A Hecke-equivariant decomposition of spaces of Drinfeld cusp forms via representation theory, and an investigation of its subfactors

There are various reasons why a naive analog of the Maeda conjecture has to fail for Drinfeld cusp forms. Focussing on double cusp forms and using the link found by Teitelbaum between Drinfeld cusp forms and certain harmonic cochains, we observed a while ago that all obvious counterexamples disappear for certain Hecke-invariant subquotients of spaces of Drinfeld cusp forms of fixed weight, which can be defined naturally via representation theory. The present work extends Teitelbaum's isomorphism to an adelic setting and to arbitrary levels, it makes precise the impact of representation theory, it relates certain intertwining maps to hyperderivatives of Bosser-Pellarin, and it begins an investigation into dimension formulas for the subquotients mentioned above. We end with some numerical data for $A=\mathbb{F}_3[t]$ that displays a new obstruction to an analog of a Maeda conjecture by discovering a conjecturally infinite supply of $\mathbb{F}_3(t)$-rational eigenforms with combinatorially given (conjectural) Hecke eigenvalues at the prime $t$.


Introduction
The conjecture of Maeda, which is now supported by much computational evidence but no theoretical insight, asserts that the spaces S cl k (SL 2 (Z)) of classical modular forms of weight k and level one consist of a single Hecke orbit under the natural action of Gal(Q/Q) on systems of Hecke eigenvalues. This suggests that S cl k (SL 2 (Z)) possesses no non-trivial decompositions into Hecke stable subspaces. Moreover the Maeda conjecture asserts that for fixed k the Galois group associated to the Hecke field is the symmetric group S m on m = dim S cl k (SL 2 (Z)) letters. Recently variants of the conjecture for levels Γ other than one have been suggested; see [DPP18]. Once the subspace spanned by CM forms has been removed from S cl k (Γ), the number of Hecke orbits seems to be related to the number of inertial types of conductor given by the level -but this does not fully explain what is observed.
For Drinfeld modular forms, the situation is different. Let A be the coordinate ring of a smooth projective curve over a finite field of characteristic p minus one point ∞ with quotient field F and let Γ ⊂ GL 2 (F ) denote a congruence subgroup (throughout this introduction). Denote by S k,l (Γ) the space of Drinfeld cusp forms of weight k, type l and level Γ. The work [BP09] of Bosser and Pellarin for A = F q [t] and Γ = GL 2 (A) gives maps S k ′ ,l ′ (Γ) → S k,l (Γ) under certain numerical conditions on (k, l) and (k ′ , l ′ ), that have to do with the vanishing of certain binomial coefficients mod p and require k − k ′ to be even. The maps are given in terms of hyperderivatives and are Hecke equivariant up to a twist by a character, which for a prime p of A is given by p (k−k ′ )/2 . Since these maps are in general neither trivial nor surjective, the spaces S k,l (Γ) can contain non-trivial Hecke stable subspaces. In particular, a direct analog of a Maeda type conjecture is not possible for Drinfeld modular forms. An even simpler argument to dispute such an analog is to use the p-power map S k,l (Γ) → S pk,pl (Γ). It is injective and, up to a Frobenius-twist, Hecke equivariant; but clearly the target has in general a strictly larger dimension than the domain. It has also been known from the beginning that there are cuspidal Hecke eigenforms that are not doubly-cuspidal. However they could play a role similar to CM forms in the classical case.
Our starting point towards a systematic study of Maeda-style conjectures in the Drinfeld setting is the Hecke-equivariant residue isomorphism S k,l (Γ) ∼ = C har (V k,l ) Γ of Teitelbaum from [Tei91] between spaces of cusp forms of a given weight k, type l and level Γ and spaces of Γ-invariant harmonic cochains with values in a certain GL 2 -representation V k,l that is a finite-dimensional F -vector space. The results in [Tei91] are only fully developed for groups Γ that have no prime-to-p torsion, and the results are not developed in an adelic setting. In [Böc02] the first author gave such an adelic setting, but only for small adelic level groups K. As the group SL 2 (A), which has prime-to-p torsion, is the most natural candidate for studying Maeda-style behaviors in our setup, and as one needs a suitable Hecke theory, Section 1 of this article is devoted to removing this restriction and to proving the following theorem which generalizes both [Tei91] and [Böc02]. Here, we denote by S k,l (K) the space of adelic Drinfeld cusp forms of weight k, type l and level K, by C ad har (V k,l , K) the corresponding space of adelic harmonic cochains and by St K the adelic Steinberg module. All three carry an action of a naturally defined Hecke algebra H K .
(a) There are Hecke-equivariant isomorphisms S k,l (K) ≃ −→ C ad har (V k,l , K) GL 2 (F ) ⊗ F C ∞ and C ad har (V k,l , K) GL 2 (F ) ≃ −→ St K ⊗ GL 2 (F ) V k,l . (b) The assignment N → C ad har (N, K) GL 2 (F ) defines an exact functor from the category of F [GL 2 (F )]modules of finite F -dimension to the category of H K -modules of finite F -dimension.
Let us mention that in the course of proving the above theorem, we provide an alternative description of the Hecke action in the adelic setting not given [Böc02], and we indicate a proof why the two actions agree. The action here is much simpler than that in [Böc02]. The latter was motivated by translating the Hecke action on (adelic) Drinfeld cusp forms via the residue isomorphism. The remaining part of the proof is based on general cohomological considerations presented in Appendix A. A main observation is that the Steinberg modules St and St K recalled in formulas (2) and (7), respectively, have nice cohomological properties for all congruence subgroups Γ or all compact open subgroups K, and not only Γ that are p ′ -torsion free, or K that are small.
Part (b) of the above theorem turns any composition series of V k,l into a Hecke-stable composition series of S k,l (K). Therefore in Section 2 we shall study in detail the representation theory of F [GL 2 (F )]-representations of finite F -dimension. Building on [Bon11], we classify the irreducible representation of GL 2 (F ) and explain how to algorithmically determine the simple factors of V k,l ; observe that this has no counterpart in characteristic zero, where the analogous GL 2 (Q)-representations are irreducible. Moreover in Proposition 2.6 we provide non-trivial maps between representations V k,l and V k ′ ,l ′ under certain conditions on (k, l) and (k ′ , l ′ ), to which we were led to by [BP09]. The kernels and images of these maps contribute to composition series of the V k,l , but in general do not give a maximal composition series. For arbitrary (k, l), we cannot describe a composition series of the V k,l , not even algorithmically; we can only describe as a quotient its Jordan-Hölder factor of highest weight.
Section 3 explains how the maps between different representations V k,l of Section 2 turn into the hyperderivatives of [BP09] as well as the Frobenius map on Drinfeld cusp forms under the functor and the isomorphism of the above theorem, see Corollary 3.8 and Proposition 3.11. As a byproduct we provide natural extensions of the maps of [BP09] to any ring A and any level subgroup K. Taking the existence of single-cuspidal eigenforms into account, our representation-theoretic approach covers all previously known obstructions to the irreducibility of S k,l (SL 2 (A)) as a Hecke-module.
Part (b) of the above theorem also makes it clear that from a representation theoretic viewpoint it is essential to understand the Hecke-modules C ad har (N, K) GL 2 (F ) for simple N . A first step is to determine their dimension. If K is small, or if Γ has no prime-to-p torsion, such formulas are implicit in [Tei91]; for instance, for any such Γ there exists a constant c Γ > 0 such that dim F C har (V ) Γ = c Γ dim F V for any F [GL 2 (F )]-module V of finite F -dimension. This does not apply, however, to groups like SL 2 (A), for which one should first study analogs of a Maeda conjecture. Therefore in Section 4 we investigate the dimension of C har (L k ) SL 2 (A) for the irreducible SL 2 (F )-modules L k , k ≥ 0, introduced in Section 2, and for A = F q [t]. If q is odd and k is odd, the dimension is zero. Otherwise we expect For q = 2, 3, 5, we present closed formulas for all k, which support this expectation, see Propositions 4.9, 4.10 and 4.11.
Having understood the impact of representation theory, in Section 5 we explore the possibility of there being a Maeda-style conjecture. We take A = F q [t] and Γ = SL 2 (A), and we consider a natural increasing sequence of weights k n for which the GL 2 (F )-representation V kn,l is irreducible (for SL 2 the types are irrelevant). Making use of computer algebra systems, we explicitly determine the characteristic polynomials of the Hecke operator T t on C har (V kn,l ) Γ at the prime (t) ⊂ A, disregarding single-cuspidal Hecke eigenforms. The outcome of our experiments in Table 1 for p = 3 caught us by surprise. In the weights k n searched, we discovered an increasing number (in n) of F -rational Hecke eigenforms of multiplicity one, that seem to be new to the literature. We have conjectural recipes for the number of such forms in weight k n and for the occurring T t -eigenvalues, see Table 2 and Conjecture 5.1. The existence of these forms beyond the range of our data is open. Putting these eigenforms and the single cuspidal eigenforms aside, our data still gives no Maeda type conjecture. In our computations we encountered one or two Hecke orbits; in all cases their associated Galois group over F is a symmetric group, as conjectured in the classical case. But we feel that more computations are needed -which is difficult, since k n grows exponentially. Our search for a conceptual explanation for the existence of these special F -rational eigenforms has failed so far, but we plan to further investigate this question in future work.

Notation and conventions
• F will denote a global function field with constant field F q of cardinality q and characteristic p, and we set e := log p q, so that q = p e .
• We fix a place ∞ of F and define A as the subring of F of functions regular away from ∞. This is the coordinate ring of a smooth affine curve with finite unit group and constant field k.
• We letÂ be the profinite completion lim ←− A/n, the limit being over all non-zero ideals n of A.
• For any non-zero prime ideal p of A, we let A p be the completion of A at p and we let F p be the fraction field of A p , i.e., the completion of F at p.
• We write A ∞ F =Â ⊗ A F for the ring of adeles away from ∞. • The completion of F at ∞ will be F ∞ , its valuation ring O ∞ ⊂ F ∞ , and a uniformizer is π ∈ F ∞ .
• The completion of an algebraic closure of F ∞ is denoted by C ∞ .

Harmonic cochains and the Steinberg module
In this section we recall results on harmonic cochains and the Steinberg module, and we briefly describe the link to Drinfeld cusp forms for GL 2 .

The local theory
Let T be the Bruhat-Tits-tree for PGL 2 for F ∞ . We consider it as a graph with vertex set T 0 and oriented edge set T or 1 . For any oriented edge e, write e * for the edge opposite to e, and write o(e) and t(e) for the origin and the terminus of e. The tree comes with a left-action of PGL 2 (F ∞ ), i.e., we have such an action on T 0 and on T or 1 and the maps o(·) and t(·) are equivariant with respect to these actions. For a precise description of the tree, see [Ser03, II.1].
Let N be an F [GL 2 (F )]-module. Define the space of N -valued harmonic cochains on T as The F -vector space C har (N ) carries an action of GL 2 (F ) by defining for γ ∈ GL 2 (F ) and c ∈ C har (N ) the map γ • c : T or 1 → N, e → γ(c(γ −1 e)).
We will be interested in invariants of C har (N ) under certain subgroups of GL 2 (F ). To define them, let P be a rank 2 projective A-submodule of F 2 , so that we have a group monomorphism Aut A (P ) → GL 2 (F ). For a non-zero ideal n of A define Aut A (P, n) as the subgroup of elements of Aut A (P ) whose induced action on P/nP is trivial.
Definition 1.1. One calls a subgroup Γ of GL 2 (F ) a congruence subgroup if there are P and n as above such that Aut A (P, n) ⊂ Γ ⊂ Aut A (P ).
If N has finite F -dimension, it is shown in [Tei91], building on [Ser03], that for any congruence subgroup Γ of GL 2 (F ) the space C har (N ) Γ of Γ-invariant N -valued harmonic cochains is a finitedimensional F -vector space.
Denote by F 2 the tautological GL 2 (F )-representation and write det for the representation on F via the determinant GL 2 (F ) → GL 1 (F ). For any F [GL 2 (F )]-module N we denote by N * the F -dual Hom F (N, F ); it is again an F [GL 2 (F )]-module. Of central interest to us are the F [GL 2 (F )]-modules (1) Let S k,l (Γ) denote the space of cusp forms of weight k and type l for Γ from [Gek88], for A = F q [t], or from [Böc02], in general. The following result is from [Tei91], though proofs in full generality are only given in [Böc02].
For A = F q [t] the isomorphism above is also one of Hecke modules. As it is more natural to discuss the Hecke action in an adelic context, we shall postpone this for the moment.
The description of Drinfeld cusp forms via harmonic cochains is a first combinatorial description of such forms. Another one is via the Steinberg module, which we recall next. We consider the projective space can be interpreted as the boundary of T , its subset P 1 (F ) is related to the pair (T , Γ). The Steinberg module for GL 2 (F ) is defined as the left Z[GL 2 (F )]-module (2) Recall that a group is called p ′ -torsion free, if all its torsion elements have order a power of p. If Γ is a congruence subgroup of GL 2 (F ) that is p ′ -torsion free, then it is shown in [Ser03, II.     Let us give an explicit description of b Γ from Lemma 1.3 (d), as it will be used in the proof of the following result. 1 As a Z-module, St is free, and a basis is given by the elements (a : 1) − (1 : 0) with a ∈ F . Thus it will suffice to describe b Γ (P ′ − P ) for any pair of distinct points P, P ′ of P 1 (F ). Let ℘ P →P ′ be the geodesic in T from P to P ′ . We think of ℘ P →P ′ as a sequence (e i ) i∈Z of edges such that t(e i−1 ) = o(e i ) for i ∈ Z. The geodesic ℘ P →P ′ is then characterized by requiring that this path has no back-tracking, and that the half lines (e i ) i≤0 and (e i ) i≥0 represent the boundary points P and P ′ , respectively. From [Ser03, II.2.9] one deduces: (3) The sum is finite: there are infinite half lines contained in (e i ) i≤0 and in (e i ) i≥0 , that end in P and P ′ , respectively, on which Stab Γ (e i ) is non-trivial (and growing in |i|) and contained in Stab Γ (P ) or Stab Γ (P ′ ), respectively. The non-vanishing is clear since b Γ is an isomorphism and P ′ − P is non-zero in St.
The following lemma will be used to prove the comparison result Proposition 1.12.
Then the following diagram is commutative: and he verifies that that this map is well-defined and if composed with∂ ] is a projective Z[Γ]-module. One obtains an induced map ι N,Γ : C har (N ) Γ → St ⊗ Γ N . 1 We do assume throughout this paragraph that the reader is familiar with [Ser03, II.2.9]. We implicitly recall b −1 Γ in the second paragraph above Lemma 1.10.
Theorem 1.6 ([Tei91, Prop. 21] 3 ). Suppose that Γ is a p ′ -torsion free congruence subgroup of GL 2 (F ). Then the map ι Γ,N is an F -vector space isomorphism In particular ι Γ,V k,l : We shall remove the constraint that Γ be p ′ -torsion free in Theorem 1.14 and show further that the functor St ⊗ Γ is exact on F [GL 2 (F )]-modules.

The global adelic theory
To define a Hecke action, for general F and ∞ with class number possibly different from 1, we adelize the above situation. We fix a compact open subgroup K ⊂ GL 2 (A ∞ F ). As before, we let N be an Let e ∈ T or 1 , v ∈ T 0 and g ∈ GL 2 (A ∞ F ). We shall write [e, g] K and [v, g] K for respective classes in T or 1,K := T or 1 × GL 2 (A ∞ F )/K and T 0,K := T 0 × GL 2 (A ∞ F )/K. Both T or 1,K and T 0,K carry a left GL 2 (F )operation by γ[s, g] K := [γs, γg] K for s a simplex and γ ∈ GL 2 (F ). The space C ad har (N, K) of harmonic cochains of level K is the set of all maps c : for all e ∈ T or 1 and It is a left F [GL 2 (F )]-module by defining for γ ∈ GL 2 (F ) and c ∈ C ad har (N, K) the map γ • c : T or 1,K → N, [e, g] K → γ(c(γ −1 [e, g] K )).
By strong approximation the determinant map induces a bijection of double cosets The right hand side is a finite extension of the class group of A, and hence a finite group. Hence there is a tuple (g c ) c∈Cl K in GL 2 (A ∞ F ), such that GL 2 (A ∞ F ) is equal to the disjoint union c∈Cl K GL 2 (F )g c K. The groups Γ gc := g c Kg −1 c ∩ GL 2 (F ) are congruence subgroups and one can construct a natural isomorphism between spaces of invariants In particular, the space C ad har (N, K) GL 2 (F ) of GL 2 (F )-invariant N -valued adelic harmonic cochains has finite F -dimension whenever N has finite F -dimension. The Hecke action clearly commutes with the left GL 2 (F )-action and thus preserves C ad har (N, K) GL 2 (F ) . It is also clear that the operator |KyK is independent of the chosen y i . The operators |KyK do in general not preserve the 'local' factors in (6), i.e., the direct summands on the right. Let S k,l (K) denote the space of adelic cusp forms of weight k and type l for K from [Böc02]. It is the natural generalization to the adelic setting of the local definition of S k,l (Γ) used by Goss, Gekeler et al. The following is obtained as in [Böc02, Sects. 5, 6], building on [Tei91] after taking into account the different normalizations regarding the action on V k,l .
which is equivariant for all Hecke operators |KyK for y ∈ GL 2 (A ∞ F ).
The map is given by an explicit residue map which we will recall in Section 3.
Next we recall the adelic Steinberg module. We define it as the kernel in the short exact sequence with deg K = deg ⊗ id GL 2 (A ∞ F )/K . The sequence carries a left Z[GL 2 (F )]-action induced from the left actions of GL 2 (F ) on P 1 (F ) as given before (2) and on GL 2 (A ∞ F )/K by left multiplication. For y ∈ GL 2 (A ∞ F ) and KyK = i y i K from above, we define a Hecke action on St K as follows: first define It is clear that deg K is equivariant for these definitions of |KyK , and that the maps are independent of the chosen representatives y i . Moreover the Hecke operation commutes with the GL 2 (F )-action since one acts from the right and the other from the left. Hence |KyK lies in End Z[GL 2 (F )] (St K ).
The advantage of the above action is that it works for any compact open subgroup K of GL 2 (A ∞ F ). To compare it with [Böc02, Sect. 6], we need a definition.
We refer the reader to Lemma 1.15 for some criteria for K to be small.
The construction of Hecke operators on St K in [Böc02, Sect. 6] was only given for small K. The construction was adapted to the adelization of the isomorphism from Theorem 1.6. To continue, the following result is important. Proposition 1.9. Suppose that K is small. Then for any y ∈ GL 2 (A ∞ F ), the operator |KyK constructed here and the operator |KyK constructed in [Böc02,Sect. 6] agree.
For the proof we first need some preparations similar to [Böc02, § 6.4], that basically amount to an adelic version of Lemma 1.3. Suppose that K is small. A simplex [s, g] K of the union of trees is trivial. Define sets of K-stable simplices T st 0,K and T or,st 1,K and observe that by definition GL 2 (F ) acts freely on these. One has a natural boundary map ∂ , and we put a bar on the induced boundary maps. The Z/(2)-action also preserves Z[T * 1,K ∩ (T 1 × {gK})], and we write Z[T * 1,K ∩ (T 1 × {gK})] for the module of coinvariants. We observe that Z[T st 1,K ] is again a free GL 2 (F )-module. Now for each g ∈ GL 2 (F ) consider the following commutative diagram with exact rows and where the horizontal maps are the natural choices. To verify the properties of the diagram and its well-definedness the reader should convince herself that the following identifications hold. One has T Choose now for each class c ∈ Cl K from (5) a representative by g c ∈ GL 2 (A ∞ F ), and choose representatives for the sets GL 2 (F )/Γ gc , c ∈ Cl K . Taking the direct sum c∈Cl K γ∈GL 2 (F )/Γg c over the above diagrams with g being replaced by γg c , we obtain the above diagram with the expressions ∩(T 1 × {gK}) removed in the top row and with ∩(T 0 × {gK}) removed in the bottom row, i.e., the diagram in the adelic setting! From Lemma 1.3 and GL 2 (F )/Γ gc ∼ = GL 2 (F )\GL 2 (F )g c K/K, we deduce Let us also describe b K . Let P, P ′ be distinct elements of P 1 (F ) and g ∈ GL 2 (A ∞ F ). Then as indicated below Lemma 1.3 in the local situation, there is a geodesic path ℘ P →P ′ ,gK from (P, gK) to (P ′ , gK). Let ([e i , g] K ) i∈Z denote the sequence of successive edges forming ℘ P →P ′ ,gK . Then (3) interpreted in the adelic situation gives Proof of Proposition 1.9. The Hecke action introduced here on St K is defined via compatible actions on the middle and right terms of the complex (7). The Hecke action in [Böc02,Sect. 6] is defined via a compatible action on the middle and right term of the complex in Lemma 1.10(a). The key for the comparison is the isomorphism b K . Note that the elements (P ′ , gK) − (P, gK) generate St K over Z, when gK traverses GL 2 (A ∞ F )/K and P, P ′ traverse all distinct pairs of points of P 1 (F ). Fix gK and distinct points P ′ , P from P 1 (F ), and represent ℘ P →P ′ ,gK by its sequence of successive edges ([e i , g] K ) i∈Z . For a stable edge [e, g] K , in [Böc02, § 6] its course crs([e, g] K ), a geodesic path of edges, is defined as follows: crs([e, g] K ) always contains the stable edge [e, g] K . If [t(e), g] K is stable, the path ends at [t(e), g] K ; otherwise there is a unique boundary point (Q ′ , gK) such that gK)), and in that case crs([e, g] K ) contains the infinite geodesic path from (Q, gK) to [o(e), g] K . An important observation concerning ℘ P →P ′ ,gK is that in the formal completion of Z[T 1,K ], one has the equality (of infinite sums) On the left, there are only finitely many i for which the inner sum is non-zero; there are always inner sums that are infinite, and so we do need to consider this in the formal completion. Also, there may be infinite cancellations on the left; such a cancellation occurs if [e i , g] K and [e i ′ , g] K are stable for some , g] K and all edges in between are unstable, because then crs([e i , g] K ) and crs([ here the right hand side is finite, because the sum is over stable edges only. It follows for the Hecke action as defined in [Böc02, § 6] applied to b K ((P ′ , gK) − (P, gK)) The Hecke action defined here on St K = Ker(deg K ) is given directly by To see that the actions agree, we need to show that b K ((P ′ , gy j K) − (P, gy j K)) = i∈Z [e i , gy j ] st K for all j ∈ J. The point is that the geodesic ℘ P →P ′ ,gy j K is the sequence of successive edges ([e i , gy j ] K ) i∈Z , and altering the label gK into gy j K maps a geodesic to a geodesic. This completes the proof.
One can now generalize the map ι Γ,N from (4) to the adelic setting. This gives the F -linear homomorphism ι K,N : for any F [GL 2 (F )]-module N of finite F -dimension. Because of Proposition 1.9 we can quote from [Böc02, § 6] the following result where the Hecke action on St K is the one defined here.
We shall remove the requirement that K is small in Corollary 1.21 and also show that the functor 1.3 Extensions of Theorem 1.6 and Theorem 1.11 In this subsection we extend Theorem 1.6 and Theorem 1.11 in the way promised after their formulation using some more general cohomological results proved in Appendix A. One motivation is that it should be clarified that the above results hold for all congruence subgroups Γ of GL 2 (F ) or all compact open subgroups K of GL 2 (A ∞ F ), respectively. Another is that the comparison to [BP08] in Section 3, as well as our computations for A = F q [t] in Section 5 towards a Maeda type conjecture are for Γ ∈ {SL 2 (A), GL 2 (A)} which are not p ′ -torsion free.
Let first Γ be a congruence subgroup of GL 2 (F ), say Aut A (P, n) ⊂ Γ ⊂ Aut A (P ) for some projective rank 2 A-submodule P of F 2 and some non-zero ideal n of A. Let p be any non-zero prime ideal of A. Then the homomorphism Aut . Let Γ p denote the intersection of this kernel with Γ; then clearly Γ p is p ′ -torsion free. By construction, it is normal in Γ, and since Aut A (P, np) ⊂ Γ p , it is a congruence subgroup.

Fix p as above and let
There is an obvious action ofΓ := Γ/Γ p on C har (N ) Γp and an action on for m ∈ St, n ∈ N and γ ∈ Γ. One verifies that ι Γp,N induces an isomorphism C har (N ) Γ → (St⊗ Γp N )Γ, using the explicit formula (4). The norm map considered in Proposition A.1 (b) gives an isomorphism Proof. We need to show that the following diagram commutes: Going up and right gives ] ⊗ Γp N , it remains invariant if we apply to it p 1,Γp⊂Γ ⊗ id N with p 1,Γp⊂Γ from Lemma 1.5. Hence Observe that in the summation index, we have Γ p -orbits of Γ-stable elements. If we follow the bottom path, then c → Now observe that eγ −1 = γe, that γc(e) = c(γe) and that Γ = ∪γ ∈Γ γΓ p . It follows that the last expression is equal to (11), which shows that the diagram commutes.
Remark 1.13. In the above construction ofι Γ,N , we can replace p by any proper ideal m ⊂ A that is non-zero, and work with Γ m instead of Γ p , and it is straightforward to observe that Proposition 1.12 also holds in that generality. It then follows in particular thatι Γ,N is independent of any choice, and that for p ′ -torsion free Γ the map is the one defined by Teitelbaum.
Thus from now on we simply write ι Γ,N instead ofι Γ,N for all congruence subgroups Γ of GL 2 (F ).
Next we state an immediate consequence of Proposition A.1(c): Theorem 1.14. Let Γ ⊂ GL 2 (F ) be a congruence subgroup, let A be a ring of characteristic p > 0 and let N be an A[Γ]-module. Then the following hold: We now pass to the adelic situation and use the notation from A.2. We denote by K ′′ ⊂ K ′ ⊂ K compact open subgroups of GL 2 (A ∞ F ) such that K ′′ is normal in K. We consider G := GL 2 (F ) as a subgroup of GL 2 (A ∞ F ) via the diagonal embedding F ֒→ A ∞ F . Thereby S := GL 2 (A ∞ F )/K ′′ carries a left action by G; it also carries an obvious right action by H := K/K ′′ ; a short computation reveals that S is free as a right H-set, so that condition (31) in Subsection A.2 is satisfied. Let M := St be the Steinberg module. Then , the action is given by for (γ, h) ∈ GL 2 (F )×K/K ′′ . Note also that for any g ∈ GL 2 (A ∞ F ) the group Stab G (gK ′′ ) = G∩gK ′′ g −1 is a congruence subgroup of GL 2 (F ).
We remind the reader that for any K there exists a free A-submodule Λ of (A ∞ F ) 2 of rank 2 that is invariant under the action of K, so that K ⊂ Aut A (Λ); this uses that for any free A-submodule of rank 2 of (A ∞ F ) 2 , the stabilizer under the action of K is open in K. We choose any such Λ for the following lemma. The lemma gathers result on smallness of K and on smallness of K ′ /K ′′ for St K ′′ . Lemma 1.15. For any proper non-zero ideal n ⊂ A, define K(n) := Ker(Aut A (Λ) → Aut A/n (Λ/nΛ)). We assume that K ′′ = K ∩ K(n) for some such n. Then the following hold.
(a) The group K(n) is small.
Proof. The proof of (a) and (b) is a standard argument: Let g ∈ GL 2 (A ∞ F ). Using the first isomorphism theorem in group theory, one sees that gK ′′ g −1 ∩ GL 2 (F ) ⊂ gK ′ g −1 ∩ GL 2 (F ) is a normal subgroup of finite p-power index. Hence to prove (b), it suffices to prove that K ′′ is small. Since smallness is inherited by compact open subgroups, it suffices to prove (a), and only in the case when n is a maximal ideal of A. Denote by g n the component of g at n and by K n the component of K(n) at n. Then K n is a compact open pro-p subgroup of GL 2 (F n ) and so is g n K n (g n ) −1 . Moreover we have GL 2 (F ) ∩ gK(n)g −1 ⊂ GL 2 (F ) ∩ g n K n (g n ) −1 as a subgroup of GL 2 (F ). Since pro-p groups do not contain elements of finite order prime to p, part (a) follows.
In light of (b), to prove (c) it suffices to show that if K ′ is small, then We do have a norm map for the adelic Steinberg module relative to K ′′ ✂ K: Proof. One deduces from (12) that the map is one of Z[G]-modules. To see the bijectivity, by (7) it is enough to see that is bijective. But this is clear, since the target space is constant on H orbits and because H = K/K ′′ .
From Lemma 1.15, the remarks preceding it and Propositions A.3 and A.4, we conclude.
Then the following hold:

modules is functorial, and the sequence in (c) is one of H K -modules.
Let n ⊂ A be any proper non-zero ideal and let K ′′ := K(n)∩K. Let N be a F [GL 2 (F )]-module of finite F -dimension. Using (9) one verifies that the isomorphism ι K ′′ ,N : C ad har (N, K ′′ ) GL 2 (F ) → St K ′′ ⊗ GL 2 (F ) N from Theorem 1.11 is equivariant for the action of H = K/K ′′ , and it therefore induces an isomorphism given by the norm map, and together we obtain an isomorphism The proposition immediately implies thatι K,N is independent of any choices. After having given its proof, we shall drop the tilde from the notation.
The proof of Proposition 1.18 requires the following analog of Lemma 1.5: g] K is stable, and let it be zero otherwise. Then the following hold.
Then the following diagram commutes is clear by definition. In the other two cases, it follows from where in the middle step we use that K ′′ is normal in K. Part (b) is clear from In part (c), the commutativity of the diagrams formed by the first two and by the last two rows is trivial. The proof of the commutativity for the diagram formed by rows 2 and 3 is analogous to that of Lemma 1.5. The exactness of the second row uses the freeness of the right H-action on St K ′′ . It follows from the explicit expressions for b K ′′ and b K-st K ′′ , and from our definitions.
Proof of Proposition 1.18. The proof is analogous to the proof of Proposition 1.12. Let c be in We need to show that Because . (15) The right hand side lies in Z[T and also in the kernel of∂ K-st K ′′ ⊗ id N ; cf. the third row of the diagram in Lemma 1.19 (c). Moreover p 1,K ′′ ⊂K maps the right expression in (14) to the right expression in (15). To conclude, we use that the diagram formed by rows 2 and 3 in Lemma 1.19 (c) commutes, and that We need one more preparatory result. For this, let Supp y : in the second definition we embed GL 2 (F p ) into GL 2 (A ∞ F ) via the component at p. Both sets are finite. One has the following lemma: For an open normal subgroup K ′′ ⊂ K let ι K,K ′′ : X K → X K ′′ be the canonical inclusion, where in the last case we take ι K,K ′′ = ν K,K ′′ ,N from Corollary 1.17 (b). Let |KyK in End(X K ) be the Hecke operator attached to some y ∈ GL 2 (A ∞ F ). Then for any non-zero ideal n ⊂ A such that (Supp y ∪ Supp K) ∩ Supp n = ∅ and with K ′′ := K ∩ K(n) as in Lemma 1.15, the following diagram commutes Proof. Set S := Supp y ∪ Supp K and F S := p∈S F p , and let pr S : GL 2 (A ∞ F ) → GL 2 (F S ) denote the canonical projection. Define K S := pr S (K) and y S := pr S (y), and let H := p∈Max A S GL 2 (A p ) and H(n) := {h ∈ H | h ≡ 1 (mod n) in GL 2 (A/n)}; we regard K S , H and H(n) as subgroups of GL 2 (A ∞ F ) by extending tuples by 1 at the missing components. We first show that Because K is compact open for the adelic topology, there exists a finite set S ′ ⊂ Max(A) such that K ⊃ p∈Max A S ′ GL 2 (A p ), and by enlarging S ′ , we may assume S ′ ⊃ S. Let p ∈ S ′ S. Then, by the definition of Supp K and because S ⊃ Supp K, we have K ⊃ GL 2 (A p ). Thus, by taking a finite product of subgroups, it follows that Because H is maximal compact in p∈Max A S GL 2 (F p ), any y ′ ∈ K can be written as a product y ′ = y ′ S · h with y ′ ∈ K S and h ∈ H. By using that in fact H ⊂ K, we obtain (16). Intersecting (16) with K(n) also gives K ′′ = K S × H(n); we refer to this formula as (16 ′′ ).
Choose now y S,i ∈ GL 2 (F S ) such that K S y S K S = i y S,i K S , using that the left hand side is compact and right translation invariant under the compact open subgroup K S ⊂ GL 2 (F S ). For g ∈ GL 2 (A ∞ F ), in the following we also write g = (g S , g S ) with g S = pr S (g) and g S the tuple of those components of g with index set Max A S; we use this notation also for subsets whenever this is permitted.
Define y i ∈ GL 2 (A ∞ F ) as the extension of y S,i given by y i = (y S,i , y S ). Then we have because y S ∈ H and H(n) is normal in H. The analogous computation for K in place of K ′′ gives Ky ′′ K = i y i K. This shows that the Hecke operators |K ′′ y ′′ K ′′ and |Ky ′′ K can be defined by the same expressions and now the commutativity is clear.
Proof. Part (b) follows from part (a) and from Corollary 1.17 (c) and (d). To prove (a), let y ∈ GL 2 (A ∞ F ). Choose a non-zero proper ideal n ⊂ A such that (Supp y ∪ Supp K) ∩ Supp n = ∅. Consider the commutative diagram By Lemma 1.20 the vertical maps are equivariant for the Hecke operator | KyK . By Theorem 1.11 the lower horizontal map has the same property. Since the vertical maps are inclusions and the horizontal maps are isomorphisms, the result follows.
Remark 1.22. Suppose that Cl K is the trivial group (which for instance happens for A = F q [t] and K maximal compact). Let Γ := GL 2 (F ) ∩ K. Then one can define meaningful Hecke actions directly on C har (V k,l ) Γ and on St ⊗ Γ V k,l , so that ι Γ,V k,l becomes a Hecke equivariant isomorphism; see [Gek88, § 6] for the case S k,l (Γ), or [Böc02, Ch. 6] (note that the actions are normalized differently, which we will discuss further at the end of Subsection 3.2). In this particular case, there is no gain of the adelic over the local situation. The Hecke actions agree and proofs from either perspective are of a similar difficulty. However, we should point out that, by definition, the Hecke action on St K ⊗ GL 2 (F ) V k,l is induced from an action on St K . In the local situation this is not true anymore: The Hecke action involves the module V k,l directly. Thus to show the Hecke equivariance for morphisms under the functor in Theorem 1.14(b) the full GL 2 (F )-action on the coefficients needs to be considered.
2 Representations of SL 2 (F ) and GL 2 (F ) By the results of the previous section, in particular Corollary 1.21, we can use representation theory to gain insights into the Hecke-module structure of spaces of Drinfeld cusp forms. In this section, we first develop the relevant representation theory of SL 2 (F ) and GL 2 (F ) building heavily on [Bon11]. Afterwards, we introduce certain explicit maps between symmetric powers which will have very natural interpretations when passing to Drinfeld cusp forms in Section 3. Finally, we also briefly mention representations of the finite groups SL 2 (F q ) and GL 2 (F q ) which will become relevant in Section 4.

Symmetric powers and irreducible representations
This subsection is mostly standard material. For an excellent reference in the case of SL 2 we refer to [Bon11]. Denote by F [X, Y ] k the subspace of homogeneous polynomials of degree k in F [X, Y ]. We let the group GL 2 (F ) act on F [X, Y ] by This action clearly preserves the homogeneous components F [X, Y ] k . If one identifies F [X, Y ] 1 with the dual vector space of F 2 in the obvious way, then Sym k ((F 2 ) * ) is naturally isomorphic to F [X, Y ] k as a GL 2 (F )-module. For shorter notation, we often abbreviate ∆ k := Sym k ((F 2 ) * ). For any k ≥ 0, we define L k ⊂ ∆ k as the F -linear span of {X i Y k−i | k i ≡ 0 (mod p)}. In Lemma 2.1 we shall show that L k is an irreducible subrepresentation of ∆ k .
To see that L k is a subrepresentation of ∆ k , and to introduce the Frobenius twist of a representation, note that τ : is a group homomorphism. Given any representation V of GL 2 (F ) on an F -vector space, it follows that it s-fold Frobenius twist V (s) , defined as the vector space V together with the action By a similar computation one also verifies the following: Write k = k 0 + k 1 p + . . . + k s p s in its base p expansion, with 0 ≤ k i ≤ p − 1, where we insist that k s = 0 -we also use the notation k = (k s . . . k 1 k 0 ) p to denote the base p-expansion of k. Then one has a monomorphism of GL 2 (F )-representations with image L k . This is a form of the Steinberg Tensor Theorem specialized to the group GL 2 ; for a direct proof see [Pel17,Lemma 8]. It follows immediately that In the sequel, for m ∈ Z we denote by det m the action of GL 2 (F ) Let us recall results on (highest) weights for GL 2 . Denote by T (F ) the subtorus a 0 0 d | a, d ∈ F × of GL 2 (F ). It is commutative, and so the restriction of any F -linear algebraic representation V of GL 2 (F ) to T (F ) decomposes as a direct sum V = ⊕ (n,m)∈Z 2 V (n, m) of F -subvector spaces where a 0 0 d acts on v ∈ V (n, m) as a n d m . Only finitely many V (n, m) are non-zero, and the corresponding pairs (n, m) are called the weights of V . From 0 1 1 0 ∈ GL 2 (F ) one deduces that with (n, m) also (m, n) is a weight. A weight (n, m) is called dominant if n ≥ m. We define a partial order on the weights of V as follows: which is easily verified. The defining formula also shows that m → m i extends to a uniformly continuous function · i : Z p → Z p for the p-adic topology. Moreover one has Lucas' Theorem. Proof. For m ∈ N 0 this is classical, and can be found for instance in [Gra97]. For m ∈ Z p , the formula follows from the uniform continuity of · i and the congruence for all m ∈ N 0 .
Proof of Lemma 2.1: We first prove (a) and (b). From the definition of L k it is straightforward to see that its weights are the pairs (−i, i − k) with 0 ≤ i ≤ k and k i ≡ 0 (mod p), and for these i each L k (−i, k − i) has dimension 1. In particular, L k has highest weight (0, −k). It is shown in [Bon11, Thm. 10.1.8. (b)] that the remaining assertions are true for the restrictions of L k and ∆ k to SL 2 (F ). Since we know that L k is a subrepresentation of ∆ k , knowing (a) and (b) for SL 2 implies the analogous assertions for GL 2 .
Part (c) follows from (a) and the classification of irreducible representations in terms of highest weights stated above. For (d) note that the dual of an irreducible representation is again irreducible, since the canonical bidual-map is an isomorphism, and thus L * k is irreducible. To determine its weights, let ξ i be the unique element of L * k that is 1 on X i Y k−i and zero on X i ′ Y k−i ′ for i ′ = i. These elements form a basis of L * k and Twisting with det −k preserves the irreducibility, but changes the weight (i, k − i) to (i − k, −i). Hence L * k ⊗ det −k has the same weights as L k and is thus isomorphic to it because both are irreducible. Part (e) is immediate from Proposition 2.2 and the definition of L k . One can also use (17) and a dimension count.
For the algebraic group G ∈ {SL 2 , GL 2 } let K 0 (G) denote the Grothendieck ring of algebraic representations of G(F ) on finite-dimensional F -vector spaces. The multiplication of the ring is given by the tensor product of representations. If G = SL 2 , then as an additive group K 0 (SL 2 ) is the free module on the symbols L k , k ≥ 0 (we regard L k as a module for SL 2 ⊂ GL 2 ). By [Bon11, (10.1.14)] ∆ k , k ≥ 0, is also a Z-basis of K 0 (SL 2 ). If G = GL 2 , then from Lemma 2.1 it easily follows that the L k ⊗ det m , k, m ∈ Z, k ≥ 0 form a Z-basis of K 0 (GL 2 ), and below we shall deduce rapidly from Bonnafé's results that the symbols ∆ k ⊗ det m , k ≥ 0, m ∈ Z also form a Z-basis of K 0 (GL 2 ). It follows from Lemma 2.1(a) that when expressing ∆ k ⊗ det m in K 0 (GL 2 ) as a linear combination in the L k ′ ⊗ det m ′ then only multiplicities 0 and 1 can occur, that L k ⊗ det m occurs with multiplicity 1, and that L k ′ ⊗ det m ′ cannot occur in ∆ k ⊗ det m if k ′ > k.
To give a recursive formulas for the coefficients in {0, 1} that occur when expressing ∆ k ⊗ det m in K 0 (GL 2 ) as a linear combination in the L k ′ ⊗ det m ′ , we follow [Bon11, Subsec. 10.1.3]: Define recursively sets E(k), k ≥ 0 as follows, where by k 0 we denote the lowest digit of k in its base p expansion, i.e., the residue of k by division by p: Corollary 2.4. In K 0 (GL 2 ) for k ≥ 0 and m ∈ Z one has

Proof.
To prove the result, by twisting with powers of det we may clearly assume m = 0. By the basis property of the L k ⊗ det m , given k, there exist unique constants e k ′ ,m ′ ∈ N 0 such that in K 0 (GL 2 ) we have This equality can be restricted to K 0 (SL 2 ). It follows from Proposition 2.3 that Hence for each k ′ ∈ k − 2E(k) there is a unique m ′ for which e k ′ ,m ′ is non-zero, and equal to 1, and for all other k ′ , all e k ′ ,m ′ are zero. It remains to determine the unique m ′ for the former k ′ . For this note that a 0 0 a , a ∈ F × acts on ∆ k ′ ⊗ det m ′ as a k ′ +2m ′ and on ∆ k as a k . This implies the claim on m ′ in the corollary, and we are done.

Hyperderivatives on symmetric powers
Motivated by results of Bosser-Pellarin on certain hyperderivatives between certain spaces of Drinfeld modular forms, we were seeking a representation theoretic description of these maps in via the residue map. This is how we arrived at the definitions in this section. The detailed relation to the work of Bosser-Pellarin will be explained Section 3.
We need the following result.   Proof. Part (a) is an immediate consequence of the last assertion of Proposition 2.2.
In (b), the equivalence of (i) and (ii) is immediate from Lucas' theorem Proposition 2.2 applied with m = k + s − 1. Next we prove (ii) ⇒ (iii). By (ii), the number k + s − 1 is divisible by p r , and (iii) follows from Suppose conversely that ( Assume that 0 < m < p r . Let m = m 0 + m 1 p + . . . + m r−1 p r−1 and s = s 0 + s 1 p + . . . + s r−1 p r−1 and define i t = max{0, s t − m t } for t = 0, . . . , r − 1, and i = i 0 + i 1 p + . . . + i r−1 p r−1 . Because m > 0 we have 0 ≤ i ≤ s − 1 and by Lucas' theorem we deduce i+m s ≡ 0 (mod p) while i s ≡ 0 (mod p), which is a contradiction. Hence we must have m = 0 and hence (ii) holds.
In particular, the F -linear hull W of B in ∆ k−2+2s ⊗ det m+s is stable under GL 2 (F ).

Proof.
To see (a) observe that if i < s, then i s = 0 (mod p), and thus D s X i Y j = 0 for i < s. Moreover if j < s, then i s = (−1) s j s = 0 (mod p) by Lemma 2.5, and it follows that D s X i Y j = 0 also for j < s.
Next let γ = 1 b 0 1 . Then the assertion follows from Finally the case γ = 1 0 c 1 is completely analogous to the previous case using Lemma 2.5 (b). Regarding (c), note first that B lies clearly in the kernel of D s . However it is also obvious that set This completes the proof of (c). Finally, the last sentence is immediate from (c).
Remark 2.7. While our main motivation to study these particular maps is tied to the results in Section 3, they are already of purely representation-theoretic interest. Note however that the image of D s can very well be (and is) 0 in many cases (for example when the source is already irreducible as a GL 2 (F )-representation.
Remark 2.8. For a fixed k ≥ 0, let B k denote the intersection of all kernels of the hyperderivatives with source ∆ k . By Lemma 2.1 (b) we have that L k ⊂ B k once B k is non-trivial. It is a natural question to ask if equality holds. The following example shows that this is in general not the case: Let q = 3 and k = 40. Then, there are hyperderivatives only for s 1 = 2, s 2 = 5 and s 3 = 14. Now, X 16 Y 24 is in B 40 , since 16 2 ≡ 0 (mod 3), 16 5 ≡ 0 (mod 3) and 16 14 ≡ 0 (mod 3) by Proposition 2.2. However, since 40 16 ≡ 0 (mod 3), X 16 Y 24 is not in L 40 .

The Cartier operator
In the spirt of the previous subsection, we were looking for a representation theoretic description of the Frobenius operator on Drinfeld modular forms. Via the residue map we arrived at the following construction. Since contrary to the previous case, our new map is only defined over a perfect field, we work with C ∞ for simplicity of notation, and we identify ∆ k ⊗ F C ∞ with C ∞ [X, Y ] k . Let σ denote the Frobenius of the perfect field C ∞ . Then, for any C ∞ -vector space M , we denote by σ * M the C ∞ -vector space with underlying abelian group M and scalar multiplication given by Proposition 2.9. Let k ≥ 2 and m ∈ Z. The map C p : is GL 2 (F )-equivariant and surjective.
Proof. Firstly, the map is C ∞ -linear by construction and clearly surjective. We proceed as in the proof of Proposition 2.6 and consider matrices γ ∈ GL 2 (F ) of the form (i) 1 b 0 1 , b ∈ F , (ii) 1 0 c 1 , c ∈ F , and (iii) diagonal matrices a 0 0 1 , a ∈ F × .
We begin with (iii) and so γ = a 0 0 1 . Here, we only need to consider the case p | i (and so p | j).
In this sum, only the terms with i − n = 0 (mod p) can possibly be non-zero by definition of C p . Let Then by Lucas's theorem We obtain The case γ = 1 0 c 1 follows in a completely similar way, thus completing the proof.
Using the notation from Section 1, we apply Proposition 2.9 in the case m = l and Lemma 2.12 to M = (∆ pk−2 ⊗ det pl−1 ) ⊗ F C ∞ to obtain an injective GL 2 (F )-equivariant map We will relate this map to the Frobenius map on Drinfeld modular forms in Section 3.
2.4 Irreducible representations of SL 2 (F q ) and GL 2 (F q ) We conclude the section with the classification of the irreducible representations of F q [SL 2 (F q )] and F q [GL 2 (F q )] following Bonnafé. This will be relevant in Section 4. Define L k as the F q -span in L k of the monomials X i Y k−i , 0 ≤ i ≤ k, k i ≡ 0 (mod p) and define ∆ k in the same manner. Observe that by definition, L k = ∆ k for 0 ≤ k ≤ p − 1. We write det m for the representation Corollary 2.14. Up to isomorphism, the irreducible representations of the ring Proof. We shall use the following result that can be found in [Wei03, §7, Thm. 1.9]: Let G be a finite group. Then the number of non-isomorphic irreducible representations of G over F p is equal to the number of p-regular conjugacy classes of G.
By the theory of the rational canonical form, any semisimple element is either (i) scalar, or conjugate to a unique companion matrix of the form 0 b is (ii) irreducible or (iii) a product of two distinct linear factors. The number of classes in (i) is q − 1, that in (iii) is 1/2(q − 1)(q − 2), that in (ii) is 1/2 deg(x q 2 − x)/(x q − x) = 1/2(q 2 − q). Therefore the number of semisimple (i.e., p-regular) conjugacy classes of GL 2 (F q ) is By Proposition 2.13 the representations L(k, m) := L k ⊗ Fq det m ⊗ Fq F p are irreducible, and their number is q(q − 1). Hence it suffices to show that they are pairwise non-isomorphic. So suppose there is an isomorphism of GL 2 (F q )-representations ϕ : L(k, m) → L(k ′ , m ′ ). By Proposition 2.13 we have k = k ′ . Thus, we can regard ϕ as an automorphism of the underlying F p -vector space; in particular it has a non-zero eigenvalue λ ∈ F p . Let µ λ : L(k, m) → L(k, m ′ ) denote the multiplication by λ. Clearly, µ λ is SL 2 (F q )-equivariant. Thus, ϕ − µ λ is a morphism of SL 2 (F q )-representations with non-zero kernel, hence by Proposition 2.13, ϕ = µ λ . Now, we consider the elements g = ξ 0 0 1 , ξ ∈ F × q . We obtain λξ m−k X k = ϕ(g · X k ) = g · ϕ(X k ) = λξ m ′ −k X k .

Hecke-equivariant maps between spaces of Drinfeld cusp forms
In this section we apply the machinery developed in Section 1 to the maps constructed in Subsection 2.2 and Subsection 2.3. As a result, we obtain Hecke equivariant maps betweens spaces of (adelic) Drinfeld cusp forms of different weights. More generally, in Proposition 3.5 we see that any filtration of the GL 2 (F )-module V k,l gives rise to a Hecke-stable filtration on Drinfeld cusp forms of weight k and type l. Afterwards, we show that our constructions coincide with previous work of Bosser-Pellarin and the well-known Frobenius map in the special case A = F q [t]. We conclude the section with some computational examples.

Maps induced from representation theory
Recall that in Proposition 2.6 we constructed certain hyperderivative maps on symmetric powers: Let k ≥ 2, s ≥ 1, m ∈ Z and suppose that k+s−1 i = 0 (mod p) for i = 1, . . . , s. Then there is an explicit Upon noting that V k,l = (∆ k−2 ⊗det l−1 ) * , we obtain (by duality) a GL 2 (F )-equivariant map D * s : V k,l → V k+2s,l+s . Now, let K be any compact open subgroup of GL 2 (A ∞ F ). We can apply Corollary 1.17(d) to obtain a Hecke-equivariant map Finally, by invoking the isomorphisms in Corollary 1.21 and Theorem 1.7, we obtain the Heckeequivariant map D * s : S k,l (K) → S k+2s,l+s (K). In particular, D * s (S k,l (K)) ⊂ S k+2s,l+s (K) is a Hecke-stable subspace. We obtain the following consequence of Proposition 2.6.
Corollary 3.1. Suppose that k+s−1 i = 0 (mod p) for i = 1, . . . , s and let W be as in Proposition 2.6. Then, we have Remark 3.2. We directly obtain that for all s ≥ 1 such that D s is defined and such that k + 2s − 2 satisfies the condition on its base p-expansion in Lemma 2.1(e), we must have D * s (St K ⊗ GL 2 (F ) V k,l ) = {0}. Indeed, for such weights ∆ k+2s−2 = L k+2s−2 is irreducible as a GL 2 (F )-module. So, suppose that there is an s ≥ 1 such that D s is defined. Since the target space will always have smaller dimension, D s must have a kernel, and as ∆ k+2s−2 = L k+2s−2 is irreducible, we deduce that D s is the zero map. Thus, W ⊥ = {0}, and hence, by Corollary 3.1, At this point it is natural to ask how many Hecke-stable subspaces arise via hyperderivatives, i.e. how many hyperderivatives are defined for a fixed target space.
is a Hecke-stable subspace of S k,l (K). Furthermore, these are all of the Hecke-stable subspaces of S k,l (K) arising from the hyperderivatives defined in Proposition 2.6.
Proof. The Hecke-stability is immediate from the above. We only need to check that D s j is defined and that these are all possible choices of s. By Lemma 2.5, we have k−1−s i ≡ 0 (mod p) for all 1 ≤ i ≤ s if and only if s is of the form s = s j := j i=0 k i p i , for some 0 ≤ j ≤ r − 1, in particular, this shows that there are at most ⌊log p (k − 1)⌋ possible such s.
Remark 3.4. It follows immediately from Proposition 3.3 that for weights of the form k = p n + 1 − cp n−1 , with 0 ≤ c < p − 1 and n ≥ 1, there are no nonzero s such that D * s S k−2s,l−s (K) ⊂ S k,l (K). Thus, there are no obstructions to the irreducibility of S k,l (K) as a Hecke module coming from the hyperderivatives considered in this paper. We will investigate a Maeda-type conjecture for Drinfeld cusp forms of weights of this form in Section 5.
We point out that for k = p n + 1 − cp n−1 the corresponding V k,l = (∆ k−2 ⊗ det l+1−k ) * is irreducible as k − 2 = p n − 1 − cp n−1 is a magic number and thus ∆ k−2 = L k−2 ; see Lemma 2.1(e). So we can even generalize the previous remark to the statement that there are no obstructions to the ireducibility of the Hecke module S k,l (K) coming from representation theory.
We can also apply the above technique to the Cartier operator constructed in Subsection 2.3: Recall that we constructed a GL 2 (F )-equivariant injective map Again, we apply Corollary 1.17 to obtain the injective Hecke-equivariant map Thus, by invoking the isomorphisms in Corollary 1.21 and Theorem 1.7 we obtain a Hecke-equivariant map α p : S k,l (K) → σ * S pk,pl (K).
While the above maps are so far the only explicit source of Hecke-stable subspaces, we are able to state the following more general result.
(a) Any filtration of the GL 2 (F )-module V k,l induces Hecke-stable filtrations of C ad har (V k,l , K) GL 2 (F ) and S k,l (K).

then any
Hecke-eigensystem in S k,l (K) corresponding to M also appears in S k ′ ,l ′ (K) Proof. Both parts are immediate from Corollary 1.21 (b).

Hyperderivatives and the work of Bosser-Pellarin
From here on we assume A = F q [t]. Denote by Ω the Drinfeld symmetric space P 1 (C ∞ ) P 1 (F ∞ ) with its natural structure as a rigid space, e.g. [Gek88,§ 5]. Let f : Ω → C ∞ be a locally analytic function and z ∈ Ω. Then the hyperderivatives D s f at z are defined by the formula (D s f )(z)ε s for ε ∈ C ∞ with |ε| sufficiently small; see [US98,Def. 2.3] or [BP08, § 3.1].
Here are some formal properties of hyperderivatives: (a) The functions D s f are locally analytic on Ω, and rigid analytic, if f is so.
(b) If f has a Laurent series expansion f = n∈Z a n (z−b) n converging on an annulus r < |z−b| < R, then D s f has the Laurent series expansion Lemma 3.6. Consider the annulus A := {z ∈ C ∞ | r < |z − c| < R} for c ∈ C ∞ and 0 < r < R rational numbers. For rigid analytic functions f, g : A → C ∞ one has Proof. By explicit computation: Write f = n∈Z a n (z − c) n and g = n∈Z b n (z − c) n . Then by (b) above we have The map D s preserves the subspaces of cusp forms.
Note that for A = F q [t] and K = GL 2 (Â), we have that Cl K is the trivial group and C ad har (N, K) GL 2 (F ) ∼ = C har (N ) Γ for Γ = GL 2 (A). Thus, upon observing that this isomorphism is functorial in N , we can reduce the results of the previous subsection to the local situation.
Denote the residue map S k,l (Γ) → C har (V k,l ) Γ by f → c k,l (f ), and recall that c k,l is defined by (c k,l (f ))(e)(X i Y j ) = Res e f z i dz := Res Ae f z i dz, where e is any oriented edge of the Bruhat-Tits tree, A e denotes the associated annulus, see [Tei91,Preliminaries], and i + j = k − 2 to the Hecke-operators: If f ∈ S k,l (Γ) is a Hecke eigenform with Hecke eigensystem (a p ) p , then the form τ p (f ) is again a Hecke eigenform with eigensystem (a p p ) p . If we denote the Frobenius morphism on the perfect field C ∞ by σ, we can reformulate the above to the following statement: Proposition 3.10. The map τ p : S k,l (Γ) → σ * S pk,pl (Γ) is Hecke-equivariant.
It what follows, we give a representation theoretic explanation for this phenomenon using the map α p constructed in (20) in Subsection 2.3.
Proof. Assume that the annulus corresponding to e is given by {z ∈ C ∞ | r < |z − c| < R} for c ∈ C ∞ and 0 < r < R. Write f = n∈Z a n (z − c) n . Then τ p (f ) = n∈Z a p n (z − c) pn . Thus, if we write On the other hand, we have c pk,pl (τ p (f ))(e)(X i−1 Y j−1 ) = Res e (f p z i−1 dz) Since pn − 1 = p(n − 1) + p − 1, we see that i−1 pn−1 = 0 if p ∤ i by Proposition 2.2, which shows that in this case c pk,pl (τ p (f ))(e)(X i−1 Y j−1 ) = 0.
Assume now that p | i. Then we have, again by Proposition 2.2, and thus, which completes the proof; note that binomials mod p lie in F p and are thus fixed by σ.
Thus, the Hecke-equivariant map constructed in Subsection 3.1, α p : S k,l (K) → σ * S pk,pl (K), is just τ p when switching to the local situation. As the final step to obtain a representation theoretic proof for Proposition 3.10, note that contrary to the situation in Subsection 3.2, the different normalizations of the local and adelic Hecke operators are not visible in this case as

Examples
In the following examples, we always fix A = F q [t] and Γ = GL 2 (A).

Non-vanishing hyperderivatives via Petrov's special family
In his thesis, A. Petrov introduced the following family (n ≥ 1) of single cuspidal Hecke eigenforms f n ∈ S 2+n(q−1),1 (Γ) with A-expansions, where for each monic a ∈ A one defines u a (z) := u(az). The importance of Petrov's family is that it gives an explicit description of all cuspforms in level 1 which vanish exactly to the order 1 at the cusp at infinity. It is not difficult to show that the analytic hyperderivatives defined in this section transform Petrov's forms in the following way where G s+1 is the s + 1-th Goss polynomial for the lattice πA as in [Gek88]; see e.g. [Pet15,(2)] for this calculation. It follows then from [Pet13, Theorem 2.2] that all of the forms D s f n for n, s ≥ 1 are non-zero functions, and thus whenever D s preserves modularity, the forms D s f n give examples of non-zero cuspidal Hecke eigenforms in the image of the analytic hyperdifferential operators.

Chains of hyperderivative maps and interactions with the Frobenius
We have observed computationally that it is possible to have chains of hyperderivative maps between spaces of various weights, but sometimes no direct map. So, for example for q = 3 we have chains of hyperderivatives Notice that one has D 2 D 16 = D 16 D 2 = 18 2 D 18 ≡ 0 in characteristic 3, and so while D 18 does not preserve modularity (i.e. D 18 does not meet the requirements of Bosser-Pellarin's result) on either of the spaces S 62,0 (Γ) or S 66,0 (Γ) when q = 3, the compositions D 2 D 16 and D 16 D 2 still do, being just the zero map in all cases. As all of these maps are between spaces of type 0, Petrov's examples from the previous section do not apply. There are also Frobenius maps We have computed the action of the Hecke operator T t on the image of all of these maps using Sage which we summarize in the following diagram.
(X − t 30 ) · (X + t 48 + t 24 ) (X 2 + t 6 X − t 96 + t 90 + t 18 ) (X 2 + t 10 X + t 92 − t 84 + t 12 ) (irreducible of degree 3) The lower part in each box displays the factorized characteristic polynomial of T t acting on the space of cusp forms displayed in the upper part of the box. The colors indicate which factors come from factors in lower weight via the maps above. We should point out that one such link is not displayed in the above diagram: The factor (X + t 48 + t 24 ) in weight 102 in fact comes via D 20 from the factor (X + t 28 + t 4 ) in weight 62, i.e. should be colored blue. However, this coloring would be misleading, because this factor can't possibly explain the factor (X + t 46 + t 22 ) in weight 98 as D 16 D 2 = 0, nevertheless it behaves exactly as one would expect assuming the composition would be non-zero, which we find to be quite an interesting observation. We plan to investigate the interactions between the Frobenius and the hyperderivatives further in future work.

Dimension formulas for SL 2 (A)-forms, A = F q [t]
Using representation theory for GL 2 (F ) we were able to gain in Corollary 2.4 some understanding of the Jordan-Hölder constituents of V k,l . By Proposition 3.5, any composition series of V k,l as an GL 2 (F )-module will induce a Hecke-stable filtration of C har (V k,l ) Γ with subquotients isomorphic to C har (L k ′ ⊗ det m ′ ) Γ for suitable (k ′ , m ′ ). Thus, the number of non-zero such subquotients yields a lower bound on the number of factors of the characteristic polynomial for each Hecke operator T p acting on S k,l (SL 2 (A)), and the dimension of each non-zero such subquotient yields an upper-bound on the degree of the factor of the characteristic polynomial for T p corresponding to the "restriction of T p to this subquotient." We will see in Section 5 that even in cases where there is just one non-zero irreducible subquotient, the characteristic polynomials of the Hecke operators can still factor further over F , and thus more work will be needed to understand this phenomenon.
Thus, in any effort to formulate an analog of the Maeda conjecture in the setting of Drinfeld modular forms, one is forced to grapple with the various modules C har (L k ′ ⊗ det m ′ ) Γ , as above. The present section explains, how for A = F q [t] and Γ = SL 2 (A), one can in principle compute the dimensions of the building blocks C har (L k ) SL 2 (A) . The outcome, which we shall make explicit for q ∈ {2, 3, 5}, looks simple; see Propositions 4.9, 4.10 and 4.11, but we have not found a simple way to prove this or to give closed formulas in general. Our explicit calculations do suggest that we have a rough estimate for the dimension given by In Proposition 4.1 we express dim C har (V ) SL 2 (A) for any SL 2 (F )-representation V that is finite dimensional over F in terms of a formula that only involves the action restricted to SL 2 (F q ) and the Steinberg module st for the latter finite group. For this we work out in detail some material from [Tei91]. In Subsection 4.2 we explain how from this and work of Reduzzi on K 0 of the representation category of F q [SL 2 (F q )], in principle, one can derive explicit formulas for dim C har (V ) SL 2 (A) . To obtain similar results for dim C har (V ) GL 2 (A) , one would need to extend [Red10] to GL 2 (F q ).

A model for SL
Let X := SL 2 (F q )/ SB 2 (F q ). One can show that st is the kernel of the F q [SL 2 (F q )]-linear map and observe that the map is split by F q → F q [X ], a → a x∈X x, because #X = q + 1 ≡ 1 (mod p).
Note that one has the F q [SL 2 (F q )]-linear isomorphism The following result is an analog of the isomorphism for type 1 Drinfeld modular forms for GL 2 (A) stated at the bottom of [Tei91,p. 507].
Proposition 4.1. One has an isomorphism of F -vector spaces The F -vector space structure on the right is induced from that on V , the SL 2 (F q )-action on V is induced from the map SL 2 (F q ) ֒→ SL 2 (A) that arises from the inclusion F q ֒→ F q [t].
Proof. The principal congruence subgroup of level t, denoted by Γ(t), is normal in GL 2 (A), and so by Lemma 1.3 (d) and Remark 1.4 we have an exact sequence of Z[GL 2 (A)]-modules Recall that the group GL 2 (F ∞ ) acts transitively on the tree T ; the action is induced by the natural left action of GL 2 (F ∞ ) on F 2 ∞ . The standard vertex v 0 of T is given by the lattice The standard oriented edge e 0 is the oriented edge with o(e 0 ) = v 0 and t(e 0 ) = v 1 . It is immediate from the definitions that one has where I denotes the Iwahori subgroup of GL 2 (O ∞ ) of matrices whose reduction modulo π is upper triangular. This gives bijections GL 2 (F ∞ )/F * ∞ GL 2 (O ∞ ) → T 0 and GL 2 (F ∞ )/F * ∞ I → T or 1 . Under this identification the map that associates to an oriented edge e its origin o(e) becomes the canonical Combining this with (24), we obtain an exact sequence of Z[SL 2 (A)]-modules, Going back through the definitions, we see that the map∂ Γ(t) is just given by the natural map g SB 2 (F q ) → gSL 2 (F q ). Since (24) and thus also (25) split as a sequence of Z[Γ(t)]-modules, we can tensor with V over Γ(t) to obtain an SL 2 (A)/Γ(t) = SL 2 (F q )-equivariant exact sequence To rewrite the middle and right term of (26), we let G = SL 2 (A), H = SL 2 (F q ), H ′ = SB 2 (F q ) and N = Γ(t), so that G = N ⋊ H. We claim that the following natural maps are H-equivariant isomorphisms: For (i), this is easily verified directly: The injectivity is immediate as Ind G H Z is a free Z[N ]-module. Similarly, the surjectivity follows immediately from G = N ⋊ H. The H-equivariance is obvious. The verification of (ii) can be done similarly upon observing that Ind G H ′ Z is a free Z[N ]-module on the basis H/H ′ . By invoking these isomorphisms, we may rewrite (26) as By tracing back through the isomorphisms, one sees that∂ Γ(t) is the map induced by (23). We conclude that Thus, by (23) we have St ⊗ Γ(t) V = st ⊗ Fq V | SL 2 (Fq) , and this yields Proof. Part (a) follows from st being a projective F q [SL 2 (F q )]-module. To see (b), apply the functor Hom SL 2 (Fq) (·, V ) to the split exact sequence 0 → st → Ind SL 2 (Fq) SB 2 (Fq) F q → F q → 0 from (23), to obtain the exact sequence Applying Shapiro's Lemma to the middle term and computing dimensions proves (b).
Because of Proposition 4.2(a), we have dim Hom SL 2 (Fq) (st, V k,l ) = dim Hom SL 2 (Fq) (st, ∆ k−2 ). Now the latter dimension can be computed using Proposition 4.2(b) and from the known structure of the SL 2 (F q )-and SB 2 (F q )-invariants of F q [X, Y ]: Define and Proof. The calculation of the SL 2 (F q )-invariants is due to Dickson, see [Dic11] or [KM86, Thms. 2.1 and 2.2]. We give the argument for the computation of the SB 2 (F q )-invariants, and observe right away that the equality on the right is straightforward.
We fix k ≥ 0 and then identify F[x, y] k with F[z] ≤k , the polynomials in z of degree at most k, by the map f (X, Y ) → f (z, 1). Then the action a b The action restricted to SB 2 (F q ) extends to all of F[z], and it is straightforward to verify that the invariants under the unipotent group 1 Fq Hence the expression is invariant under all a ∈ F × q if and only if 2i ≡ k (mod q − 1). Let i 0 ≥ 0 be minimal such that 2i 0 ≡ k (mod q − 1), let q ′ = q − 1 if q is even and q ′ = 1 2 (q − 1) if q is odd, and let h = (z q − z) q ′ . Then we have
From the above, it follows that an F-basis of F[X, Y ] SB 2 (Fq) is given by Counting invariants in a fixed homogeneous degree, and using deg f = q + 1, the corresponding generating function for the dimensions of .
Subtracting from this the generating function for the dimensions of F[X, Y ] SL 2 (Fq) gives .
Noting that V k+2,l is related to F [X, Y ] k , the following is an immediate consequence.
(a) All factors of K 0 (G) ⊗ Z Q are totally real fields (Reduzzi).
The factors of K 0 (G)⊗ Z Q are abelian extensions of Q; they only ramify at primes dividing q 2 −1.
We now explain, how in principle Theorem 4.6 allows one to give explicit expressions for the dimension of Hom SL 2 (Fq) (st, L k ) and carry this through for q = 2, 3, 5. For this, observe first that L k | G = L k ⊗ Fq F and thus dim F Hom SL 2 (Fq) (st, L k ) = dim Fq Hom SL 2 (Fq) (st, L k ).
For simplicity of exposition, we assume K 0 (G) ⊗ Z R ∼ = R q , and we let σ 0 , . . . , σ q−1 be the embeddings K 0 (G) ⊗ Z Q → R -we could also work with embeddings into Q q , alternatively.
A Z-basis of K 0 (G) is given by ∆ k , k = 0, . . . , q − 1, or alternatively by L k , k = 0, . . . , q − 1. Observe that the formulas of Bonnafé in Proposition 2.3 that allow one to go back an forth between the two bases continue to hold for F q [SL 2 (F q )]-representations and k = 0, . . . , q −1. Recall also that st = L q−1 .
Note next that for an F q [G]-representation V one has V (e) ∼ = V for the e-th Frobenius twist. Let now k be arbitrary in Z ≥0 and write k = i≥0 k i q i in its base q expansion with k i ∈ {0, . . . , q − 1}. Define furthermore n k (d) := #{i ≥ 0 | k i = d} for d = 0, . . . , q − 1. Then as an F q [G]-representation one has To see the first isomorphism, observe that by the Steinberg Tensor Theorem, this holds for the base p expansion, and if we insert twists. However after applying this theorem, we can group the tensor products in packages of e consecutive digits and then use V = V (e) and run the Tensor Theorem backwards. The second isomorphism follows by simply regrouping the factors.
The expression on the left may look complicated. But it simply says that after evaluating the basic polynomials for the L (r) k at the roots α i , computing L k amounts to raising these values (separately) to suitable powers. A main point of Reduzzi's result is that it makes the a priori inexplicit product structure on K 0 (G) explicit after passing to the ring R q .
To solve equation (27) one only needs linear algebra, since the images of the ∆ k , k = 0, . . . , q − 1 form an R-basis of R q . If the α i are complicated, one has to think properly of how to solve the above system to the needed precision. This should not be too hard, since the tuple of λ k (d), d = 0, . . . , q − 1 lies in Z d , and we only need to approximate the λ k (d) to less than 1/2. Also note that we are only interested in the coefficient of ∆ q−1 . This element is invariant under Frobenius twist. Hence one Summing over the Frobenius twists may allow one to work over the trace field, i.e., the field that arises for q = p. We did not pursue this, since shall only present formulas for q = 2, 3, 5. We give details for q = 3, and only the solution in the other two cases.
Remark 4.12. Dimension formulas for spaces of classical cusp forms of weight k for SL 2 (Z) can be written in a case-by-case way using congruences of k − 1 = dim Q Sym k−2 Q 2 modulo 12. The formulas in the above propositions can be written in a similar form using congruences of dim L k modulo (q 2 − 1)/ gcd(2, q 2 − 1) and of k modulo 2. However, our notation is more compact.
We think that the quantity (q 2 −1)/ gcd(2, q 2 −1) should be interpreted as the index [PSL 2 (A) : Γ 1 (t)] where Γ 1 (t) denotes the image of Γ 1 (t) in PSL 2 (A), in analogy to the classical case. There the term [PSL 2 (Z) : Γ] occurs as a factor in dimension formulas for classical modular forms of weight k for congruence subgroups Γ (up to an error O(1)), where now Γ denotes the image of Γ in PSL 2 (Z).
Remark 4.13. An alternative approach to obtain the results in this subsection is to work out the Brauer characters of the L k and use the orthogonality relations for such to compute the multiplicity of st in L k . This allows one to recover the results of the present subsection. We plan to explore this further in future work.
For the sake of completeness, we note the following easy result for general odd q.
The latter is zero by Corollary 4.4.
Remark 4.15. The above proposition can also easily proved directly by considering the action of the matrix −1 0 0 −1 ∈ SL 2 (A) directly on C har (L k ) similar to the classical case.
5 Investigations towards a Maeda-style conjecture for q = 3 and special weights As indicated in Section 3, the weights of the form k = 1 + p n − cp n−1 for n ≥ 0 and 0 ≤ c < p − 1 are the most natural candidates for a simple formulation of a conjecture of Maeda-type as the relevant GL 2 (F )-representation V k,l is irreducible (for any l). In particular, there are no non-zero hyperderivatives from lower weights. In this section, we want to investigate such a possible conjecture in the special case q = 3. This is based on explicit computations using the computer algebra systems Magma and Sage. In the sequel, we consider A = F 3 [t] and Γ = GL 2 (A). We set k n = 1 + 3 n and compute the characteristic polynomial P t,n,l ∈ F [X] of the Hecke operator T t associated to t ∈ A acting on the spaces S kn,l (Γ) for the two types l = 0, 1. We denote by d n,l the dimension of these spaces. Note that the weights of the form 1+ 3 n − 3 n−1 are not relevant for us as there are no non-zero Drinfeld cusp forms of odd weight for the group Γ.
Note that the factorizations in Table 1 take place in F [X]. Note also that all irreducible factors of degree d > 1 that appear in Table 1 have Galois group S d .
Thus, we observe that aside the from the large number of linear factors appearing and the splitting into two large factors in weight 244, we have a Maeda-style behavior. The factor (X − t) showing up in all spaces of type 1 corresponds to the unique cuspidal, but non double cuspidal form by [Pet13, Theorem 3.2] and can therefore be omitted from the discussion. However, the other linear terms and their corresponding eigenvalues show remarkable symmetries, which we are going to explore further.
In the sequel, we refer to these eigenvalues as special eigenvalues.
(see [Ser79,VIII.1]) that one has a well-defined norm map ν X,G : H 0 (G, X) = X G −→ H 0 (G, X) = X G , x → g∈G gx. (28) Its kernel and cokernel are the Tate cohomology groupsĤ −1 (G, X) andĤ 0 (G, X), respectively. Recall also from [Ser79, IX.3] that X is called cohomologically trivial ifĤ i (H, X) = 0 for all i ∈ Z and for all subgroups H of G; observe that ν X,G is an isomorphism if X is cohomologically trivial.
Let now Γ be any group, let Γ ′ ⊂ Γ be a normal subgroup of finite index and setΓ = Γ/Γ ′ .
for m ∈ M , n ∈ N and γ ∈ Γ, and that the map is an A-module isomorphism. We also use the term norm map for the composition ν  Proof. Observe first that we may assume that M is a free Z[Γ]-module; this holds because cohomological triviality is inherited by direct summands and because any projective module is a direct summand of a free module. Next note that cohomological triviality is also preserved under filtered direct limits and under direct sums. Since our M will be a limit of modules of the form Z[Γ] n , it will thus suffice to prove the lemma assuming that