Introduction
Motivation
In particular, we find desired cuspidal regular irreducible representations of the quasi-split group U(2,2) over a finite field, which decompose on restriction to SU(2,2). We note that the reason why the character theory of the finite group of Lie type SU(2,2) defined over a finite field is arithmetic in nature is because the underlying group SL4(Fp) has non-connected center .
Overview of the results
We call an irreducible representation of GeF regular if it is a component of the Gelfand-Graev representation. Let π be an irreducible representation of GeF that splits into d0 components with constraint on GF, which is π|.
Structure of the thesis
Again, this mimics the case of SL2, where the non-zero contribution to π+−π− comes from the unipotent conjugation classes. In Chapter 5 we find regular cuspid irreducible representations of GeF that split when restricted to GF for the case of SU(2,2).
Concluding remarks
The action of F on groups of signs and cosigns of the maximally split torus T is given by. In light of the above proposal, we will sometimes use the notation RTG*(s) for RGT(θ).
Preliminaries
We consider several distinctive irreducible representations of G=SL2(Z/DZ) described in Section 2.2; for now, it suffices to say that these representations are (up to isomorphism) separated along tuples of the form t|e), where i ∈ {±} and e ∈ {±1}. We compute the fixed points ofg in X(D) in Section 2.4, which allows us to compute the holomorphic Lefschetz number of the map g in Section 2.5.
Ingredients of the main theorem
There is, up to twist equivalence, a unique irreducible cuspidal representationπ that when restricted to SL2(Fp) splits into a pair of two irreducible π+, π− of the same dimension; the representationπλ is given by the unique nontrivial quadratic characterλ of order p+ 1. The signs of the representations appearing in a distinctive G-representation on these conjugation classes are as in Table 2.3 above.
A key virtual character
Note that we have used the fact that -u1 and u−1 are in the same conjugacy class of SL2(Z/4Z). When not specified, xi above takes values in{1, ηpi}, where ηpi is a non-square modpi for i ∈ [0, t] an integer.
Fixed points on the modular curve
The holomorphic Lefschetz number
Let dgκ: Tκ(M) → Tκ(M) be the difference induced by the map g on the holomorphic tangent space at the fixed point κ. We calculate the Lefschetz numbers using the fixed points in Lemma 2.4.2, which in turn give us the characters χS2(Γ(D)).
Some useful lemmas
Proof of the main theorem
A maximal torus of GF is defined as a subgroup of the form TF, where T is an F-stable maximal torus of G. In light of the semi-simple classes that do not contribute to the alternate sum of multiplicities ∆Mπ, we want to show that the regular unipotent classes are non-zero on the alternate sum. A formula for π1, π2 on the regular unipotent elements in this particular case will be given in the proof of the main result of Chapter 7.
Then the GF conjugation class of the pair (T, θ) corresponds to the G∗F∗ conjugation class of the pair (T∗, s), where s is a regular semisimple element of G∗F∗ and T ∗ is a rational maximal torus containing s. Thus, since by Lemma 6.2.1 the restriction of the irreducible cuspidal representation π of GeF introduced at the beginning of Section 6.2, is given by π|.
Study case SU(1, 1): Reframing Hecke’s original setting for
The associated symmetric domain G/K
The goal is to realize it as a bounded symmetric domain and look at the Gondas action given by the usual SL2(R) action on the upper half plane. By the Harish-Chandra decomposition, we know that there exists a bounded open subset D⊆P+ such that the mapping G→D is given by. This action is actually the action given by the piecewise linear transformation we saw earlier.
The action of G on D viewed as the action of SL2(R) on the top half-plane. One can easily verify that the induced action of D0 on D0 is the usual action of SL2(R) on the upper half plane.
Rational boundary components
Now let c denote the composition of the Cayley transformations corresponding to a maximal set of strongly orthogonal roots. We make a further comment on obtaining constrained roots from successful applications of the Cayley transformations. Note that the restricted roots are obtained by projecting all roots in ∆ onto the linear span of the maximal set of strongly orthogonal, non-compact imaginary roots.
As a result, projecting onto the linear space of the maximally strong orthogonal set is equivalent to the above description of finding the restricted roots. We know that the rational boundary components F of D=G/K are parametrized in terms of the subsets S r}, where r is the rank of g.
First, it is well known that the coupled component of the centralizer CG∗(s)◦ is reductive, with Weyl group W∗◦(s) generated by reflections wα∗ with α∗ ∈Φ∗ such that α∗(s) = 1 GeF = 1, where Γ is the Gelfand-Graev character of the group GeF, i.e. we want π to be a regular character. We will see that this means that the restriction of the cuspid representation π to GF will be given by χ(s).
Recall that the construction of the bijection depends on the fact that if T is an F-stable maximal torus that puts (G, F) and (G∗, F∗) in duality, then we have an isomorphism between T∗ F∗ and the character group(TF)∧ as seen in proposition 4.1.7. Since in the case of G we are simply connected, we have X(T) = Ω, so γ is in fact an element of the guards' lattice.
Preliminaries on Deligne-Lusztig theory and character val-
Basic notions on algebraic groups and maximal tori
Note that while every maximal torus of G lies in a Borel subgroup of G, it need not be true that every F-stable maximal torus of G lies in an F-stable Borel. As a result, it may happen that there is a maximal torus of GF that does not lie in a Borel subgroup of GF. We call an F-stable maximal torus that lies in an F-stable Borel subgroup of Gmaximally split.
We also know that there exists an F-stable Borel subgroup and thus an F-stable maximal torus of G. Let T be an F-stable maximal torus of G and T∗ an F∗-stable maximal torus of G∗ such that T, T∗ in duality in the sense of Proposition 4.1.6.
Deligne-Lusztig representations
When L=T is a maximal torus, RTG(θ), for θ an irreducible character of TF, is precisely the induction from BF to GF, mentioned above, of a character of BF induced by θ. Note that the Harish-Chandra induction gives us an RLG functional when L is a rational Levi of a rational parabolic P of G. We have seen so far that RGT(θ) for maximal rational T and θ an irreducible character of TF , is a generalized character of GF.
We will first give a formula for the scalar product hRGT(θ), RTG0(θ0)iGF which will tell us, up to a sign, when RGT(θ) is an irreducible character of GF. Let T be an F-stable maximal torus of Gand, let θ be an irreducible character of TF in general position.
Character values on regular unipotent classes
Following the direct generalization to the case of SL2, we want this alternative sum to be non-zero on the regular impoverished classes, so we require. Since π is an irreducible cuspidal representation of GeF which is a component of the Gelfand-Graev representation Γ, we have seen by the result of Theorem 5.1.4 that π must be of the form π = ±RGe. In the following we will finalize the first step in calculating the values of the irreducible components of π|GF on regular unipotent elements, as detailed in the discussion after Theorem 4.3.15.
The purpose of the following lemmas is to describe the isomorphism γ as rewriting equation (6.3) in terms of Gaussian sums. Maintaining the above notation, a lifting of the type presented in Lemma 6.3.3 for the isomorphism γ :H1(F, Z)→µd is given by.
Study case SU (2, 2), part I: On finding desired cuspidal
On regular cuspidal irreducible representations
Let V =E4 be the 4-dimensional vector space over E with standard basis, and L ⊂ V the standard OE lattice in V. Note that U(2,2) is a connected reductive group with connected center such that SU(2, 2) contains. , while SU(2,2) is a rational form of G=SL4(Fp). We note that for regular characters π, that is, irreducible components of the Gelfand-Graev representation of GeF, the constraint π|.
If π is cuspid, irreducible, and a component of Γ, the Gelfand-Graev representation of GF, then π has the form π = ±RTG(θ), where T is an F-stable maximal torus of G and θ is a non- -singular character of TF. The statement of the lemma follows simply: if two charactersθi,θj fori6=j are equal, we get θwe =θforwij ∈W(Tew)F given by the transposition(ij) inS4.
Zero contribution for ∆M π on the semisimple conjugacy classes
As a result, since CG(x)/CG(x)◦ is trivial, Theorem 5.2.1 tells us that there is only one GF conjugation class at the intersection of the geometric conjugation class of x with GF. First we will write these characters in terms of the complex numbers σz; this is done in Theorem 6.2.10 when d0 =d. Note: Gaussian sums of the above form can be defined over any finite field with a primary feature.
As outlined by the result of Theorem 4.3.15, we will see what is the value of the inner product of the double DG(χ) with a Gelfand-Graev representation. On the other hand, T is a maximal F-stable torus in G, contained in a maximal F-stable torus Te of Ge, so that Te = TZe0 and using the same ideas as in the proof of the result in Lemma 4.2.6, we get Te =TZe.
Towards a formula for multiplicity defect for G F
Some basic results on Gauss sums
This section is intended to recall basic results on Gaussian sums that will be instrumental in the calculation of σζ. Note that in the definition above,φ is a multiplicative character of F×q|I| is, while χ0 is an additive character of FqN which as a character of Fq|I| be considered. Let Fq be a finite field with q elements, χ a non-trivial additive character and φ a non-trivial multiplicative character of Fq.
Note that in our case χ0 is non-trivial, so we have G(φ) :=G(φ, χ0) modulo pq|I| if φ is a multiplicative character of F×p|I|.
Step I: On a final form of character values of regular cuspidal
Although the following analysis will not deal with this situation in detail, exactly the same ideas of the proof apply, with minor modifications related to the fact that the irreducible components of a certain character χ(s) in more than one Gelfand-Graev representation Γz of GF, z ∈ H1(F, Z). A useful overview of some of the background ingredients used in our proofs is also detailed in Chapters 1,3 and 4 of [4] and Chapter 2 in [17]. Let's look at the embedding G → Ge and reason at the level of the group G, which connected midpoint.e θ is the restriction to TF of the character θevan TeF which is in general position, and therefore non-singular since Ge midpoint connected .
So we have π = χ( . es) and by general properties of the Deligne-Lusztig inductionResGGeFF(χ( . es)) = χ(s), so we come to the same conclusion that π|GF =χ(s), where is regular semisimple in G∗F∗. Besides proving the desired multiplicity defect, this suggests possible interpretations of ∆Mπ for SU(2,2) in terms of certain arithmetic invariants that would be analogous to the class number found in Hecke's original problem for SL2. Then the multiplicative characters ζI1 and ζI2 introduced in Theorem 6.3.5 are the unique character of degree two of the cyclic group F×p2, respectively a character of F×p that takes the value 1on the square modp.
Let A be the fundamental group of the derived group of∗, i.e. its weight network modulo its root network.
