A note on cusp forms and representations of SL₂(𝔽_𝑝)

1 Introduction

Given a prime p, the cusp forms of weight k for the principal congruence subgroup Γ(p):=Ker(SL2(ℤ)→SL2(𝔽p)) form a finite-dimensional linear space over ℂ, denoted by Sk⁢(Γ⁢(p)); these holomorphic functions defined on the upper half-plane are objects of considerable interest in number theory. Here we focus on the case k=2. The space S2⁢(Γ⁢(p)) is acted on by SL2⁢(𝔽p) in a natural way. We want to understand this space by viewing SL2⁢(𝔽p) as a finite reductive group.

There is a finite field analogue of the upper half-plane, ℙ1\ℙ1⁢(𝔽p), which is an algebraic curve over 𝔽¯p. The group SL2⁢(𝔽p) also acts on this curve and its ℓ-adic cohomology in a natural way, and Drinfeld found that all the discrete series representations of SL2⁢(𝔽p) appeared in this cohomology. This is one of the starting points of Deligne–Lusztig theory, a geometric approach to the representations of reductive groups over finite fields.

Let G=SL2⁢(𝔽¯p) with F being the standard Frobenius endomorphism on G over 𝔽p (so GF=SL2⁢(𝔽p)). And let Z be the centre of G.

We recall our basic objects briefly. (The details can be found e.g. in [5, 1].) First, S2⁢(Γ⁢(p)) can be identified with the space of differential 1-forms on the modular curve in a natural way, and GF acts on S2⁢(Γ⁢(p)) via this identification by Möbius transformation. Let S2⁢(Γ⁢(p))¯ be the dual space of S2⁢(Γ⁢(p)), and denote the character of S2⁢(Γ⁢(p))+S2⁢(Γ⁢(p))¯ by S2,p. Now fix an anisotropic torus Ta, and fix a split torus Ts. Note that Ta∩Ts=Z. For an irreducible character θs∈TsF^, we put RTsθ:=IndBFGFθs~, where B is an F-stable Borel subgroup containing Ts, and θs~ is the trivial extension of θs; this gives roughly half of the irreducible characters of GF, and the other ones come from the geometry of ℙ1\ℙ1⁢(𝔽p): fix a prime ℓ≠p; each θa∈TaF^ corresponds to an ℓ-adic local system on ℙ1\ℙ1⁢(𝔽p), and the trace of the alternating sum of the corresponding cohomology is a virtual character RTaθa. These RTaθa and RTsθs are called Deligne–Lusztig characters of GF.

We show that (see Theorem 2.1) the character S2,p of SL2⁢(𝔽p) is a sum of Deligne–Lusztig characters with interesting coefficients: the coefficients are linear polynomials in p and only depend on the residue of p modulo 12. Moreover (see Corollary 2.6 and Corollary 2.8), these coefficients imply that the single space S2⁢(Γ⁢(p)) usually does not admit such a decomposition, and that every non-trivial irreducible character of PSL2⁢(𝔽p) is a summand of S2,p when p is big enough. Our argument is based on a formula due to Jared Weinstein and a tensor product property of the Steinberg character.

2 Comparing the spaces

For convenience, we assume p≥7. (This avoids the case that dim⁡S2⁢(Γ⁢(p))=0.)

We shall start with a Mackey formula for a tensor product with the Steinberg character. (Indeed, the study concerning tensoring with the Steinberg character has a long history; see [7, 8] for more recent works.) Our original approach is an explicit computation, which is now replaced by the following general lemma due to an anonymous referee.

From now on, we let 𝐆 be G=SL2 over 𝔽p. (So 𝐙=Z.) Let T be an F-stable maximal torus of G and θ∈TF^; we always assume T is Ta or Ts (a fixed anisotropic torus and a fixed split torus). We denote the unique character of TF of order 2 by α. Note that α is the “quadratic residue symbol”, i.e. α⁢(t)=1 if and only if t is a square in TF; in particular, if T=Ts, then α|Z=1 if and only if p=1mod4; if T=Ta, then α|Z=1 if and only if p=3mod4.

Let the following subgroups be given:

Then Weinstein’s argument in [15, p. 31] implies that S2,p is a sum of the permutation characters

(See also [14, Theorem 4.3], in which the formula is established in the framework of parabolic cohomology.)

In order to put the space of cusp forms into the picture of representation theory of a finite reductive group, we need to decompose the above characters of large degree; for this purpose, we shall use the following nice formula of tensor product by Steinberg character (see also [8, Lemma 2.1] for a relevant formula with general class functions).

Now let G~* be the preimage of G* along the surjection GF→GF/Z for each *∈{x,y,z}. Then (2.2) becomes

Here a basic observation is that the generators of Gx and Gy are semisimple, so we can conjugate G~x and G~y into TsF≅𝔽p× or TaF≅μp+1, which depends on pmod12.

For *∈{x,y}, let T* be one of Ts and Ta, and suppose G~* lies in T*. Then (2.3) becomes (note that BF/G~z=TsF/Z)

where the second equality follows from Lemma 2.4.

The coefficients cθ above also imply that, unlike the sum S2,p, the single space S2⁢(Γ⁢(p)) is usually not uniform (in the sense of [10, Section 2.15]). For instance, we have the following result.

3 A further remark

It would be interesting to know whether there are similar results for the principal congruence subgroups Γ⁢(pr) for all r∈ℤ>0, in which case the representations of SL2⁢(ℤ/pr)≅SL2⁢(ℤp/pr) (i.e. the smooth representations of SL2⁢(ℤp)) are involved. Note that there are generalisations of Deligne–Lusztig theory to this setting; see e.g. [11, 12, 13, 3]. Also note that, while we are yet lacking a good knowledge of values of the generalised Deligne–Lusztig characters, Weinstein’s formula (2.2) still holds, and there are also possible candidates of the Steinberg representation, like the ones in [9, 2].