conjugacy classes of GSp4, Class Shinoda Class Shinoda A k, A 0, a = C i, k, D 0, a =, b = +i i T, A 2k, A, a = C 2i, k H 0, a =, i T, a 2 = +i, b 2 = A 3k, A 2, a = C 22i, k I 0, a = +/2, i T, b = γ 2k+i A 32k, A 22, a = C 3i, k, D, a =, b = +i i T, A 5k, A 3, a = C 4i, k, E, a = γ i, c = i T, B k, B 0, a = C 5i, k, E 0, a = γ i, c = i T, B 2k, C 0, a = +/2 C 6i, j, k, H 0, a =, i, j T, i < j, a 2 = +i, b 2 = +j B 2k, D 0, a =, b = D i, k, J 0, a = θ i, c = i R 2, B 22k, F 0, a = +/2 D 2i, k, F 0, a = θ i i T 2, B 3k, B 2, a = D 3i, k, i, I 0, a = θ i, b = γ 2k B 3 k + 2, B, a = D 32i, k, i, L 0, a = +/2 η i, a 2 = +/2 η i
IRREDUCIBLE CHARACTERS OF GSp4, Table 2 Continued Class Shinoda Class Shinoda B 4k, B 3, a = D 4i, j, k, I 0, a = θ i, b = γ 2k+j i T 2, j T, B 42k, B 32, a = D 5i, k, F, a = θ i i T 2, B 43k, C 2, a = +/2 D 6i, k, G 0, a =, u = η i i T 2, B 44k, C, a = +/2 D 7i, j, k L 0, a = η i, i, j T 2, i < j, a 2 = η j B 5k, D, a =, b = D 8i, k, G, a =, u = η i i T 2, B 52k, F, a = +/2 D 9i, k, K 0, a = ζ i i R, Now we compute the conjugacy classes of the Borel subgroup, the Siegel parabolic subgroup, the Klingen parabolic subgroup, and the unipotent radical of the Borel. In each of the following tables of conjugacy classes of our important subgroups, the notation will indicate which conjugacy classes of GSp4, occur in the subgroup and how many components the splitting has if the class splits into multiple classes in the subgroup. For example, the Borel subgroup is denoted by B. The class A 2 k has a non-empty intersection with B, splitting into two conjugacy classes, denoted by BA 2k and BA 2 2k. Notation Table 3: Conjugacy classes of the Borel subgroup Class representative Order of centralizer in B BA k, I 4 4 3 BA 2k, BA 2 2k, BA 3k, 3 2 4 2 2 3
2 JEFFERY BREEDING II Notation BA 2 3k, BA 3 3k, BA 32 k, BA 5 k, Table 3 Continued Class representative + Order of centralizer in B 3 2 2 2 2 3 2 BB k, diag,,, 2 3 BB 2 k, diag,,, 2 3 BB 2k, diag,,, 3 BB 2 2k, diag,,, 3 BB 3 2k, diag,,, 3 BB2k, 4 diag,,, 3 BB 3k, BB 2 3k, BB 4k, BB 2 4k, BB 42k, BB 2 42k, 2 2 4 γk + 4 γk+ 2 2 2 2 2 2 2 2 2 2
Notation BB 5k, BB 2 5k, BB 3 5k, BB 4 5k, IRREDUCIBLE CHARACTERS OF GSp4, 3 Table 3 Continued Class representative Order of centralizer in B 2 2 2 2 BC i, k, i T, diag,, +i, +i 3 BC 2 i, k, i T, diag+i, +i,, 3 BC 3 i, k, i T, diag+i,, +i, 3 BC 4 i, k, i T, diag, +i,, +i 3 BC 2i, k, i T, diag,, +i, +i 3 BC 2 2i, k, i T, diag+i, +i,, 3 BC 3 2i, k, i T, diag+i,, +i, 3 BC 4 2i, k, i T, diag +i, +i,, 3 BC 5 2i, k, i T, diag +i,, +i, 3 BC 6 2i, k, i T, diag, +i,, +i 3 BC 7 2i, k, i T, diag, +i,, +i 3 BC2i, 8 k, diag,, +i, +i 3 i T, BC 3i, k, i T, BC 2 3i, k, i T, +i +i +i +i +i +i 2 2
4 JEFFERY BREEDING II Notation BC 3 3i, k, i T, BC 4 3i, k, i T, BC 4i, k, i T, BC 2 4i, k, i T, BC 3 4i, k, i T, BC 4 4i, k, i T, Table 3 Continued Class representative +i +i +i +i +i +i i +i Order of centralizer in B 2 i i +i +i i +i +i i 2 2 2 2 2 BC 5i, k, i T, diag+i,,, i 3 BC 2 5i, k, i T, diag, +i, i, 3 BC 3 5i, k, i T, diag, i, +i, 3 BC 4 5i, k, i T, diag i,,, +i 3 BC6i, j, k, i, j T, i < j, diag, +i, +j, +i+j 3 BC6i, 2 j, k, i, j T, i < j, diag+i+j, +j, +i, 3 BC6i, 3 j, k, i, j T, i < j, diag+i+j, +i, +j, 3 BC6i, 4 j, k, i, j T, i < j, diag+j,, +i+j, +i 3 BC6i, 5 j, k, i, j T, i < j, diag+j, +i+j,, +i 3 BC6i, 6 j, k, i, j T, i < j, diag+i, +i+j,, +j 3
Notation IRREDUCIBLE CHARACTERS OF GSp4, 5 Table 3 Continued Class representative Order of centralizer in B BC6i, 7 j, k, i, j T, i < j, diag+i,, +i+j, +j 3 BC6i, 8 j, k, i, j T, i < j, diag, +j, +i, +i+j 3 Notation Table 4: Conjugacy classes of the Siegel parabolic subgroup Class representative Order of centralizer in P P A k, I 4 4 2 2 P A 2 k, P A 3k, P A 2 3k, P A 32 k, P A 5 k, + 4 2 2 3 2 2 2 2 3 2 2 P B k, diag,,, 2 3 P B 2 k, + + 2 2 P B 2k, diag,,, 2 2 P B 2 2k, diag,,, 2 2 P B2k, 3 diag,,, 3 P B 22 k, + + 2
6 JEFFERY BREEDING II Notation P B 3 k, P B 4 k, P B 42 k, P B 43 k, P B 44 k, P B 5k, P B 2 5k, P B 3 5k, P B 52 k, Table 4 Continued Class representative Order of centralizer in P 2 2 + aγk 2 + aγk+ 2 b+ + c + 2+ + 2 2 2 2 2 2 2 2 2 γ k + γ + 2 2 2 P C i, k, i T, diag,, +i, +i 2 2 P C 2 i, k, i T, diag+i, +i,, 2 2 P C 3 i, k, i T, diag+i,, +i, 3 P C 2i, k, i T, diag,, +i, +i 3 P C 2 2i, k, i T, diag+i, +i,, 3 P C 3 2i, k, i T, diag+i,, +i, 3 P C2i, 4 k, diag +i,, +i, 3 i T, P C 22i, k, i T, γ k+ +i +i+ 2
Notation P C 2 22i, k, i T, P C 3i, k, i T, P C 2 3i, k, i T, P C 3 3i, k, i T, P C 4i, k, i T, P C 2 4i, k, i T, IRREDUCIBLE CHARACTERS OF GSp4, 7 Table 4 Continued Class representative +i γ k+i+ +i +i + +i +i +i +i +i +i +i i +i i i +i +i Order of centralizer in P 2 2 2 2 2 2 P C 5i, k, i T, diag+i,,, i 3 P C 2 5i, k, i T, diag, i, +i, 3 P C6i, j, k, i, j T, i < j, diag, +i, +j, +i+j 3 P C6i, 2 j, k, i, j T, i < j, diag+i+j, +j, +i, 3 P C6i, 3 j, k, i, j T, i < j, diag+i+j, +i, +j, 3 P C6i, 4 j, k, i, j T, i < j, diag+j,, +i+j, +i 3 P D 2 i, k, i T 2, diag θ i, θ i, θ i, θ i 2 P D 3i, k, i, diag θ i, θ i, θ i, θ i 2 P D 2 3i, k, i, diag θ i, θ i, θ i, θ i 2 P D4i, j, k, i T 2, j T, diag θ i, θ i, +j θ i, +j θ i 2 P D4i, 2 j, k, i T 2, j T, diag+j θ i, +j θ i, θ i, θ i 2
8 JEFFERY BREEDING II Notation P D 5 i, k, i T 2, Table 4 Continued Class representative θ i θ i θ i θ i θ i θ i Order of centralizer in P 2 Table 5: Conjugacy classes of the Klingen parabolic subgroup Notation Class representative Order of centralizer in Q QA k, I 4 4 2 2 QA 2k, QA 2 2k, QA 3k, QA 2 3k, QA 32 k, QA 5 k, 3 2 4 2 2 3 + 3 2 2 3 2 QB k, diag,,, 2 2 2 QB 2 k, diag,,, 2 2 2 QB 2k, diag,,, 3 QB2k, 2 diag,,, 3 QB 3k, QB 2 3k, 2 2 2 2
Notation QB 4k, QB 2 4k, QB 42k, QB 2 42k, QB 5k, QB 2 5k, IRREDUCIBLE CHARACTERS OF GSp4, 9 Table 5 Continued Class representative 4 γk + 4 γk+ Order of centralizer in Q 2 2 2 2 2 2 2 2 2 2 QC i, k, i T, diag,, +i, +i 3 QC 2 i, k, i T, diag+i, +i,, 3 QC 2i, k, i T, diag,, +i, +i 3 QC 2 2i, k, i T, diag+i, +i,, 3 QC 3 2i, k, i T, diag +i, +i,, 3 QC2i, 4 k, diag, +i,, +i 3 i T, QC 3i, k, i T, QC 2 3i, k, i T, QC 4i, k, i T, QC 2 4i, k, i T, +i +i +i +i +i +i +i i i +i 2 2 2 2
20 JEFFERY BREEDING II Notation QC 3 4i, k, i T, Table 5 Continued Class representative +i i Order of centralizer in Q 2 QC 5i, k, i T, diag+i,,, i 2 2 QC 2 5i, k, i T, diag, +i, i, 3 QC5i, 3 k, i T, diag i,,, +i 2 2 QC6i, j, k, i, j T, i < j, diag, +i, +j, +i+j 3 QC6i, 2 j, k, i, j T, i < j, diag+i+j, +j, +i, 3 QC6i, 3 j, k, i, j T, i < j, diag+j,, +i+j, +i 3 QC6i, 4 j, k, i, j T, i < j, diag+i, +i+j,, +j 3 QD i, k, i R 2, diag, θ i, θ i, +i 2 QD 2 i, k, i R 2, diag+i, θ i, θ i, 2 QD 6 i, k, diag, η i, η i, 2 i T 2, QD 8 i, k, i T 2, η i η i 2 Table 6: Conjugacy classes of N GSp4 Notation Class representative Order of centralizer in N GSp4 NA I 4 4 NA 2k, 3 k γ NA 2 2k, NA 3i, j, y, i, j T 3, γ 2i γ j y = γ 2n for some n T 3 γ i y γ j γ i 4 3
IRREDUCIBLE CHARACTERS OF GSp4, 2 Notation NA 2 3k, NA 3 3K, NA 32 i, j, y, i, j T 3, γ 2i γ j y = γ 2n+ for some n T 3 NA 5 i, j, i, j T 3 Table 6 Continued Class representative γ k γ k γ i y γ j γ i γ i γ j γ i+j γ i Order of centralizer in N GSp4 2 3 3 2 4. Induced characters This section contains the character tables of representations induced from representations of N GSp4, B, P, and Q. If a conjugacy class of GSp4, is not listed in the table, the character takes the value 0 on that conjugacy class. 4.. Generic characters. Let ψ and ψ 2 be non-trivial characters of F and let ψ N be the character of N GSp4 defined by y ψ x N = ψ xψ 2 y. y This defines a representation of N GSp4. The representation Ind G N GSp4 ψ N will be denoted by G and its character by χ G. If π, V is an irreducible representation that can be embedded into G, we call its image a Whittaker model of π and say that π is generic. The character table of G is Table 7. Table 7: G character values Class G character value A 4 2 A 2 2 A 3 2 A 32 2 A 5 We compute χ G, χ G = 2. So there are precisely 2 irreducible generic representations of GSp4, since Gelfand-Graev representations of finite groups of Lie type are multiplicity free, [6], [8], [9]. 4.2. Borel. Let χ, χ 2, and σ be characters of the multiplicative group F. Define a representation π on the Borel subgroup B by a b χ aχ 2 bσc. cb ca
22 JEFFERY BREEDING II The character of this representation is given by χ aχ 2 bσc. The representation Ind G Bπ will be denoted by χ χ 2 σ. The central character of χ χ 2 σ is χ χ 2 σ 2. The character table of χ χ 2 σ is Table 8. Class Table 8: χ χ 2 σ character values χ χ 2 σ character value A k 2 + + 2 χ χ 2 σγ 2k A 2k + 2 χ χ 2 σγ 2k A 3k 3 + χ χ 2 σγ 2k A 32k + χ χ 2 σγ 2k A 5k χ χ 2 σγ 2k B k + 2 χ χ 2 σγ 2k χ + χ 2 B 2k + χ χ 2 σ γ 2k + χ χ 2 + χ + χ 2 B 3k + χ χ 2 σγ 2k χ + χ 2 B 4k χ χ 2 σγ 2k χ + χ 2 B 42k χ χ 2 σγ 2k χ + χ 2 B 5k χ χ 2 σ γ 2k + χ χ 2 + χ + χ 2 C i, k + χ χ 2 σγ 2k+i + χ γ i χ 2γ i + χ γ i + χ 2γ i χ χ 2 σ γ 2k+i χ + χ 2 + C 2i, k χ χ 2 χ γ i + χ 2γ i + χ + χ 2 χ γ i χ 2γ i + χ γ i + χ 2γ i C 3i, k χ χ 2 σγ 2k+i + χ γ i χ 2γ i + χ γ i + χ 2γ i C 4i, k χ χ 2 σγ 2k χ γ i + χ γ i + χ 2γ i + χ 2γ i C 5i, k + χ χ 2 σγ 2k χ γ i + χ γ i + χ 2γ i + χ 2γ i C 6i, j, k χ χ 2 σγ 2k+i+j χ γ i + χ γ j + χ 2γ i + χ 2γ j + χ γ i+j χ 2γ i + χ 2γ j + χ 2γ i+j χ γ i + χ γ j 4.3. Siegel. Let π, V be an irreducible representation of GL2,, and let σ be a character of F. Define a representation π of the Siegel parabolic subgroup P on V by A λa σλπa. The character of this representation is given by σλχ π A, where χ π is the character of π. The representation Ind G P π will be denoted by π σ. If π has central character ω π, then the central character of π σ is ω π σ 2. The character table of π σ is Table 9. Class Table 9: π σ character values A k 2 + + σγ 2k ω π π σ character value
Class IRREDUCIBLE CHARACTERS OF GSp4, 23 Table 9 Continued π σ character value A 2k + σγ 2k ω π A 3k σγ 2k ω π + 2χ πm A 32k σγ 2k ω π A 5k σγ 2k χ πm B k + 2 σγ 2k χ πdiag, B 2k 2 + σγ 2k+ χ πdiag+/2, +/2 B 2k σ γ 2k ω π + ω π + + χ πdiag, B 22k + σ γ 2k+ χ πdiag+/2, +/2 B 3k + σγ 2k χ πdiag, B 4k σγ 2k χ πdiag, B 42k σγ 2k χ πdiag, B 43k σγ 2k+ χ πdiag+/2, +/2 B 44k σγ 2k+ χ πdiag+/2, +/2 B 5k σ γ 2k χ πdiag, + χ πm + χ πm B 52k σ γ 2k+ χ πdiag+/2, +/2 C i, k σγ 2k+i ω π + ω π+i + + χ πdiag, +i C 2i, k σ γ 2k+i χ πdiag, + χ πdiag+i, +i +χ πdiag, +i + χ πdiag, +i C 22i, k σ γ 2k+i+ χ πdiag+/2, +/2 +χ πdiag+i+/2, +i+/2 C 3i, k σγ 2k+i χ πm + χ πm+i +χ πdiag, +i C 4i, k σγ 2k χ πdiag, +i + χ πdiag, i C 5i, k + σγ 2k χ πdiag, +i + χ πdiag, i C 6i, j, k σγ 2k+i+j χ πdiag, +i + χ πdiag, +j +χ πdiag+i, +i+j + χ πdiag+j, +i+j D 2i, k + σγ 2k+i χ πdiag θ i, θ i D 3i, k σ γ 2k+i χ πdiag θ i, θ i + χ πdiag θ i, θ i D 4i, j, k σγ 2k+i+j χ πdiag θ i, θ i + χ πdiag+j θ i, +j θ i D 5i, k σγ 2k+i χ πdiag θ i, θ i 4.4. Klingen. Let χ be a character of F, and let π, V be an irreducible representation of GL2,. Define a representation π of the Klingen parabolic subgroup Q on V by t a b c d t χtπ a b c d, where = ad bc. The character of this representation is given by χtχ π a b c d, where χπ is the character of π. The representation Ind G Kπ will be denoted by χ π. If π has
24 JEFFERY BREEDING II central character ω π, then the central character of χ π is χω π. The character table of χ π is Table 0. Class Table 0: χ π character values χ π character value A k 2 + + χ ω π A 2k χ ω π + + χ πm A 3k 2χ ω π + χ πm 2 A 32k + χ χ πm A 5k χ χ πmγ k B k + χ χ ω π + ω π B 2k + χ + χ χ πdiag, B 3k χ χ ω π + + χ πm B 4k χ χ χ πm + χ πm B 42k χ χ χ πm + χ πm B 5k χ + χ χ πdiag, C i, k + χ + χγ i χ πdiag, +i C 2i, k χ χγ i + χ χ πdiag, +i + + χ γ i χ πdiag, +i C 3i, k χ + χγ i χ πdiag, +i C 4i, k χ χγ i + χγ i χ πm + χ πdiag+i, i C 5i, k χ χγ i + χγ i ω π + + χ πdiag+i, i C 6i, j, k χ + χγ i+j χ πdiag+i, +j +χγ i + χγ j χ πdiag, +i+j D i, k χ + χγ i χ πdiag θ i, θ i D 6i, k + χ χ πdiag η i, η i D 8i, k χ χ πdiag η i, η i 5. Irreducible characters All of the non-trivial irreducible characters of Sp4, were determined by Srinivasan, [8]. Her list of characters can be used to determine the list of all of the irreducible characters of GSp4, and some of their character tables. This list will help to determine the irreducible constituents of representations which are induced from the Borel, Siegel parabolic, and Klingen parabolic subgroups. Such constituents are precisely the non-cuspidal representations of GSp4,. The list of all of the nontrivial irreducible characters of GSp4, is given below in terms of Srinivasan s list of irreducible characters of Sp4,. Note that we use the same monomorphism from F into C that we chose at the beginning of this 4
IRREDUCIBLE CHARACTERS OF GSp4, 25 paper to determine values of characters of Sp4,. See [8] for the definitions 2 of the characters χ,..., χ 9, ξ,..., ξ 4, Φ,..., Φ 9, and θ,..., θ 3. 5.. Constructing characters of GSp4,. Define the group GSp4, + := Z Sp4,. Let χ be an irreducible character of Sp4, and let α be a character of F = Z such that α = χ I 4. We extend the character χ to an irreducible character of GSp4, + by defining αχzg := αzχg, for z F = Z and g Sp4,. Since α = χ I 4, we have that αχ is well-defined. The character of GSp4, induced from αχ is denoted by Indαχ or simply by Indχ if α is the trivial character. Theorem 5.. All irreducible characters of GSp4, are listed in Table. The characters in the table are pairwise ineuivalent. In each row in the table, α is any character of F = Z with the specified value of α. Genericity of a character is indicated by a bullet,, in the g column. The abbreviation t = 2 is also used. In the cases where the induced character decomposes, say into χ a and χ b, we have that χ b = ξχ a, where ξ : F C is the uniue non-trivial uadratic character defined by ξγ =. Note that there are more characters of Sp4, than those given explicitly in Srinivasan s tables, [8]. The missing characters are the trivial character, ξ 22, ξ 22, ξ 42, ξ 42, φ i, and θ i for i = 2, 4, 6, 8. Srinivasan explains how to compute the missing non-trivial characters using the tables in her paper. Note that we do not obtain any new characters of GSp4, with these characters, because Indαξ 2 = Indαξ 22, Indαξ 2 = Indαξ 22, Indαξ 4 = Indαξ 42, Indαξ 4 = Indαξ 42, Indαφ i = Indαφ i+, and Indαθ i = Indαθ i+ for i = 2, 4, 6, 8. Table : Irreducible characters of GSp4, Character α Constituents Dimension g α GSp4, no condition α GSp4, Indαχ n Indαχ n a 2 2 n R n Indαχ n b 2 2 Indαχ 2n Indαχ 2n a 4 n R 2 n Indαχ 2n b 4 Indαχ 3n, m Indαχ 3n, m a 2 + + 2 n, m T, n < m n+m Indαχ 3n, m b 2 + + 2 Indαχ 4n, m Indαχ 4n, m a 2 + 2 n, m T 2, n m n+m Indαχ 4n, m b 2 + 2 Indαχ 5n, m Indαχ 5n, m a 4 n T 2, m T n+m Indαχ 5n, m b 4 Indαχ 6n Indαχ 6n a 2 + n T 2 Indαχ 6n b 2 + Indαχ 7n Indαχ 7n a 2 + n T 2 Indαχ 7n b 2 + 2 There is a misprint in [8], page 523. Note that dimθ is indeed 2 2 2 +.
26 JEFFERY BREEDING II Table Continued Character α Constituents Dimension g Indαχ 8n Indαχ 8n a 2 + + n T Indαχ 8n b 2 + + Indαχ 9n Indαχ 9n a 2 + + n T Indαχ 9n b 2 + + Indαξ n Indαξ n a 2 + n T 2 n Indαξ n b 2 + Indαξ n Indαξ n a 2 + n T 2 n Indαξ n b 2 + Indαξ 2n n T 2 n+t Indαξ 2n 4 Indαξ 2n n T 2 n+t+ Indαξ 2n 2 + 2 Indαξ 3n Indαξ 3n a 2 + + n T n Indαξ 3n b 2 + + Indαξ 3n Indαξ 3n a 2 + + n T n Indαξ 3n b 2 + + Indαξ 4n n T n+t Indαξ 4n 2 + + 2 Indαξ 4n n T n+t+ Indαξ 4n 4 IndαΦ t+ IndαΦ 2 + IndαΦ 3 t+ IndαΦ 3 2 + IndαΦ 5 t IndαΦ 5 2 + + IndαΦ 7 t IndαΦ 7 2 + + IndαΦ 9 a 2 + IndαΦ 9 IndαΦ 9 b 2 + Indαθ Indαθ 2 2 + Indαθ 3 Indαθ 3 2 + Indαθ 5 Indαθ 5 2 2 Indαθ 7 Indαθ 7 2 Indαθ 9 a + 2 2 Indαθ 9 Indαθ 0 Indαθ Indαθ 2 Indαθ 3 Indαθ 9 b 2 Indαθ 0 a 2 + 2 2 Indαθ 0 b 2 2 Indαθ a 2 2 + Indαθ b 2 2 + Indαθ 2 a 2 2 + Indαθ 2 b 2 2 + Indαθ 3 a 4 Indαθ 3 b 4
IRREDUCIBLE CHARACTERS OF GSp4, 27 Before we begin the proof of Theorem 5., we note the following useful result from Fulton-Harris, [7]. Let H be a subgroup of index 2 in a group G. Then H is a normal subgroup of G and G/H is a group of order 2. Let U and U denote the trivial and non-trivial representations of G, respectively, obtained from the two irreducible representations of G/H. For any representation V of G, let V = V U. The character of V is the same as the character of V on elements of H, but takes opposite values on elements not in H. In particular, Res G HV = Res G HV. If W is any representation of H, there is a conjugate representation defined by conjugating by any element t G such that t / H; if ψ is the character of W, the character of the conjugate is h ψtht. Since t is uniue up to multiplication by an element of H, the conjugate representation is uniue up to isomorphism. Proposition 5.2 [7], Proposition 5.. Let V be an irreducible representation of G. Let W = Res G HV be the restriction of V to H. Then exactly one of the following holds: V is not isomorphic to V ; W is irreducible and isomorphic to its conjugate; Ind G HW = V V. 2 V = V ; W = W W, where W and W are irreducible and conjugate but not isomorphic; Ind G HW = Ind G HW = V. Each irreducible representation of H arises uniuely in this way, noting that in case the restrictions of V and V to H determine the same representation. Proof. of Theorem 5.. Let π be an irreducible representation of Sp4,. Let α be a character of Z. Define a representation π + of GSp4, + by π + z g := αzπg. By Schur s Lemma, elements of Z act as scalars on vectors in the space of π. It is clear that this representation is well-defined if and only if α± acts as π±i on the space of π. The character of this new representation is αzχ π g, where χ π is the character of π. By taking the inner product of the character with itself, one sees that this representation is irreducible. Consider the induced representation Ind GSp4, GSp4, π +. By Proposition 5.2, the + induced representation is either irreducible or it has precisely two irreducible constituents, since GSp4, + is an index two subgroup of GSp4,. Moreover, the induced representation has precisely two irreducible constituents if and only if χ π g = χ π γ g γ γ γ = χ π g for all g Sp4,. Otherwise, it is irreducible. So one only needs to check which characters of Sp4, are eual to their conjugate. For those characters χ such that χ χ, both χ and χ yield the same irreducible representation of GSp4, via the induction of the extension of these characters to GSp4, +. In total, 2 + 2 + 4 uniue irreducible characters are obtained in this manner, which is the same amount as the number of conjugacy classes of GSp4,. This completes the proof. The computation of the character values was performed using the values from Srinivasan s character tables, [8]. The conjugacy classes we obtained for GSp4, are, due to the manner of their computation, organized in a way which is uite different from the classes of Sp4,. This naturally presented some problems in
28 JEFFERY BREEDING II bookkeeping. Moreover, Srinivasan defined Sp4, using a different J matrix, which she calls A. The classes of Sp4, with respect to our J matrix will be denoted using Srinivasan s notation: A n, B n, B n, C n i, C 6 i, j, D n i. These are the images of Srinivasan s conjugacy classes under the isomorphism that sends an element g to g. In Table 2, we describe how to compute character values of the induced characters Ind GSp4, GSp4, + χ +, where χ + = αχ with χ a character of Sp4, and α a character of the center Z of GSp4,. Table 2: Indαχ character values Class Indαχ character value A k 2χA α A 2k χa 2 + χa 22α A 3k 2χA 3α A 32k 2χA 32α A 5k 2χA 4 + χa 42α B k 2χD α B 2k 0 { 2χB B 8 αγk+ 4 2k 4 if 4 0 if 34 B 22k { 0 if 4 2χB 6 + + αγk+ 4 4 if 34 B 3k χd 2 + χd 22α B 4k χd 3 + χd 34α B 42k χd 32 + χd 33α B 43k 0 B 44k 0 B 5k { 2χB 9 αγk+ 4 4 if 4 0 if 34 B 52k C i, k C 2i, k { 0 if 4 2χB 7 + + αγk+ 4 if 34 4 { 0 if i is odd 2χB 8 i i 2 αγk+ 2 if i is even 2χB 3 i t, t i 2 2 2χB 3 i t, t i 2 2 0 otherwise αγ 2k+i+t 2 if 4 and i is even αγ 2k+i+t 2 if 34 and i is odd
Class C 22i, k C 3i, k IRREDUCIBLE CHARACTERS OF GSp4, 29 Table 2 Continued 2χB 2 + t i 2 Indαχ character value αγk+ + 4 + i 2 if 4 and i odd 2χB 2 + t i + αγk+ 4 + 2 i 2 if 34 and i even 0 otherwise { 0 if i is odd 2χB 9 i 2 αγk+ i 2, j i 2 2 if i is even C 4i, k χc 4i + χc 42iα C 5i, k 2χC 3iα { 0 if i + j is odd C 6i, j, k 2χB 3 i j i+j αγk+ 2 if i + j is even D i, k { 0 if i is odd 2χB 5 i 2, i 2 αγk+ i 2 D 2i, k χb 6iα D 32i, k 0 D 5i, k χb 7iα D 6i, k 2χC iα D 8i, k χc 2i + χc 22iα if i is even We omit the values of Indαχ on the classes D 3 i, k, D 4 i, j, k, D 7 i, j, k, and D 9 i, k since they turn out to not be necessary for our determination of the irreducible non-cuspidal characters. Note that the group GSp4, + only has elements with suare multipliers and so the induced character takes the value 0 on the non-suare multiplier classes of GSp4, by the induced character formula. If the induced character decomposes into two constituents, then we know that the sum of the values of the constituent characters on the non-suare multiplier classes is 0, but the particular values each constituent takes on these classes are unknown. However, we can say that the values of the constituent characters on the suare multiplier classes are half the values of the induced character on those classes. We now give a correspondence between our notations and Shinoda s notations for the irreducible characters of GSp4,. Here, ω 0 is the non-trivial uadratic character on < η > and Λ is the character Λ : F C such that Λ 2 θ = ω 0 θ = ω 0 η. Table 3: Notations for irreducible characters of GSp4, Shinoda Our notation Shinoda Our notation X λ, µ, Indλµ χ 3n, m a, χ 6ω, Indχ 7n a X λ, µ ξx λ, µ ω = n
30 JEFFERY BREEDING II Table 3 Continued Shinoda Our notation Shinoda Our notation X λ, µ, Indλµ ξ 4n χ 7Λ, IndΛ F ξ n a X λ, µ = ξx λ, µ Λ Λ X 2Λ, IndΛ F χ 2n a, χ 7Λ IndΛ F Φ X 3Λ, ν, ν ξ IndΛ F ν χ 5n, m a χ 8Λ, IndΛ F ξ n a Λ Λ X 3Λ, ξ, Λ Λ IndΛ F ξ ξ 2n χ 8Λ IndΛ F Φ 3 X 3Λ, ξ IndΛ F ξ ξ 4n τ Indθ 3 X 4Θ IndΘ F χ n a τ 2 IndΦ 9 a ω = n and Λ not uadratic X 5Λ, ω, IndΛ F χ 4n, m a τ 3 Indθ ω = n and Λ uadratic X 5Λ, ω, IndΛ F ξ 2n τ 4λ Indλ F θ 7 χ λ, λ ξ Indλ ξ 3n a τ 5λ Indλ F θ 5 χ ξ Indξ Φ 5 θ Indθ 9 a χ 2λ, λ ξ Indλ ξ 3n a θ 2 Indθ 0 a χ 2ξ Indξ Φ 7 θ 3 Indθ a χ 3λ Indλ 2 χ 8n a θ 4 Indθ 2 a χ 4λ Indλ 2 χ 9n a θ 5 Indθ 3 a χ 5ω, ω = n Indχ 6n a θ 0λ λ GSp4, For our families of characters with parameters n or n, m, we note that the corresponding character from [6] depends on our fixed embedding of F 4 into C. For example, the n such that Indλ 2 χ 8 n a = χ 3 λ must, in particular, satisfy γ in + γ in = λγ i + λγ i. The conditions on the parameters are suppressed in Table 3. 5.2. Irreducible non-cuspidal representations. The irreducible non-cuspidal representations can now be determined by decomposing the parabolically induced representations of the standard Borel, Siegel, and Klingen subgroups defined above into irreducible constituents. In the theorem, we also describe our characters in terms of the irreducible characters from Table where possible. Precisely which character from Table a particular irreducible non-cuspidal character is sometimes depends on our embedding of F 4 into C. Theorem 5.3. Let σ be a character of F. Every irreducible non-cuspidal representation of GSp4, is one of the following types.
IRREDUCIBLE CHARACTERS OF GSp4, 3 Type I. These are the irreducible representations obtained by parabolic induction from the Borel subgroup. More precisely, they are irreducible representations of the form χ χ 2 σ, where χ and χ 2, are characters of F. Representations of this form are irreducible if and only if χ F, χ 2 F and χ χ ± 2, where F is the trivial character of F. Let α = χ χ 2 σ 2. In terms of the irreducible characters that appear in Table, these are the characters Indαχ 3 n, m a, Indαχ 3 n, m b, Indαξ 4 n, where n, m T, n < m. Type II. Let χ be a character of F with χ 2 F. Then the induced representation χ χ σ decomposes into two irreducible constituents IIa : χst GL2 σ and IIb: χ GL2 σ. Let α = χσ 2. Characters of Type IIa are the characters Indαχ 9 n a or Indαχ 9 n b, where n T. Characters of Type IIb are the characters Indαχ 8 n a or Indαχ 8 n b, where n T. Type III. Let χ be a character of F such that χ F. Then χ F σ decomposes into two irreducible constituents IIIa : χ σst GSp2 and IIIb: χ σ GSp2. Let α = χσ 2. Characters of Type IIIa are the characters Indαξ 3n a, Indαξ 3n b, or IndαΦ 7, where n T. Characters of Type IIIb are the characters Indαξ 3 n a, Indαξ 3 n b, or IndαΦ 5, where n T. Type V. ξ ξ σ decomposes into four irreducible constituents Va: σindθ Vb: σindφ 9 a Vc: σindφ 9 b Vd: σindθ 3. Type VI*. F F σ decomposes into six irreducible constituents VI*a: σst GSp4 VI*b: σindθ 9 a VI*c: σindθ 9 a VI*d: σindθ a VI*e: σindθ 2 a VI*f: σ GSp4, where St GSp4 := Indθ 3 a is the Steinberg representation of GSp4, and GSp4 is the trivial representation of GSp4,. Type VII. These are the irreducible representations of the form χ π, where π is an irreducible cuspidal representation of GL2, and χ is a character of F. Representations of this form are irreducible if and only if χ F and χ ξ, where ξ is the non-trivial uadratic character such that ξπ = π. Let ω π be the central character of π and let α = χω π. Characters of Type VII are the characters Indαχ 5 n, m, Indαξ 2 n, and Indαξ 4m, where n T 2, m T. Type VIII. Let π be an irreducible cuspidal representation of GL2, with central character ω π. Then F π decomposes into two irreducible constituents or VIIIa: Indω π Φ 3 and VIIIb: Indω π Φ. VIIIa: Indω π ξ n and VIIIb: Indω π ξ n. for some n T 2. Type IX. Let π be an irreducible cuspidal representation of GL2, with central character ω π such that ξπ = π. Then ξ π decomposes into two irreducible constituents IXa: Indξω π θ 5 and IXb: Indξω π θ 7.
32 JEFFERY BREEDING II Type X. These are the irreducible representations of the form π σ, where π is an irreducible cuspidal representation of GL2,. Representations of this form are irreducible if and only if the central character ω π of π is not trivial. Let α = ω π σ 2. Characters of Type X are the characters Indαχ 2 n a or Indαχ 2 n b, where n R 2. Type XI. Let π be an irreducible cuspidal representation of GL2, with trivial central character. Then π σ decomposes into two irreducible constituents XIa: σindχ 7 n a and XIb: σindχ 6 n a. Note that the irreducible constituents in Theorem 5.3 are sometimes written as twists of an irreducible character rather than in the form that is in Table. The reader may have noticed that the type notation IV is not present. The representations in our type VI* turn out to be the irreducible representations that appear in the image of the non-supercuspidal types IV and VI under the functor F we will define in Section 6. Type VI* representations also occur as the images of non-supercuspidal representations of type V in the case that the uadratic character ξ is unramified. See [3] and [4] for a description of the different types of nonsupercuspidal representations of GSp4, F, where F is a non-archimedean local field of characteristic zero. Corollary 5.4. The irreducible cuspidal characters of GSp4, are the following characters in Table : Indαχ n a, Indαχ n b, Indαχ 4 n, m a, Indαχ 4 n, m b, Indαξ 2n, Indαθ 0 a, and Indαθ 0 b. We remark that for our purposes of determining cuspidality of characters we only needed the portion of the character table of Indαχ that is given in Table 2. Since, in many cases, only the values on the classes with suare multiplier are found with our method, we could not verify irreducible constituents, except for certain types, using the inner product. On the other hand, because we had the complete character list, the character values we could compute were enough to show that a particular irreducible character was a constituent. This can be done by listing the irreducible characters with the correct dimension and comparing enough character values. The complete character tables of the irreducible non cuspidal characters σindχ 6 n a, σindχ 7 n a, Indω π Φ, Indω π Φ 3, σindφ 9, σindθ, σindθ 3, Indξω π θ 5, Indξω π θ 7, σindθ 9 a, σindθ a, σindθ 2 a, σst GSp4, Indω π ξ n a, Indω π ξ n a, Indχω π ξ 2 n, and Indχω π ξ 4n are given at the end of this paper. These tables were completed using both [8] and [6]. Proof of Theorem 5.3 The irreducible non-cuspidal representations are supported in the Borel, the Siegel parabolic, or the Klingen parabolic. We first consider those supported in the Borel. Borel: Let χ, χ 2, and σ be characters of F. As in 4.2, these characters are used to define a representation of the Borel subgroup and induced to GSp4, to obtain the representation χ χ 2 σ. From its character table, we have χ χ 2 σ = χ 2 χ σ. We also have χ χ 2 σ, χ G =,
IRREDUCIBLE CHARACTERS OF GSp4, 33 indicating that exactly one irreducible constituent of χ χ 2 σ is generic. We also compute that the possible values of χ χ 2 σ, χ χ 2 σ are, 2, 4, and 8 according to the conditions in the following table. Value Conditions χ F, χ 2 F, χ 2 χ ± 2 χ 2 = χ ± and χ 2 F 2 χ i = F, χ j F for i j 4 χ = χ 2 = ξ 8 χ = χ 2 = F Using either the character inner product or by adding character values, the constituents in types II VI* are verified. Klingen: Let χ be a character of F and let π be an irreducible cuspidal representation of GL2, with central character ω π. As in 4.4, define a representation of the Klingen parabolic subgroup and induce to GSp4, to obtain the representation χ π. Let ρ χ π be the character of χ π. Then ρ χ π, χ G =, indicating that exactly one irreducible constituent of χ π is generic. We also compute that the possible values of ρ χ π, ρ χ π are and 2. By adding character values, if χ π is reducible then either χ = F and F π = Indω π Φ 3 + Indω π Φ or F π = Indω π ξ n + Indω π ξ n or 2 χ = ξ and π is such that ξπ = π and we have ξ π = Indξω π θ 5 + Indξω π θ 7. Siegel: Let σ be a character of F and let π be an irreducible cuspidal representation of GL2, with central character ω π. As in 4.3, define a representation of the Siegel parabolic subgroup and induce to GSp4, to obtain the representation π σ. We have π σ, χ G = for all such representations π σ, indicating that exactly one irreducible constituent of π σ is generic. Let χ π σ be the character of π σ. Then the possible values of χ π σ, χ π σ are and 2. We have χ π σ, χ π σ = 2 precisely when ω π is trivial. By adding character values, if π σ is reducible then π σ = σindχ 7 n a + σindχ 6 n a for some n T 2, depending on π and on our embedding of F into C. 4 5.2.. Decompositions for types V and VI*. We give more information on the decompositions of the non-cuspidal representations supported in the Borel subgroup for types V and VI*. These decompositions can be verified with the character tables provided in sections 4 and 7. Type V: Constituents of ξ ξ σ. ξ ξ σ = ξ St GL2 σ + ξ GL2 σ = ξ St GL2 ξσ + ξ GL2 ξσ. Each of the four representations on the right side is reducible and has two constituents which are shown in the following table.
34 JEFFERY BREEDING II Table 4 Continued ξ St GL2 ξσ ξ GL2 ξσ Table 4: Type V constituents ξ St GL2 ξσ ξ GL2 ξσ ξ St GL2 σ σindθ σindφ 9 a ξ GL2 σ σindφ 9 b σindθ 3 Type VI*: Constituents of F F σ. F F σ = St GL2 σ + GL2 σ = F σst GSp2 + F σ GSp2. Each of the four representations given on the right is reducible and has three irreducible constituents, which are shown in the following table. The common factor σindθ 9 a occurs as a constituent of each of the four representations St GL2 σ, GL2 σ, F σst GSp2, and F σ GSp2. Table 5: Type VI* constituents St GL2 σ common GL2 σ F σst GSp2 σst GSp4 σindθ 9 a σindθ a common factor σindθ 9 a σindθ 9 a F σ GSp2 σindθ 2 a σindθ 9 a σ GSp4 6. Dimension formulas A representation π, V of a group G defined over a non-archimedean local field is called smooth if every vector in V is fixed by an open-compact subgroup K of G. A representation π, V is called admissible if the space V K is finite dimensional for any open compact subgroup K. A fundamental problem for representations of groups defined over local fields is that they are not always semi-simple, i.e., they do not decompose into a direct sum of irreducible constituents. For such reducible representations, one must determine how they can be constructed from irreducible representations as extensions of a uotient by a subrepresentation. Consider the group G = GSp4, F, where F is a non-archimedean local field of characteristic zero with ring of integers o and maximal ideal p such that o/p is a finite field with elements. Fix a generator ϖ of p. If x is in F, then define vx to be the uniue integer such that x = uϖ vx for some unit u in o. Write νx or x for the normalized absolute value of x such that νϖ =. The principal congruence subgroup of level p n of G, denoted by Γp n, is Γp n = {g G : g I 4 mod p n }. We have the following short exact seuence Γp K GSp4, o/p,