UNIVERSITY OF OKLAHOMA GRADUATE COLLEGE IRREDUCIBLE NON-CUSPIDAL CHARACTERS OF GSp(4, F q ) A DISSERTATION SUBMITTED TO THE GRADUATE FACULTY in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY By JEFFERY BREEDING II Norman, Oklahoma 0
IRREDUCIBLE NON-CUSPIDAL CHARACTERS OF GSp(4, F q ) A DISSERTATION APPROVED FOR THE DEPARTMENT OF MATHEMATICS BY Dr. Ralf Schmidt, Chair Dr. Andy Magid Dr. Tomasz Przebinda Dr. Alan Roche Dr. Deborah Trytten
c Copyright by JEFFERY BREEDING II 0 All rights reserved.
Acknowledgements I wish to thank my advisor Ralf Schmidt for his help and useful suggestions in completing and improving this dissertation. In addition, I am very grateful for his many recommendation letters for travel support so that I could attend several enlightening number theory conferences and workshops. I would also like to thank the Department of Mathematics and the College of Arts and Sciences of the University of Oklahoma for providing additional financial support. I would especially like to thank my fiancée Aimee Acree for her patience and support. iv
Contents Introduction. Basic definitions and notations...................... 3. Character theory............................. 5.3 Representations of GL(, F q )....................... 9.4 Generic representations of GSp(4, F q ).................. 3 Conjugacy classes 5. An isomorphism from PGSp(4, F q ) to SO(5, F q )............ 5. SO(5, F q ) and PGSp(4, F q )....................... 0.3 GSp(4, F q )................................. 9.4 Borel, Siegel, and Klingen subgroups.................. 4.4. Borel................................ 4.4. Siegel............................... 57.4.3 Klingen.............................. 70.5 N GSp(4)................................... 8 3 Induced characters 85 3. Parabolic induction............................ 86 3.. Borel................................ 86 3.. Siegel............................... 88 3..3 Klingen.............................. 9 3. G...................................... 94 4 Irreducible characters and cuspidality 96 4. Irreducible characters........................... 96 4. Irreducible non cuspidal representations................ 0 4.. Decompositions for types V and VI*.............. 5 4.3 Irreducible non supercuspidal representations............. 7 4.3. Decompositions for types IV, V, and VI............ 9 4.4 Subspaces of admissible non supercuspidal representations...... 3 References 36 v
List of Tables. X φ character values............................. Conjugacy classes of SO(5, F q ) and PGSp(4, F q )............. Conjugacy classes of GSp(4, F q )..................... 33.3 Conjugacy classes of the Borel subgroup................ 43.4 Conjugacy classes of the Siegel parabolic subgroup........... 58.5 Conjugacy classes of the Klingen parabolic subgroup......... 7.6 Conjugacy classes of N GSp(4)....................... 8 3. χ χ σ character values....................... 87 3. π σ character values.......................... 89 3.3 χ π character values.......................... 9 3.4 G character values............................. 95 4. Irreducible characters of GSp(4, F q )................... 99 4. σind(χ 6 (n)) character values....................... 0 4.3 σind(χ 7 (n)) character values....................... 03 4.4 σind(ω π Φ ) character values....................... 04 4.5 σind(ω π Φ 3 ) character values....................... 06 4.6 σind(φ 9 ) character values........................ 07 4.7 σind(θ ) character values......................... 08 4.8 σind(θ 3 ) character values......................... 09 4.9 σind(ξω π θ 5 ) character values....................... 0 4.0 σind(ξω π θ 7 ) character values....................... 4. σind(θ 9 ) character values......................... 4. σind(θ ) character values........................ 3 4.3 σind(θ ) character values........................ 4 4.4 σst GSp(4) character values........................ 5 4.5 σ GSp(4) character values......................... 6 4.6 Irreducible non cuspidal representations................ 9 4.7 Group V constituents........................... 6 4.8 Group VI* constituents.......................... 6 4.9 Irreducible non supercuspidal representations............. 7 4.0 Group IV constituents.......................... 9 vi
4. Group V constituents........................... 30 4. Group VI constituents.......................... 30 4.3 Dimensions of Γ(p) fixed vectors.................... 33 vii
Abstract Admissible non supercuspidal representations of GSp(4, F ), where F is a local field of characteristic zero with an odd-ordered residue field F q, have finite dimensional spaces of fixed vectors under the action of principal congruence subgroups. We can say precisely what these dimensions are for nearly all local fields and principal congruence subgroups of level p by understanding the non cuspidal representation theory of the finite group GSp(4, F q ). The conjugacy classes and the list of irreducible characters of this group are given. Genericity and cuspidality of the irreducible characters are also determined. viii
Chapter Introduction Representations of linear groups over a finite or a local field F are constructed from cuspidal representations, as noted in [8] by Harish-Chandra, in a way analogous to the construction of Eisenstein series from cusp forms. The representation theory of these groups can be thought of as having two kinds of representations: cuspidal and non cuspidal. For reductive algebraic groups over F, non cuspidal representations are constructed from cuspidal representations by parabolic induction. One parabolically induces a cuspidal representation defined on a parabolic subgroup and then decomposes the induced representation into irreducible constituents. When all of the parabolic subgroups and cuspidal representations of those subgroups have been exhausted, the result is a list of every irreducible non cuspidal representation of the group. The only irreducible representations of the group remaining are those that cannot be obtained through parabolic induction. These are the cuspidal representations. Let G be a connected algebraic group over a finite field F. A maximal Zariskiconnected solvable algebraic subgroup of G is called a Borel subgroup of G. Let B, B be Borel subgroups. Then B = gbg for some g G. A parabolic subgroup P
of G is a subgroup that contains the Borel subgroup (or one of its conjugates) as a subgroup. A Levi subgroup M of P is a maximal reductive subgroup (determined up to conjugacy) of P. The unipotent radical U of P is its maximal unipotent subgroup. The group we are interested in is the general symplectic group GSp(4, F q ) over a finite field F q of odd characteristic. We look at cuspidal representations defined on the Borel subgroup consisting of upper triangular matrices and two parabolic subgroups called the Siegel parabolic and the Klingen parabolic, which are each block upper triangular. Parabolically inducing cuspidal representations of these three subgroups turns out to yield the complete collection of non cuspidal representations of GSp(4, F q ). Character theory is used to determine the non cuspidal representations by decomposing parabolically induced cuspidal representations into irreducible constituents. Our first step is to find the list of conjugacy classes of GSp(4, F q ). This list is used to compute the classes of the Borel, the Siegel parabolic, and the Klingen parabolic subgroups. The main tool we use for determining the conjugacy classes of GSp(4, F q ) is a paper of Wall [5], which was also used to determine the conjugacy classes of the symplectic group Sp(4, F q ) in Srinivasan s paper [4]. An isomorphism between GSp(4, F q ) modulo its center and a special orthogonal group is used to solve the conjugacy class problem for GSp(4, F q ). We then determine how the conjugacy classes split in the Borel, the Siegel parabolic, and the Klingen parabolic subgroups. All of the irreducible characters of the finite group GSp(4, F q ) and their cuspidality and genericity are determined. Cuspidality is determined by defining cuspidal representations on the Borel, the Siegel parabolic, and the Klingen parabolic subgroup and then inducing. The irreducible non cuspidal representations are precisely the irreducible constituents of these induced representation. Criteria are determined for these induced characters to be irreducible. If an induced character is reducible,
then the constituents are found. Before doing the computations, we already have some idea as to when the induced character is irreducible and what the irreducible constituents are if it is reducible. It is expected that the results will be similar to those in Sally and Tadić s paper []. In [], the irreducible non supercuspidal representations are given for GSp(4, F ), where F is a non archimedean local field of characteristic 0. These results are summarized in a table given in []. In the local field case, the terms supercuspidal and non supercuspidal are normally used instead of cuspidal and non cuspidal. The natural analogous representation in the finite field case turns out to be correct and is verified by computing the character values. However, some of the results in [] don t have a clear analogue in the finite field case.. Basic definitions and notations Let F q denote the finite field with q = p n elements, with p an odd prime. Definition... The group G = GSp(4, F q ) is defined as GSp(4, F q ):= {g GL(4, F q ) : t gjg = λj}, where J = - - for some λ F q, which will be denoted by λ(g) and called the multiplier of g. The set of all g GSp(4, F q ) such that λ(g) = is a subgroup and is denoted by Sp(4, F q ). Note that for any g G, we can uniquely write g as g = λ(g) g, λ(g) 3
where g Sp(4, F q ). The order of Sp(4, F q ), as computed by Wall, is q 4 (q 4 )(q ). So the order of GSp(4, F q ) is q 4 (q 4 )(q )(q ). We recall some basic definitions in representation theory. By a group, we mean a finite group and by a vector space, we mean a finite-dimensional complex vector space. Definition... A representation (π, V ) of a group G is a group homomorphism π : G GL(V ) and a vector space V, where GL(V ) is the group of all invertible linear automorphisms of V. The representation (π, V ) will be referred to by either π or V. It is also said that the group G acts on V by the action of π. The dimension of the representation π is defined as the dimension of the vector space V. A linear representation is a one-dimensional representation. Definition..3. Two representations (π, V ) and (π, V ) of a group G are called equivalent or isomorphic if there exists an invertible map h : V V such that for all g G π(g) = h π (g)h. Definition..4. Let π be a representation of the group G. A subrepresentation of π is the restriction of the action of π to a subspace U V such that U is invariant under the action π. Definition..5. A representation is called irreducible if there is no nontrivial invariant subspace. The set of all equivalence classes of irreducible representations of a group G is denoted by Irr(G). 4
Given a representation V of a group G, a representation of any subgroup H of G is obtained simply by restricting the representation to H, denoted by Res G HV or by ResV when the group G and subgroup H are clear from context. The vector space V in these notations may be replaced by the action π of the representation. When V is replaced by its character χ, Res G Hχ or Resχ will denote the character of the restricted representation. A representation of G can be obtained from a representation on a subgroup H of G through the process of induction. The representation of G induced from a representation V of H is denoted by Ind G HV, or, if the group G and subgroup H are clear from context, by IndV. Again, the vector space V in these notations may be replaced by the action π of the representation and when V is replaced by its character χ, Ind G Hχ or Indχ will denote the character of the induced representation. The induced representation Ind G HV can be realized by the following construction. Let G be a group and H a subgroup of G. Let (π, V ) be a representation of H. The induced representation Ind G HV is isomorphic to the space of functions V G given by V G = {f : G V : f(hg) = π(h)f(g), for h H, g G}, with the group G acting on this space by right translation.. Character theory When the representation (π, V ) is finite dimensional, the group GL(V ) can be viewed as the group of invertible n n matrices, where n is the dimension of V. Under this identification, the trace of π(g) is defined for any g G. Definition... Let π be a finite dimensional representation of a group G. The 5
character of π is a function χ : G C defined by χ(g) = tr(π(g)), where tr denotes the trace map. It follows from properties of the trace map that characters are class functions, i.e., they are constant on conjugacy classes. Note that for the identity element I G, χ(i) = tr(π(i)) is the trace of the identity map of the space of π, which is precisely its dimension. The character χ also takes the same values on equivalent representations. Indeed, for two equivalent representations π and π of G, we have, for some invertible map h from the space of π to the space of π, π(g) = h π(g)h for all g G and tr(π(g)) = tr(h π (g)h) = tr(π (g)) for all g G. Definition... Let G be a group. The character table of G is a square array of complex numbers with rows indexed by the inequivalent irreducible characters of G and the columns indexed by the conjugacy classes. The entry in row χ and column C is the value of χ on the conjugacy class C. C C C n χ χ (C ) χ (C ) χ (C n ) χ χ (C ) χ (C ) χ (C n )....... χ n χ n (C ) χ n (C ) χ n (C n ) 6
For a particular character χ of G, the character table of χ is defined to be a table of complex numbers with rows indexed by the conjugacy classes of G with the entry in row C denoting the value of χ on the conjugacy class C. C χ(c ) C χ(c ).. C n χ n (C n ) Definition..3. Let χ and χ be characters of a group G. The inner product of χ and χ is defined as (χ, χ ) G = G χ (g)χ (g) g G The subscript G will be dropped from the notation of the inner product (, ) G when the group G is clear from context. Theorem..4. Let G be a group. The irreducible characters of G form an orthonormal basis for the vector space of all class functions of G with respect to the inner product (, ). Corollary..5. Let χ be a character of a group G. Then χ is irreducible if and only if (χ, χ) =. When a representation π of a subgroup H is induced to the group G, the induced character χ G can be determined using the character χ of π in the following way. Ind G H(χ)(g) = χ G (g) = H x G, xgx H χ(xgx ), for g G. As given in [4], the induced character s values on conjugacy classes of the group G can also be found. Let C be a conjugacy class of a group G. Then the conjugacy 7
class C splits into distinct conjugacy classes of a subgroup H, say C = D... D r. The value of the induced character is given by the formula Ind G H(χ)(C) = χ G (C) = G H r i= Another important standard result is the following. D i C χ(d i). Lemma..6. (Frobenius Reciprocity) Let H G and let χ be a character of G and ψ be a character of H. Then (χ, Ind G Hψ) G = (Res G Hχ, ψ) H An irreducible character on a subgroup H might not retain its irreducibility when induced. It is important to note that finite groups have the complete reducibility property, i.e., every representation of the group decomposes into a direct sum of irreducible representations. A representation (π, V ) of a finite group G is said to be multiplicity free if in its decomposition into irreducibles, no irreducible representation occurs more than once. By Schur s Lemma, the center of G acts by scalars in an irreducible representation (π, V ). In particular, if G = GL(n, F q ) or G = GSp(4, F q ), then the center of G consists of non-zero scalar multiples of the identity matrix. The center of G is isomorphic to F q so there exists a character ω π : F q C, known as the central character of π, such that z π... v = ω π(z)v z 8
for every z F q, v V. If χ is a character of F q and (π, V ) is a representation of GSp(4, F q ), then a new representation on V can be defined by (χπ)(g) = χ(λ(g))π(g), where λ(g) is the multiplier of g. This representation is denoted by χπ and called the twist of the representation π by the character χ. If ω π is the central character of π, then the central character of χπ is ω π χ..3 Representations of GL(, F q ) The representation theory of GL(, F q ) is the first object of study in order to understand the irreducible non cuspidal representations of GSp(4, F q ). A nice treatment of the representation theory of GL(, F q ) is given in [] and in [4]. The methods used to study the representation theory of GL(, F q ) in [] can be extended with some modification to study the representation theory of GL(, F), where F is a local field. Let B GL() be the subgroup of GL(, F q ) consisting of all upper triangular matrices. B GL() is called the Borel subgroup of GL(, F q ). B GL() = y x GL(, F q ). y The subgroup of B GL() consisting of matrices with diagonal entries equal to will be denoted by N GL() and the subgroup of B GL() consisting of diagonal matrices will be denoted by T GL(). 9
Let χ, χ be characters of F q. Define a character χ of B GL() by χ y x = χ (y )χ (y ). y Denote the representation of GL(, F q ) induced from this character of B GL() by χ χ. Theorem.3.. Let χ, χ, µ and µ be characters of F q. Then χ χ is an irreducible representation of degree q + of GL(, F q ) unless χ = χ, in which case it is the direct sum of two irreducible representations having degrees and q. We have χ χ = µ µ if and only if either χ = µ and χ = µ or χ = µ and χ = µ. Proof: See []. The irreducible representation of dimension contained in χ χ is the character g χ(det(g)). The other q-dimensional representation is obtained by taking the tensor of this character with the q-dimensional irreducible subrepresentation of F q F q, where F q denotes the trivial character of F q. This q-dimensional representation is called the Steinberg representation of GL(, F q ). The Steinberg representation of GL(, F q ) will play a role in the representation theory of GSp(4, F q ). The irreducible representations χ χ are called representations of the principal series. A representation (π, V ) of GL(, F q ) is said to be cuspidal if there does not exist 0
a non-zero linear functional l on V such that l π x v = l(v) for all v V, x F q. If F is a local field and if (π, V ) is a representation of GL(, F ) such that there does not exist a non-zero linear functional l on V satisfying the condition above, then the representation (π, V ) is said to be supercuspidal. Proposition.3.. Let (π, V ) be a cuspidal representation of GL(, F q ). Then the dimension of V is a multiple of q. Proof: See []. The irreducible cuspidal representations of GL(, F q ) and their character tables are found in [4]. Let be a generator of F q and let φ be a character of F q such that φ φ q. All irreducible cuspidal representations of GL(, F q ) are of the form X φ where the character table of X φ is given by the table below. Table.: X φ character values Conjugacy class ( ) x a x = x X φ character value (q )φ(x) ( ) x x b x = φ(x) ( ) x c x,y = y 0
Table. Continued Conjugacy class ( ) x y d x,y = y x X φ character value ( φ(x + y ) + φ(x y ) ) Let ψ be a non-trivial character of F q and let ψ N be the character of N GL() defined by ψ N x = ψ(x). This defines a representation of N GL(). The representation of GL(, F q ) induced from ψ N is denoted by G. Theorem.3.3. (Uniqueness of Whittaker models) The representation G is multiplicity free. Every irreducible representation of GL(, F q ) that is not one dimensional occurs in G with multiplicity precisely. Proof: See []. If (π, V ) is an irreducible representation that can be embedded into G, we call its image a Whittaker model of π. A Whittaker model of π is then a space W(π) of functions W : GL(, F q ) C having the property that W x g = ψ(x)w (g). The functions W are invariant under right translation and give a representation of GL(, F q ) which is isomorphic to π. Representations that have a Whittaker model
are called generic..4 Generic representations of GSp(4, F q ) Whittaker models for representations of GSp(4, F q ) can also be defined. Consider the subgroup N GSp(4) of GSp(4, F q ) defined as y N GSp(4) = x y GSp(4, F q). Let ψ and ψ be non-trivial characters of F q and let ψ N be the character of N GSp(4) defined by ψ N y x y = ψ (x)ψ (y). This defines a representation of N GSp(4). Denote the representation of GSp(4, F q ) induced from ψ N 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. A Whittaker model of π is then a space W(π) of functions W : GSp(4, F q ) C having the property that y W x y g = ψ (x)ψ (y)w (g). Genericity of an irreducible representation π of GSp(4, F q ) is easy to determine 3
using character theory. Indeed, for a particular irreducible non cuspidal representation π of GSp(4, F q ) with character χ π, one computes the inner product (χ π, χ G ). If (χ π, χ G ) = 0, then π is not generic. If (χ π, χ G ) 0, then π is generic. Moreover, when (χ π, χ G ) 0, then (χ π, χ G ) =, i.e., Whittaker models are unique. The uniqueness of Whittaker models is known in general, but it is verified computationally. Therefore, to determine genericity, we compute the conjugacy classes of N GSp(4) and the character table of G. Note that the character of the representation of N GSp(4) defined above is ψ (x)ψ (y). 4
Chapter Conjugacy classes The conjugacy classes and their orders of the unitary, symplectic and orthogonal groups can be determined using the results of Wall [5]. Srinivasan, in [4], used Wall s results to explicitly determine the conjugacy classes and orders of centralizers of elements of Sp(4, F q ). Wall s results cannot be directly used to determine the conjugacy classes of the group GSp(4, F q ) but they can be used to find the classes of SO(5, F q ). This is particularly useful because SO(5, F q ) is isomorphic to PGSp(4, F q ) := GSp(4, F q )/Z, where Z is the center of GSp(4, F q ). These classes are then used to determine the conjugacy classes of GSp(4, F q ).. An isomorphism from PGSp(4, F q ) to SO(5, F q ) We follow the method in [] to define an isomorphism. Definition... O(5, F q ) is defined by {g GL(5, F q ) : t gg = I 5 }, where I 5 is the 5 5 identity matrix and SO(5, F q ) := {g O(5, F q ) : det(g) = }. Define 5
O(5, F q ):= {g GL(5, F q ) : t gj 5 g = J 5 }, where J 5 = and SO(5, F q ):= {g O(5, F q ) : det(g) = }. We will show PGSp(4, F q ) = SO(5, F q ) first by showing that PGSp(4, F q ) is isomorphic to SO(5, F q ), then showing that SO(5, F q ) is isomorphic to SO(5, F q ). First note that the center of GSp(4, F q ) consists of diagonal matrices, with non-zero entries on the diagonal. Recall that the characteristic of the field F q is p. Let V = F 4 q and e, e, e 3, e 4 be the standard basis vectors of V. The group GSp(4, F q ) acts on V by matrix multiplication from the left. GSp(4, F q ) also acts on the tensor V V twisted with the inverse of the multiplier. So ρ(g)(v w) = λ(g) (gv) (gw). The action ρ of GSp(4, F q ) on V V is trivial on the center of GSp(4, F q ) and so there is an action of PGSp(4, F q ) on V V. Define a symplectic form on V by (v, v ) := t v - - v, for v, v V. Now define a symmetric bilinear form on V V, given on pure tensors by v w, v w := (v, v )(w, w ), 6
for v w, v w V V. Both of these bilinear forms are invariant under the action of Sp(4, F q ) and the symmetric bilinear form, is preserved by the action ρ of GSp(4, F q ). We embed V V in V V by the map v w (v w w v). The restriction of our symmetric bilinear form, to the wedge V V is given by v w, v w = ((v, v )(w, w ) (v, w )(w, v )). Let X be the image of the 5-dimensional subspace spanned by e e, e e 3, e e 4 e e 3, e e 4, e 4 e 3. Explicitly, X is spanned by the vectors x = (e e e e ), x = e e 3 e 3 e, x 3 = (e e 4 e 4 e e e 3 + e 3 e ), x 4 = (e e 4 e 4 e ), x 5 = e 4 e 3 e 3 e 4. It is straightforward to check that the matrix of, with respect to this basis is J 5 and that X is invariant under the action ρ of GSp(4, F q ). Since, is preserved by this action so there is a homomorphism ρ 5 : GSp(4, F q ) SO(5, F q ), ρ 5 (g) := (a ij ) where the a ij are determined by the action ρ on V V, i.e., 7
ρ(g)x j = a j x + a j x + a 3j x 3 + a 4j x 4 + a 5j x 5. The kernel of ρ 5 is the center Z of GSp(4, F q ) and so there is an isomorphism ρ 5 : P GSp(4, F q ) SO(5, F q ) We will now give an isomorphism from SO(5, F q ) to SO(5, F q ). First note that J 5 is the matrix of a non-degenerate symmetric bilinear form. Define the quadratic form Q (x) := x, x = t xj 5 x, where, is the form defined above. This form is equivalent to the bilinear form Q(x) := t xx = I 5, i.e., there exists a P such that P J 5 t P = I 5. To see this, choose a new basis for X such the matrix of the bilinear form is I 5. For x X, x is of the form x = c x + c x + c 3 x 3 + c 4 x 4 + c 5 x 5. Then, Q (x) = x x 5 + x x 4 + x 3. Define a new basis v, v, v 3, v 4, v 5 of X as follows. v = x + x 5, v = x + x 4, v 3 = x 3. To choose v 4, v 5, we look at two cases. 8
Case : p (mod 4) Then is a square. Say b = for some b F q. Then we choose v 4 = b( x x 5 ), v 5 = b( x x 4 ). Case : p 3 (mod 4) Over a finite field, is the sum of two squares, say a + a =. Choose v 4 = a ( x x 5 ) + a ( x x 4 ), v 5 = a ( x x 5 ) a ( x x 4 ). With this new basis, it is clear that an equivalent non-degenerate symmetric bilinear form of J 5 is I 5. Define the matrices P = b b b b, P = a a a a a a a a. In either case, P i J 5 t P i = I 5. Let P = P i, for the appropriate P i. Define the map φ P : SO(5, F q ) SO(5, F q ) φ P (g) := t P g t P, and φ P is an isomorphism. The maps defined above are composed to get an isomorphism from GSp(4, F q ) to SO(5, F q ), which will be called ρ 5,P := φ P ρ 5. The conjugacy classes of PGSp(4, F q ) can now be determined by computing either the conjugacy classes of SO(5, F q ) or of 9
SO(5, F q ), then using the appropriate isomorphism.. SO(5, F q ) and PGSp(4, F q ) We can determine the conjugacy classes of O(5, F q ) using Wall [5]. To do this, we first find the Jordan canonical forms of elements of GL(5, F q ) whose conjugacy class has a nonempty intersection with O(5, F q ). Once the possible Jordan canonical forms of elements in O(5, F q ) are found, it is straightforward to find class representatives in O(5, F q ). Wall [5] also gives a formula for the number of conjugacy classes of O(5, F q ), which is determined to be q + 6q + 4. Once the complete list of the classes of O(5, F q ) is computed, the class representatives and orders of centralizers for SO(5, F q ) are easily found. Let κ be a generator of F q and let ζ = κ q, θ = κ q +, η = θ q, and = θ q+. 4 The element η is the generator of the set of elements in F q whose norm over F q is and ζ is the generator of the set of elements in F q 4 whose norm over F q is. Let a, b F q be such that a + b is a square. Let c = +. Define the sets R = {,..., 4 (q )}, R is a set of 4 (q ) distinct positive integers i such that θ i, θ i, θ qi, and θ qi are all distinct, T = {,..., (q 3)}, T = {,..., (q )}, T 3 = {,..., q }. The table below lists all of the conjugacy classes of SO(5, F q ) and PGSp(4, F q ) together with the order of their centralizers in each group. Note that the class representatives are written in a form that may not be in SO(5, F q ), respectively PGSp(4, F q ), 0
but will belong to SO(5, F q 4), respectively PGSp(4, F q 4). This is done to indicate their eigenvalues, which are used to determine the orders of their centralizers. Table.: Conjugacy classes of SO(5, F q ) and PGSp(4, F q ) Notation Class representative in SO(5, F q ) Class representative in PGSp(4, F q ) Order of centralizer A q 4 (q 4 )(q ) A q 4 (q ) A 3 q 3 (q )
Table. Continued Notation Class representative in SO(5, F q ) Class representative in PGSp(4, F q ) Order of centralizer A 3 q 3 (q + ) A 5 q B q (q ) B q (q 4 )
Table. Continued Notation Class representative in SO(5, F q ) Class representative in PGSp(4, F q ) Order of centralizer B q(q )(q ) B q(q )(q + ) B 3 q (q ) 3
Table. Continued Notation Class representative in SO(5, F q ) Class representative in PGSp(4, F q ) Order of centralizer B 4 B 4 B 43 a b a a b a a b 4q B 44 c 4c c 4 c B 5 q(q ) 4
Table. Continued Notation Class representative in SO(5, F q ) Class representative in PGSp(4, F q ) Order of centralizer B 5 q(q + ) C (i), i T i i i i q(q )(q ) C (i), i T i i i i (q ) C (i), i T i i i+ i (q ) 5
Table. Continued Notation Class representative in SO(5, F q ) Class representative in PGSp(4, F q ) Order of centralizer C 3 (i), i T i i i i i q(q ) C 4 (i) i T i i i i i i i i q(q ) C 5 (i), i T i i i i i i q(q )(q ) C 6 (i, j), i, j T, i < j j i i j i j i+j (q ) 6
Table. Continued Notation Class representative in SO(5, F q ) Class representative in PGSp(4, F q ) Order of centralizer D (i), i R θ qi θ i θ i θ qi θ i θ qi i q D (i), i T η i η i θ i θ qi θ i θ qi q(q )(q + ) D 3 (i), i T η i η i θ i θ qi θ i θ qi (q ) D 3 (i), i T η i η i D 3 (i) (q + ) 7
Table. Continued Notation Class representative in SO(5, F q ) Class representative in PGSp(4, F q ) Order of centralizer D 4 (i, j), i T, j T j η i η i j θ qi θ i j θ qi j θ i q D 5 (i), i T η i η i θ i θ i θ i θ qi θ qi θ qi q(q + ) D 6 (i), i T η i η i η i η i η i η i q(q )(q + ) D 7 (i, j), i, j T, i < j η j η i η i η j D 7 (i, j) (q + ) 8
Table. Continued Notation Class representative in SO(5, F q ) Class representative in PGSp(4, F q ) Order of centralizer D 8 (i) i T η i η i η i η i η i η i η i η i q(q + ) D 9 (i), i R ζ qi ζ i ζ i ζ qi D 9 (i) q + Explicit forms of the classes D 3 (i), D 7 (i, j), and D 9 (i) are omitted since they have a more complicated form and are not classes of the Borel, the Siegel parabolic, or the Klingen parabolic subgroup..3 GSp(4, F q ) The list of conjugacy classes of PGSp(4, F q ) is used to determine the conjugacy classes of GSp(4, F q ). Let s investigate how the class representatives of PGSp(4, F q ) lead to representatives for the classes of GSp(4, F q ). Consider the natural projection map from GSp(4, F q ) to PGSp(4, F q ) given by GSp(4, F q ) PGSp(4, F q ), g g. 9
Let g, h GSp(4, F q ). If g = xhx, then, by taking multipliers on each side, it is clear that the multiplier of g is equal to the multiplier of h, i.e., λ(g) = λ(h). Moreover, under the projection map, g = xhx = x h x. So if two elements are conjugate in GSp(4, F q ), they must be conjugate in PGSp(4, F q ). The list of class representatives in PGSp(4, F q ), when pulled back to GSp(4, F q ), hit class representatives of all the conjugacy classes of GSp(4, F q ). Suppose now that two elements g, h GSp(4, F q ) are conjugate in PGSp(4, F q ), i.e. g = x h x, for some x PGSp(4, F q ). Then, for some i F q, g = i i i i xhx. Taking multipliers on both sides of the equation above, we have λ(g) = i λ(h). So if the multiplier of g is a square, then the multiplier of h is a square and if the multiplier of g is a non-square, then the multiplier of h is a non-square. Write g and h in the following way g = ig ig. jg jg jg jg.g, h = i h i h. j h j h j h j h.h, 30
with g, h Sp(4, F q ), i g, i h {0, }, and j g, j h T 3. So λ(g) = ig+jg, λ(h) = i h+j h. If g and h are conjugate, then ig = i h. Then j h = j h, or ( j g ) = ( j h ). So λ(h) = ±λ(g), i.e., the multipliers can only differ by a minus sign. It is possible that an element g of GSp(4, F q ) is conjugate to g. An example is.. =. The centralizers are somewhat affected when we pull back our representatives in PGSp(4, F q ) to GSp(4, F q ). There are two types of pullbacks. The first type consists of elements g such that g x( g)x for any x GSp(4, F q ). The second type consists of elements g such that g = x( g)x for some x GSp(4, F q ). Let Cent PGSp (g) denote the centralizer of g in PGSp(4, F q ) and let Cent GSp (g) denote the centralizer of g in GSp(4, F q ). Type Let g GSp(4, F q ) be of the first type, i.e., g is not conjugate to g. Let h Cent PGSp (g). Then g = h g h. When pulled back to GSp(4, F q ), g = z 0 hgh, with z 0 = ±I. z 0 I since g is not conjugate to g. So z 0 = I, g = hgh, and h Cent GSp (g). We get a short exact sequence Z Cent GSp (g) Cent PGSp (g). Therefore #Cent GSp (g) = (q ) #Cent PGSp (g). Type Let g GSp(4, F q ) be of the second type, i.e., g is conjugate to g. Define the set S g = {h GSp(4, F q ) : hgh = g}. Fix s 0 S g. S g is not a group, but 3
there is a bijection of sets S g Cent GSp (g), given by the map h s 0 h. Given h Cent PGSp (g), either h Cent GSp (g) or h S g. S g Cent GSp (g) maps onto Cent PGSp (g) via the projection map. Moreover, Cent GSp (g) := S g Cent GSp (g) is a group with respect to matrix multiplication and the projection map is a group homomorphism. Cent GSp (g) is a subgroup of Cent GSp (g) of index. We get a short exact sequence Z Cent GSp(g) Cent PGSp (g). Then #Cent GSp (g) = (q ) #Cent PGSp(g). Also, #Cent GSp (g) = #Cent GSp (g). So Cent GSp (g) = q #Cent PGSp (g). Thus, given a class representative g PGSp(4, F q ), we pull it back to GSp(4, F q ). Then, we determine if g is of Type or Type. If it is of Type, then there are q #GSp(4, F q ) conjugacy classes zg, for z Z, each of order. If the pullback (q ) #Cent PGSp (g) is of Type, then there are q conjugacy classes z i g, with z i = i i i i, i T. Each of these classes is of order #GSp(4, F q ) (q ) #Cent PGSp (g). Explicitly, the list of conjugacy classes of GSp(4, F q ) is given in the following table. 3
Table.: Conjugacy classes of GSp(4, F q ) Notation Class representative in GSp(4, F q ) Number of classes Order of centralizer A (k), q #GSp(4, F q ) A (k), k q q 4 (q )(q ) A 3 (k), q q 3 (q ) A 3 (k), + k q q 3 (q ) 33
Table. Continued Notation Class representative in GSp(4, F q ) Number of classes Order of centralizer A 5 (k), k q q (q ) B (k), k T q q (q ) (q ) B (k), k T + + q q (q 4 )(q ) B (k), k T q q(q )(q ) 34
Table. Continued Notation Class representative in GSp(4, F q ) Number of classes Order of centralizer B (k), k T + + q q(q ) B 3 (k), q q (q )(q ) 35
Table. Continued Notation Class representative in GSp(4, F q ) Number of classes Order of centralizer B 4 (k), k T q, B 4 (k), k T + q, B 43 (k), k T a k k+ a k+ b+ + q, q (q ) B 44 (k), k T c k+ + + q B 5 (k), k T q q(q ) 36
Table. Continued Notation Class representative in GSp(4, F q ) Number of classes Order of centralizer B 5 (k), k T + k k + k q q(q ) C (i, k), i T +i +i (q )(q 3) q(q )(q ) C (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 C (i, k), i T k T + +i+ +i (q )(q 3) 4 (q )(q ) 37
Table. Continued Notation Class representative in GSp(4, F q ) Number of classes Order of centralizer C 3 (i, k), i T +i +i +i (q )(q 3) q(q ) C 4 (i, k), i T +i i (q )(q 3) q(q ) C 5 (i, k), i T +i i (q )(q 3) q(q )(q ) C 6 (i, j, k), i, j T i < j +i +j +i+j (q )(q 3)(q 5) 8 (q ) 3 38
Table. Continued Notation Class representative in GSp(4, F q ) Number of classes Order of centralizer D (i, k), i R θ i θ qi +i (q ) 3 4 (q )(q ) D (i, k), i T θ i θ qi θ i θ qi (q ) q(q ) D 3 (i, k), i, k T θ i θ qi θ i θ qi (q ) 4 (q )(q ) D 3 (i, k), i, k T D 3 (i, k) (q ) 4 (q )(q + ) 39
Table. Continued Notation Class representative in GSp(4, F q ) Number of classes Order of centralizer D 4 (i, j, k), i T j T θ qi θ i +j θ qi +j θ i (q ) (q 3) 4 (q )(q ) D 5 (i, k), i T θ i θ i θ i θ qi θ qi θ qi (q ) q(q ) D 6 (i, k), i T η i η i (q ) q(q ) D 7 (i, j, k), i, j T i < j D 7 (i, j, k) (q ) (q 3) 8 (q )(q + ) 40
Table. Continued Notation Class representative in GSp(4, F q ) Number of classes Order of centralizer D 8 (i, k), i T η i η i (q ) q(q ) D 9 (i, k), i R D 9 (i, k) (q )(q ) 4 (q + )(q ) There are (q + q + 4)(q ) conjugacy classes..4 Borel, Siegel, and Klingen subgroups The conjugacy classes of the Borel, the Siegel parabolic, and the Klingen parabolic subgroup can now be easily computed using Table.. One does this by determining which conjugacy classes have a non-empty intersection with the subgroup, how each class splits, and computing the order of the centralizer of the class in the subgroup. In each of the following tables of conjugacy classes, the notation will indicate which conjugacy classes of GSp(4, F q ) 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 (k) has a non-empty intersection 4
with B, splitting into two conjugacy classes, denoted by BA (k) and BA (k)..4. Borel The Borel subgroup B of GSp(4, F q ) is the set of all of the upper triangular matrices, B = GSp(4, F q). Every element g B can be written uniquely in the form a g = b cb ca. x λ µ κ. µ λ, with a, b, c F q and x, κ, λ, µ F q. The order of B is therefore q 4 (q ) 3. The multiplier of the matrix g given above is λ(g) = c. The subgroup of B of elements which have on every entry on the main diagonal is N GSp(4). The conjugacy classes of the Borel subgroup B are given in the following table. 4
Table.3: Conjugacy classes of the Borel subgroup Notation Class representative Number of classes in B Order of centralizer in B BA (k), q q 4 (q ) 3 BA (k), k q q 3 (q ) BA (k), q q 4 (q ) BA 3(k), q q 3 (q ) 43
Table.3 Continued Notation Class representative Number of classes in B Order of centralizer in B BA 3(k), q q 3 (q ) BA 3 3(k), q q (q ) BA 3 (k), + k q q 3 (q ) BA 5 (k), k q q (q ) 44
Table.3 Continued Notation Class representative Number of classes in B Order of centralizer in B BB (k), k T q q (q ) 3 BB (k), k T q q (q ) 3 BB (k), k T q q(q ) 3 BB (k), k T q q(q ) 3 45
Table.3 Continued Notation Class representative Number of classes in B Order of centralizer in B BB 3 (k), k T q q(q ) 3 BB 4 (k), k T q q(q ) 3 BB 3(k), q q (q ) BB 3(k), k q q (q ) 46
Table.3 Continued Notation Class representative Number of classes in B Order of centralizer in B BB 4(k), k T q q (q ) BB 4(k), k T 4 k q q (q ) BB 4(k), k T + q q (q ) BB 4(k), k T 4 k+ q q (q ) 47
Table.3 Continued Notation Class representative Number of classes in B Order of centralizer in B BB 5(k), k T q q(q ) BB 5(k), k T q q(q ) BB 3 5(k), k T q q(q ) BB 4 5(k), k T q q(q ) 48
Table.3 Continued Notation Class representative Number of classes in B Order of centralizer in B BC (i, k), i T +i +i (q )(q 3) q(q ) 3 BC (i, k), i T +i +i (q )(q 3) q(q ) 3 BC 3 (i, k), i T +i +i (q )(q 3) q(q ) 3 BC 4 (i, k), i T +i +i (q )(q 3) q(q ) 3 49
Table.3 Continued Notation Class representative Number of classes in B Order of centralizer in B BC (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 BC (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 BC 3 (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 BC 4 (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 50
Table.3 Continued Notation Class representative Number of classes in B Order of centralizer in B BC 5 (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 BC 6 (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 BC 7 (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 BC 8 (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 5
Table.3 Continued Notation Class representative Number of classes in B Order of centralizer in B BC 3(i, k), i T +i +i +i (q )(q 3) q(q ) BC 3(i, k), i T +i +i +i (q )(q 3) q(q ) BC 3 3(i, k), i T +i +i +i (q )(q 3) q(q ) BC 4 3(i, k), i T +i +i +i (q )(q 3) q(q ) 5
Table.3 Continued Notation Class representative Number of classes in B Order of centralizer in B BC 4(i, k), i T +i i (q )(q 3) q(q ) BC 4(i, k), i T k i +i i +i (q )(q 3) q(q ) BC 3 4(i, k), i T i k +i (q )(q 3) q(q ) BC 4 4(i, k), i T +i k i (q )(q 3) q(q ) 53
Table.3 Continued Notation Class representative Number of classes in B Order of centralizer in B BC 5(i, k), i T +i i (q )(q 3) q(q ) 3 BC 5(i, k), i T +i i (q )(q 3) q(q ) 3 BC 3 5(i, k), i T i +i (q )(q 3) q(q ) 3 BC 4 5(i, k), i T i +i (q )(q 3) q(q ) 3 54
Table.3 Continued Notation Class representative Number of classes in B Order of centralizer in B BC6(i, j, k), i, j T i < j +i +j +i+j (q )(q 3)(q 5) 8 (q ) 3 BC6(i, j, k), i, j T i < j +i+j +j +i (q )(q 3)(q 5) 8 (q ) 3 BC6(i, 3 j, k), i, j T i < j +i+j +i +j (q )(q 3)(q 5) 8 (q ) 3 BC6(i, 4 j, k), i, j T i < j +j +i+j +i (q )(q 3)(q 5) 8 (q ) 3 55
Table.3 Continued Notation Class representative Number of classes in B Order of centralizer in B BC6(i, 5 j, k), i, j T i < j +j +i+j +i (q )(q 3)(q 5) 8 (q ) 3 BC6(i, 6 j, k), i, j T i < j +i +i+j +j (q )(q 3)(q 5) 8 (q ) 3 BC6(i, 7 j, k), i, j T i < j +i +i+j +j (q )(q 3)(q 5) 8 (q ) 3 56
Table.3 Continued Notation Class representative Number of classes in B Order of centralizer in B BC6(i, 8 j, k), i, j T i < j +j +i +i+j (q )(q 3)(q 5) 8 (q ) 3.4. Siegel The Siegel parabolic subgroup P of GSp(4, F q ) is defined as P = GSp(4, F q). Every element p P can be written uniquely in the form a p = c b d λa/ λc/ µ κ λb/. x µ, λd/ 57
with = ad bc F q, λ F q and x, κ, µ F q. The order of P is therefore q 4 (q )(q ). The multiplier of p is λ(p) = λ. We also define A = t A for any A GL(, F q ). The conjugacy classes of the Siegel parabolic subgroup P are given in the following table. Table.4: Conjugacy classes of the Siegel parabolic subgroup Notation Class representative Number of classes in P Order of centralizer in P P A (k), q q 4 (q )(q ) P A (k), k q q 4 (q ) 58
Table.4 Continued Notation Class representative Number of classes in P Order of centralizer in P P A 3(k), q q 3 (q ) P A 3(k), q q (q ) P A 3 (k), + k q q 3 (q ) P A 5 (k), k q q (q ) 59
Table.4 Continued Notation Class representative Number of classes in P Order of centralizer in P P B (k), k T q q (q ) 3 P B (k), k T + + q q (q )(q ) P B (k), k T q q(q )(q ) P B (k), k T q q(q )(q ) 60
Table.4 Continued Notation Class representative Number of classes in P Order of centralizer in P P B 3 (k), k T q q(q ) 3 P B (k), k T + + q q(q )(q ) P B 3 (k), q q (q ) P B 4 (k), k T q q (q ) 6
Table.4 Continued Notation Class representative Number of classes in P Order of centralizer in P P B 4 (k), k T + q q (q ) P B 43 (k), k T + ak ak+ b+ + q q (q ) P B 44 (k), k T c k+ + + q q (q ) P B 5(k), k T q q(q ) 6
Table.4 Continued Notation Class representative Number of classes in P Order of centralizer in P P B 5(k), k T q q(q ) P B 3 5(k), k T k k q q(q ) P B 5 (k), k T + k k + k q q(q ) P C (i, k), i T +i +i (q )(q 3) q(q )(q ) 63
Table.4 Continued Notation Class representative Number of classes in P Order of centralizer in P P C (i, k), i T +i +i (q )(q 3) q(q )(q ) P C 3 (i, k), i T +i +i (q )(q 3) q(q ) 3 P C (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 P C (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 64
Table.4 Continued Notation Class representative Number of classes in P Order of centralizer in P P C 3 (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 P C 4 (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 P C (i, k), i T k T + +i+ +i (q )(q 3) 4 (q )(q ) P C (i, k), i T k T +i+ +i + (q )(q 3) 4 (q )(q ) 65
Table.4 Continued Notation Class representative Number of classes in P Order of centralizer in P P C 3(i, k), i T +i +i +i (q )(q 3) q(q ) P C 3(i, k), i T +i +i +i (q )(q 3) q(q ) P C 3 3(i, k), i T +i +i +i (q )(q 3) q(q ) P C 4(i, k), i T +i i (q )(q 3) q(q ) 66
Table.4 Continued Notation Class representative Number of classes in P Order of centralizer in P P C 4(i, k), i T k i +i i +i (q )(q 3) q(q ) P C 5(i, k), i T +i i (q )(q 3) q(q ) 3 P C 5(i, k), i T i +i (q )(q 3) q(q ) 3 P C6(i, j, k), i, j T i < j +i +j +i+j (q )(q 3)(q 5) 8 (q ) 3 67
Table.4 Continued Notation Class representative Number of classes in P Order of centralizer in P P C6(i, j, k), i, j T i < j +i+j +j +i (q )(q 3)(q 5) 8 (q ) 3 P C6(i, 3 j, k), i, j T i < j +i+j +i +j (q )(q 3)(q 5) 8 (q ) 3 P C6(i, 4 j, k), i, j T i < j +j +i+j +i (q )(q 3)(q 5) 8 (q ) 3 68
Table.4 Continued Notation Class representative Number of classes in P Order of centralizer in P P D (i, k), i T θ i θ qi θ i θ qi (q ) q(q )(q ) P D 3(i, k), i, k T θ i θ qi θ i θ qi (q ) 4 (q )(q ) P D 3(i, k), i, k T θ i θ qi θ i θ qi (q ) 4 (q )(q ) P D4(i, j, k), i T j T θ qi θ i +j θ qi +j θ i (q ) (q 3) 4 (q )(q ) 69
Table.4 Continued Notation Class representative Number of classes in P Order of centralizer in P P D4(i, j, k), i T j T +j θ qi +j θ i θ qi θ i (q ) (q 3) 4 (q )(q ) P D 5 (i, k), i T θ i θ i θ i θ qi θ qi θ qi (q ) q(q ).4.3 Klingen The Klingen parabolic subgroup Q of GSp(4, F q ) is defined as Q = GSp(4, F q). 70
Every element g Q can be written uniquely in the form t g = a c b d t λ µ κ. µ λ, with = ad bc F q, t F q, and κ, λ, µ F q. The order of Q is therefore q 4 (q )(q ). The multiplier of g given above is λ(g) =. The conjugacy classes of the Klingen parabolic subgroup Q are given in the following table. Table.5: Conjugacy classes of the Klingen parabolic subgroup Notation Class representative Number of classes in Q Order of centralizer in Q QA (k), q q 4 (q )(q ) QA (k), k q q 3 (q ) 7
Table.5 Continued Notation Class representative Number of classes in Q Order of centralizer in Q QA (k), q q 4 (q )(q ) QA 3(k), q q 3 (q ) QA 3(k), q q 3 (q ) QA 3 (k), + k q q 3 (q ) 7
Table.5 Continued Notation Class representative Number of classes in Q Order of centralizer in Q QA 5 (k), k q q (q ) QB (k), k T q q (q )(q ) QB (k), k T q q (q )(q ) QB (k), k T q q(q ) 3 73
Table.5 Continued Notation Class representative Number of classes in Q Order of centralizer in Q QB (k), k T q q(q ) 3 QB 3(k), q q (q )(q ) QB 3(k), k q q (q ) QB 4(k), k T q q (q ) 74
Table.5 Continued Notation Class representative Number of classes in Q Order of centralizer in Q QB 4(k), k T 4 k q q (q ) QB 4(k), k T + q q (q ) QB 4(k), k T 4 k+ q q (q ) QB 5(k), k T q q(q ) 75
Table.5 Continued Notation Class representative Number of classes in Q Order of centralizer in Q QB 5(k), k T q q(q ) QC (i, k), i T +i +i (q )(q 3) q(q ) 3 QC (i, k), i T +i +i (q )(q 3) q(q ) 3 QC (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 76
Table.5 Continued Notation Class representative Number of classes in Q Order of centralizer in Q QC (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 QC 3 (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 QC 4 (i, k), i T k T +i +i (q )(q 3) 4 (q ) 3 QC 3(i, k), i T +i +i +i (q )(q 3) q(q ) 77
Table.5 Continued Notation Class representative Number of classes in Q Order of centralizer in Q QC 3(i, k), i T +i +i +i (q )(q 3) q(q ) QC 4(i, k), i T +i i (q )(q 3) q(q ) QC 4(i, k), i T i k +i (q )(q 3) q(q ) QC 3 4(i, k), i T +i k i (q )(q 3) q(q ) 78
Table.5 Continued Notation Class representative Number of classes in Q Order of centralizer in Q QC 5(i, k), i T +i i (q )(q 3) q(q )(q ) QC 5(i, k), i T +i i (q )(q 3) q(q ) 3 QC 3 5(i, k), i T i +i (q )(q 3) q(q )(q ) QC6(i, j, k), i, j T i < j +i +j +i+j (q )(q 3)(q 5) 8 (q ) 3 79
Table.5 Continued Notation Class representative Number of classes in Q Order of centralizer in Q QC6(i, j, k), i, j T i < j +i+j +j +i (q )(q 3)(q 5) 8 (q ) 3 QC6(i, 3 j, k), i, j T i < j +j +i+j +i (q )(q 3)(q 5) 8 (q ) 3 QC6(i, 4 j, k), i, j T i < j +i +i+j +j (q )(q 3)(q 5) 8 (q ) 3 QD (i, k), i R θ i θ qi +i (q ) 3 4 (q )(q ) 80
Table.5 Continued Notation Class representative Number of classes in Q Order of centralizer in Q QD (i, k), i R +i θ i θ qi (q ) 3 4 (q )(q ) QD 6 (i, k), i T η i η i (q ) q(q )(q ) QD 8 (i, k), i T η i η i (q ) q(q ).5 N GSp(4) Recall that the subgroup N GSp(4) of GSp(4, F q ) is defined as y N GSp(4) = x y GSp(4, F q). 8
Every element g N GSp(4) can be written uniquely in the form g = x λ µ κ. µ λ, with x, κ, λ, µ F q. The order of N is therefore q 4. The multiplier of the matrix g given above is λ(g) =. The conjugacy classes of N = N GSp(4) are listed in the following table. Note that the notation is slightly different than the standard notation we have been using for conjugacy classes of subgroups of GSp(4, F q ). Table.6: Conjugacy classes of N GSp(4) Notation Class representative Number of classes in N Order of centralizer in N NA q 4 NA (k), q q 3 8
Table.6 Continued Notation Class Representative Number of Classes in N Order of Centralizer in N NA (k), q q 4 NA 3(i, j, κ), i, j T 3 i j κ = n for some n T 3 i κ j i (q ) q 3 NA 3(k), q q NA 3 3(K), k q q 3 83
Table.6 Continued Notation Class Representative Number of Classes in N Order of Centralizer in N NA 3 (i, j, κ), i, j T 3 i j κ = n+ for some n T 3 i κ j i (q ) q 3 NA 5 (i, j), i, j T 3 i j i+j i (q ) q 84
Chapter 3 Induced characters Let C be a conjugacy class of a group G. Then the conjugacy class C splits into distinct conjugacy classes of a subgroup H, say C = D... D r. The value of the induced character is given by the formula Ind G H(χ)(C) = χ G (C) = G H r i= D i C χ(d i). This formula is used to find the induced character values of representations defined on the Borel, the Siegel parabolic, and the Klingen parabolic subgroups. This will lead to the complete list of the irreducible non cuspidal characters of GSp(4, F q ). This formula is also used to find the character values of the representation G to determine the genericity of characters. This chapter includes the character tables of representations induced from B, P, and Q as well as the character table of G. If a conjugacy class of GSp(4, F q ) is not listed in the table, the character takes the value 0 on that conjugacy class. 85
3. Parabolic induction The conjugacy classes of GSp(4, F q ),B, P, and Q have been determined so one can easily find all of the parabolically induced character values. 3.. Borel Let χ, χ, and σ be characters of the multiplicative group F q. Define a character on the Borel subgroup B by a b cb χ (a)χ (b)σ(c). ca The character of this representation is given by χ (a)χ (b)σ(c). This representation is induced to obtain a representation of GSp(4, F q ), denoted by χ χ σ. The standard model of this representation is the space of functions f : GSp(4, F q ) C satisfying a f(hg) = χ (a)χ (b)σ(c)f(g), for all h = b cb B. ca The group action is by right translation. The central character of χ χ σ is χ χ σ. The character table of χ χ σ is the following. 86