Euphotic representations and rigid automorphic data

We propose a new method to construct rigid $G$-automorphic representations and rigid $\widehat{G}$-local systems for reductive groups $G$. The construction involves the notion of euphotic representations, and the proof for rigidity involves the geometry of certain Hessenberg varieties.

1. Introduction 1.1. Rigid local systems and automorphic representations. Rigid local systems on a punctured curve are those that don't admit deformations that preserve the local monodromy at the punctures. Many well-known local systems in arithmetic are rigid, e.g., Kloosterman sheaves and hypergeometric sheaves. Katz ([Kat88] and [Kat96]) studied rigid local systems systematically, and he gave an algorithm for producing all tame rigid local systems of arbitrary rank. This algorithm has been extended by Arinkin [Ari10] (and Deligne) to cover all rigid local systems.
For a reductive group H over Q ℓ an H-local systems on a curve U is a homomorphism from π 1 (U, u) to H(Q ℓ ). There is a notion of rigidity for H-local systems generalizing the rigidity for rank n local systems. Much less is known about rigid H-local systems for general H.
Rigid local systems have seen application in inverse Galois theory and in the construction of motives with exceptional Galois groups, see [Yun14], [DR00], [DR10]. In particular it is of interest to construct many examples of rigid H-local systems, especially for exceptional groups H.
While [DR10] use the Katz algorithm to construct and classify rigid G 2 -local systems, this algorithm is unavailable for a general reductive group H. Even for rank n local systems it is often computationally and technically involved. In a series of works ( [HNY13], [Yun14] and [Yun16]) a new method of constructing rigid H-local systems is developed, and many examples are given. The key new ingredient in that method is to use the Langlands correspondence for function fields to transport the problem of constructing rigid H-local systems into constructing rigid G-automorphic representations. Here H is identified with the Langlands dual group of G.
This method has several advantages. When trying construct a rigid H-local system using the Katz algorithm one has to construct a rank n local system and impose conditions on it to force its global monodromy to lie in H. Not all rigid H-local systems can be obtained in this way.
In the geometric Langlands approach one may directly construct H-local systems. In addition it turns out that rigid automorphic representations are sometimes easier to obtain, and techniques from the geometric Langlands program are crucial in passing from rigid automorphic representations to local systems.
This paper aims to expand the zoo of rigid G-automorphic representations and rigid G-local systems by generalizing the construction of [Yun16]. We consider F = k(t), the function field of P 1 over k = F q . The G-automorphic representations we construct have depth zero at 0 and positive depth 1/m at ∞. The corresponding G-local systems over P 1 \{0, ∞} are tamely ramified at 0 and wildly ramified at ∞, and are expected to be rigid.

Euphotic representations.
In this introduction let G be a split almost simple group over the function field F = k(t) of P 1 k . Let F 0 , F ∞ be the local fields of F at 0, ∞ ∈ P 1 . The starting point of [Yun16] is a class of supercuspidal representations of G(F ∞ ) introduced by Reeder and Yu [RY14] called epipelagic representations. They generalize an earlier construction of simple supercuspidal representations by Gross and Reeder [GR10] that motivated the construction of Kloosterman sheaves in [HNY13].
In this paper, we define a more general class of representations of the p-adic group G(F ∞ ) than epipelagic ones which we call euphotic representations. 1 The data needed to construct a euphotic representation is a triple (P ∞ , ψ, χ). Here • P ∞ is a parahoric subgroup of G(F ∞ ); • ψ is a linear function on the vector space V P = P + ∞ /P ++ ∞ (the first nontrivial associated graded of the pro-unipotent radical P + ∞ under the Moy-Prasad filtration). We require ψ to be semisimple in the sense that its orbit under L P = P ∞ /P + ∞ is closed. • Let L ψ be the stabilizer of ψ under L P , and B ψ be a Borel subgroup of L ψ with quotient Cartan T ψ . Then χ is a character χ : T ψ (k) → Q × ℓ . A euphotic representation π of type (P ∞ , ψ, χ) is an irreducible representation of G(F ∞ ) that contains an eigenvector under B ψ (k)P + ∞ on which B ψ (k) acts via χ (inflated from T ψ (k)) and P + ∞ acts via ψ k • ψ (inflated from V P , and ψ k : k → Q × ℓ is a fixed nontrivial additive character). Compared to the notion of epipelagic representations in [RY14], we have relaxed the condition on ψ: it is only required to have a closed orbit under L P and not required to have finite stabilizer under L P . Functionals on V P with closed orbit that are not stable are also encountered in work of Kamgarpour and Yi on the geometric Langlands correspondence for hypergeometric sheaves [KY20].
1.3. Euphotic automorphic data. To construct rigid G-automorphic representations, we start with a triple (P ∞ , ψ, χ) as above, and choose a parahoric subgroup Q 0 of G(F 0 ). We impose several conditions on these data (see Definition 3.4.1) which are of geometric nature (i.e., they only have to do with the situation over k). Among these conditions is the requirement that certain Hessenberg varieties coming from cyclic gradings on g have a stabilizer property under a group action, which we call "spectrally meager" (see Definition 3.3.1), a notion that we believe is of independent interest.
We prove that a euphotic automorphic datum (P ∞ , ψ, χ, Q 0 ) is weakly rigid in the following sense: there is a small (but nonzero) number of irreducible cuspidal automorphic representations π of G(A F ) such that π ∞ is euphotic of type (P ∞ , ψ, χ), π 0 contains a nonzero Q 0 -fixed vector, and π is unramified otherwise; and the number of such cuspidal automorphic representations is uniformly bounded when k is 1 In oceanography euphotic is synonymous to epipelagic, stressing the role of light. Depending on the transparency of the ocean water the euphotic zone may vary in depth -in analogy there are more possibilities for the depth of a euphotic representation than for the depth of an epipelagic representation.
replaced with any finite extension. This is proved by analyzing the space of automorphic functions cut out by the eigen-conditions at 0 and ∞, and Hessenberg varieties naturally show up in this analysis. 1.4. Hecke eigensheaves and local systems. To construct the G-local systems out of these automorphic representations, we consider automorphic sheaves instead of functions. The automorphic datum (P ∞ , ψ, χ, Q 0 ) gives rise to an abelian category P = P(ψ, χ) of perverse sheaves on a certain moduli stack of G-bundles on P 1 k with level structures given by P ++ ∞ and Q 0 . This category has only finitely many simple objects, which is an indication of rigidity.
Here comes a crucial difference with all previous work on rigid automorphic representations. Previously the analogous categories P always decomposed into Hecke-stable pieces with one simple object (a Hecke eigensheaf) in each piece, and the framework of [Yun14] allowed us to extract a G-local system from each Hecke eigensheaf. In the euphotic situation, the category P sometimes has more than one simple object yet there isn't an obvious way to decompose it further. We remark that this is a feature rather than a bug: it is likely that these more complicated categories P give nontrivial global L-packets. To deal with this situation, we extend the framework of [Yun14] to extract eigen local systems from a Hecke eigencategory rather than a Hecke eigensheaf. The extra work needed is of categorical nature: we need to analyze the structure of semisimple factorizable module categories under a neutral Tannakian category. We give a self-contained treatment of this issue in Appendix A, proving a classification result in Theorem A.4.1.
The main results of general nature in this paper can be summarized as follows. For simplicity we state the results in the case G is split. For notations, see §4.
(2) There are finitely many semisimple G-local systems {E σ } σ∈Σ over G m,k (for some index set Σ), and a decomposition of P ss (ψ, χ) (semisimple objects in P(ψ, χ)) P ss (ψ, χ) = σ∈Σ P σ into Hecke-stable subcategories, such that each P σ is an E 2 -module category under Rep( G σ ) for G σ = Aut G (E σ ) (whose action on P σ is denoted by •), and the action of the geometric Hecke operator T V (where V ∈ Rep( G)) on A ∈ P σ is given by Here the direct sum is over all irreducible local systems E over G m,k , E σ (V ) ∈ Loc(G m,k ) is the (semisimple) local system on G m,k associated to E σ and V , [E σ (V ) : E] is the multiplicity space of E in E σ (V ), viewed as an object in Rep( G σ ). (3) The geometric monodromy of each G-local system E σ is tame and unipotent at 0. Under Lusztig's bijection between unipotent classes in G and two-sided cells of the affine Weyl group W aff , the unipotent monodromy of E σ at 0 corresponds to the two-sided cell c Q containing the longest element of W Q0 .
1.5. Examples. More than half of the paper is devoted to various examples of euphotic automorphic data that are not epipelagic. The starting point of our work is a new rigid G 2 -connection on P 1 \{0, ∞} found by the first-named author [Jak20]. We looked for the automorphic representation corresponding to the ℓ-adic counterpart of that G 2 -connection, and arrived at the notion of euphotic automorphic data in general. This G 2 example is presented in detail in §5.
In §6 we give a complete list of euphotic automorphic data when the parahoric subgroup P ∞ is the hyperspecial parahoric G(O ∞ ). The list in this case turns out to be closely related to the classification of double partial flag varieties G/P 1 × G/P 2 that are spherical as a G-variety. The latter problem has been solved by Stembridge [Ste03], and we use his results. We then study in §7 the Hessenberg varieties that appear in these examples in more detail, in order to conclude that they are spectrally meager, thereby verifying that the list in §6 indeed gives euphotic automorphic data.
In §8 we give some potential examples of euphotic automorphic data, mostly for exceptional groups. For these groups we have only checked one of the conditions in the definition of euphotic automorphic data.
1.6. Questions for further study.
(1) The most complicated part in verifying a euphotic automorphic datum is to show that certain Hessenberg varieties are spectrally meager, a condition on the stabilizers of a certain solvable group action. In §7 we deal with Hessenberg varieties arising from the adjoint representation of G, and we show they are spectrally meager by relating them to Springer fibers. What is still missing is an effective criterion for Hessenberg varieties to be spectrally meager in general. (3) Give a complete list of (P ∞ , ψ, Q 0 ) for each almost simple quasi-split G, such that there exists χ making (P ∞ , ψ, χ, Q 0 ) into a euphotic automorphic datum.
1.7.1. The curve X. Fix a finite field k of characteristic p and let F be the field of rational functions on X = P 1 k . Fix an affine coordinate t on X\{∞}, and we identify F = k(t). The closed points |X| of X are in bijection with the places of F . For x ∈ |X| we denote by O x the completed local ring of X at x and by F x its field of fractions. The ring of addèles of F is the restricted product 1.7.2. The split group G. Let G be a split, connected semisimple group over k which is almost simple over k. Fix a maximal split torus T ⊂ G, a Borel subgroup B containing T, and extend these choices into a pinning † = (T, B, {x i } i∈I ) of G (where I indexes the set of simple roots). Let Aut † (G) be the finite group of pinned automorphisms of G, which is identified with the outer automorphism group of G. Let g = Lie G.
We make the following assumption: There exists a non-degenerate Ad(G)-invariant symmetric bilinear form on g.
This assumption is satisfied when char(k) is sufficiently large.
1.7.3. The quasi-split group G. Let e ∈ {1, 2, 3}. Assume p = e and that k contains all e-th roots of unity. Fix an injective homomorphism θ Out : µ e (k) → Aut † (G). Let G be the quasi-split form of G over G m = X − {0, ∞} determined by θ Out . More precisely, let G m = Spec k[t 1/e , t −1/e ] → G m = Spec k[t, t −1 ] be the µ e -torsor, and consider the Weil restriction Res Gm/Gm G. Then G is the fixed point subgroup of Res Gm/Gm G under the diagonal action of µ e on both G m and on G via θ Out .
The base changes G F0 and G F∞ are quasi-split forms of G over the respective local fields determined by the same homomorphism θ Out , viewing µ e (k) as the the quotient of Gal(F s 0 /F 0 ) and Gal(F s ∞ /F ∞ ) realized by the tamely ramified extensions k((t 1/e )) and k((t −1/e )).
Let A = T µe,• . From the construction of the group scheme G over G m , the constant torus A × G m is a maximal split torus of G. Similarly, A F0 and A F∞ are maximal split tori of G F0 and G F∞ . 1.7.4. Coefficient field. We fix a prime ℓ = char(k). The representations of p-adic groups and adèlic groups will be on Q ℓ -vector spaces. Sheaves considered in this paper areétale complexes with Q ℓ -coefficients over k-stacks or k-stacks. All sheaf-theoretic functors are understood to be derived.
Acknowledgement. The authors are grateful to Masoud Kamgarpour and Lingfei Yi for discussions. KJ wishes to thank Michael Dettweiler and Stefan Reiter for teaching him rigid local systems. He especially thanks Jochen Heinloth for his continued support and lots of discussions about the geometric Langlands program.

Euphotic representations
In this section we introduce a class of representations of the p-adic group G(F ∞ ) that generalize the epipelagic representations introduced by Reeder and Yu in [RY14].
This section concerns only the quasi-split group G F∞ over F ∞ . We denote G F∞ simply by G in this section. Using the affine coordinate t on X \ {∞}, we identify F ∞ with K = k((t −1 )) and write K e = k((t −1/e )).

Parahorics and gradings.
2.1.1. Affine roots and root subgroups. The Lie algebra Lie G is the µ e -invariants on g ⊗ K e (where µ e acts both on K e by Galois action and on g via θ Out ). The torus A = T µe,• (see §1.7.3) acts on Lie G by the adjoint action and additionally this algebra carries a G m -action given by scaling the uniformizer t −1/e of K e . The set of affine roots Ψ aff with respect to A can also be identified with the weights of A × G m for this action on Lie G. The set of real roots Ψ re ⊂ Ψ aff consists of those affine roots which are non-trivial on the torus A.
Denote by LG the loop group of G. For any real root α ∈ Ψ re there is a subgroup U α ⊂ LG which is isomorphic to G a over k and whose Lie algebra is the α-eigenspace under the action of The Borel subgroup B fixed in §1.7.3 gives a set of simple affine roots ∆ aff ⊂ Ψ aff and positive affine roots Ψ + aff ⊂ Ψ aff . 2.1.2. Parahoric subgroups. The maximal split torus A ⊗ K of G fixed in §1.7.3 defines an apartment A of the building of G(K). Then Ψ aff can be identified with a set of affine functions on A, whose vanishing affine hyperplanes give a stratification of A into facets. There is a unique set of positive integers {n α } α∈∆ aff such that α∈∆ aff n α α = 1 as functions on A.
The fundamental alcove C ⊂ A is cut out by the inequalities α > 0 for all α ∈ ∆ aff . Let F ⊂ C be a facet in the closure of C. Let J = {α ∈ ∆ aff |α| F = 0}. Let In the rest of the paper we also assume The characteristic p = char(k) does not divide m.
Let P ⊂ G(K) be the (standard) parahoric subgroup corresponding to F. Let P ⊃ P + ⊃ P ++ be the first three steps in the Moy-Prasad filtration of P with respect to x P , the barycenter of F. In other words P + = G(K) xP,1/m , P ++ = G(K) xP,2/m . Let L P = P/P + be the Levi factor of P (a connected reductive group over k). There is a canonical section L P ֒→ P whose image contains A; we identify L P with the image of this section. The quotient V P = P + /P ++ is a representation of L P over k.

2.1.3.
Cyclic grading on the Lie algebra. Let g = Lie G. Then the barycenter x P gives a Z/mZ-grading on g g = i∈Z/mZ g(i) compatible with the Z/eZ-grading on g obtained from θ Out (under the reduction map Z/mZ → Z/eZ) and such that g(0) can be canonically identified with Lie (L P ), and g(i) can be canonically identified with for i = 0. In particular we have V P ∼ = g(1) as L P -modules. For more details, see [RY14, Theorem 4.1].
2.2. Euphotic representations. Let P be a standard parahoric subgroup of G(K). Let ψ ∈ V * P be a vector whose L P -orbit is closed. Using an Ad(G)-invariant form on g, which exists by our assumption in §1.7.2, the L P -module V * P may be identified with g(−1). Then the L P -orbit of ψ is closed if and only if ψ ∈ g(−1) is semisimple as an element of g.
Fix an additive character ψ k : P be a vector whose L P -orbit is closed. Let (π, V) be an irreducible admissible representation of G(K). We say that it is euphotic with respect to (P, ψ) if V (P + ,ψ) = 0.

2.2.2.
Action of L ψ . Let L ψ be the stabilizer of ψ under L P . Then L ψ is a (not necessarily connected) reductive group over k. In fact, since ψ can be identified with a semisimple element of g lying in g(−1), its centralizer G ψ in G is a reductive group whose Lie algebra g ψ is stable under the Z/mZ-grading. Viewing the Z/mZ-grading on g as an action of µ m on G, L P is the neutral component of G µm . Then G ψ is stable under the µ m -action, and L ψ ⊂ (G ψ ) µm is the union of components that lie in L P .
Let (π, V) be a euphotic representation of G(K). There is an action of L ψ (k) on V (P + ,ψ) . We will be interested in those (π, V) such that V (P + ,ψ) contains a principal series representation of L ψ (k). More precisely, let B ψ ⊂ L • ψ be a Borel subgroup of the neutral component L • ψ of L ψ . Let T ψ be the quotient torus of B ψ .

2.2.3.
Definition. Let ψ ∈ g(−1) be semisimple and let χ : T ψ (k) → Q × ℓ be a character. Let (π, V) be an irreducible admissible representation of G(K). We say that it is euphotic with respect to (P, ψ, χ) if the action of B ψ (k) on V (P + ,ψ) contains a nonzero eigenvector on which B ψ (k) acts via the character By Frobenius reciprocity, an irreducible admissible (π, V) is euphotic with respect to (P, ψ, χ) if and only if it is a quotient of the compact induction Relation with epipelagic representations. For simplicity in this subsection we assume G is split over F (i.e., e = 1) and L • ψ is split over k. We lift T ψ to a maximal split torus T ψ ⊂ L ψ over k. Up to changing ψ by an element in the same L P -orbit, we may assume T ψ ⊂ A (A is a maximal split torus of L P ). Then H = C G (T ψ ) is a Levi subgroup of G containing T. From the construction, H is stable under the µ m -action on G which gives the Z/mZ-grading.
We claim that the induced Z/mZ-grading on h = Lie H is stable in the sense of [RLYG12,§5.3]. Indeed, it suffices to show that the stabilizer (H ψ ) µm is finite modulo T ψ , but this is true because (H ψ ) µm is a reductive group containing T ψ as a maximal torus which at the same time is central, hence (H ψ ) µm /T ψ is finite. The theory developed in [RLYG12, Corollary 15] then attaches a regular elliptic conjugacy class [w] in the extended Weyl group W ext (H, T) ( component group of the normalizer of T in Aut(H)). One checks that [w] indeed is a well-defined conjugacy class in W (G, H, T) = (N G (H) ∩ N G (T))/T.
Let H = H ⊗ k K, a Levi subgroup of G. The Z/mZ-grading on h gives a standard parahoric subgroup P H ⊂ H and ψ ∈ h(−1) can be viewed as a linear function on P + H /P ++ H . We expect that a euphotic representation of G(K) with respect to (P, ψ, χ) should be a composition factor of a parabolic induction from an epipelagic representation of H(K) with respect to (P H , ψ) in the sense of [RY14], whose central character restricts to χ on T ψ ⊂ ZH.

2.4.
Predictions on the Langlands parameter. Again for simplicity we assume e = 1, i.e., G is split, and that L ψ is split over k.
2.4.1. Let G be the Langlands dual of G (over Q ℓ ) equipped with a maximal torus T and an isomorphism X * ( T ) ∼ = X * (T). Then the roots Φ( G, T ) are identified with the coroots Φ ∨ (G, T). Recall the Levi subgroup K be the Weil group, inertia group and wild inertia of the local field K. Let π be a euphotic representation of G(K) with respect to (P, ψ, χ). Let ρ π : W K → G be the Langlands parameter of π.
For p = char(k) large, we make the following predictions on ρ π : (1) Consider the torus S = [ H, H] ∩ T . Then, up to G-conjugacy, one should be able to arrange that ρ π (I + K ) ⊂ S. We also expect that ρ π (I + K ) is regular in S, i.e., its centralizer in G is C G ( S). This implies that ρ π (I K ) lies in the normalizer N G ( S) of S in G.
(2) By (1), ρ π induces a homomorphism ρ tame π : I tame  (2), up to conjugacy ρ tame,ss π should have image in N G ( H, T ) ′ / S, which is an extension of w by T / S. Since w acts trivially on T ψ , the projection T → T / S → T ψ dual to the inclusion T ψ ⊂ T extends to a homomorphism N G ( H, T ) ′ / S → T ψ . Then the composition should correspond to the character χ of T ψ (k) under local class field theory. (4) The slopes of the adjoint representation Ad(ρ π ) : W K → Aut( g) are either 0 or 1/m, and The prediction on Swan conductors (2.1) is based on the following heuristics. The Swan conductor should be 1/m for each root α ∨ ∈ Φ( G, T ) = Φ ∨ (G, T) which is nontrivial on the image ρ(I + K ), and should be zero on other root spaces and on t. Those α ∨ such that α ∨ | ρ(I + K ) = 1 correspond exactly to the coroots of the centralizer G ψ with respect to T. Let R ′ = {α ∈ Φ(G, T)|α(ψ) = 0}, then Swan(Ad(ρ π )) = #R ′ /m. On the other hand, the µ m -action on g preserves R ′ and freely permutes R ′ . We get that g(0) = g µm is the direct sum of g ψ ∩ g(0) = Lie L ψ and 1-dimension from the sum of root spaces for each µ m -orbit of R ′ . Therefore the number of µ m -orbits on R ′ , which is #R ′ /m, is the same as dim L − dim L ψ , hence the prediction (2.1).
Let I tame be the unique subgroup of index m. Assume the character χ is sufficiently generic, then ρ tame π will be semisimple and by (3) above, ρ tame π (I tame K (m)) should be conjugated into T / S. The genericity of χ should imply that the centralizer of ρ tame π (I tame K (m)) in C G ( S)/ S is T / S. Therefore, for χ sufficiently generic, we should have G ρπ (IK ) = C G ( S) ρ tame π (I tame K ) = T w . Therefore we expect to have (2.2) dim g ρπ (IK ) = dim t w = dim T ψ = rkL ψ .

Euphotic automorphic data
In this section we define euphotic automorphic data, and give a criterion for them to be rigid.
3.1. Pre-euphotic automorphic data. corresponding to the parahoric G(O 0 ). Let us denote both A ∞ and A 0 by A under this identification. Let P 0 be the parahoric subgroup of G(F 0 ) corresponding to the same facet F that we used to define P ∞ . We say P 0 thus constructed is opposite to P ∞ . Then the Levi factors L P∞ and L P0 can be canonically identified, which we denote by L P , or simply L.
• Let P 0 ⊂ G(F 0 ) be the parahoric subgroup opposite to P ∞ . Then Q 0 is a parahoric subgroup of G(F 0 ) which is contained in P 0 and contains the torus A.
The parahoric Q 0 ⊂ P 0 corresponds to a facet F ′ in A 0 whose closure contains F. Let x Q be the barycenter of F ′ . The parahoric Q 0 also determines a parabolic subgroup Q of L = L P0 (containing A) such that Q 0 is the preimage of Q under the projection P 0 → L.
3.1.3. Weyl groups. Let W be the Weyl group of G with respect to T. The Weyl group W of G with respect to A can be identified with the fixed points W µe . The Iwahori-Weyl groups of G(F ∞ ) and G(F 0 ) with respect to A can be identified under the identification A ∞ = A 0 ; we denote it by W = X * (A) ⋊ W . For w ∈ W , choose a lifting of it in N G (T)(k) µe ; for an arbitrary element w = (λ, w 1 ) ∈ X * (A) ⋊ W = W , we have its liftingẇ = t λẇ 1 . Let W aff ⊂ W be the affine Weyl group generated by affine simple reflections. Let Ω = Stab W (C), the stabilizer of the fundamental alcove under W . Then the projection induces an isomorphism Ω ∼ → W /W aff , and Ω is a finite abelian group.
Let W P (resp. W Q ) denote the Weyl group of L (resp. L Q , the Levi of Q 0 or Q), both as subgroups of W aff . We have W Q ⊂ W P .
3.2. The space of automorphic forms.
Any such automorphic representation π contains a nonzero vector in the following space of Q ℓ -valued functions on which B ψ P + ∞ ⊂ P ∞ acts via the character µ defined by µ| B ψ = χ and µ| P + From this we get a decomposition F = Let f ∈ F w . Assume f is not identically zero on the double coset P + ∞ ℓẇΓ 0 for some ℓ ∈ L. For any α ∈ Ψ(V P ) (i.e., α(x P ) = 1/m) and u ∈ U α we have If we furthermore assume that α(wx Q ) ≤ 0, then U w −1 α ⊂ Γ 0 and we find that This implies that ℓ −1 ψ vanishes on the space In other words Note that V ⊥ w is stable under Q w . 3.2.3. Definition. Let Y w be the closed subscheme of the partial flag variety L/Q w defined by

Remark.
(1) The variety Y w is a Hessenberg variety in the sense of [GKM06], attached to the L-module V * P and the Q w -stable subspace V ⊥ w . It carries an action of the stabilizer L ψ by right translation.
(2) If w = 1, or more generally if F P is in the closure of wF Q , then Y w = L/Q w .
(3) If we change w to ww 1 for some w 1 ∈ W Q , then both Q w and V ⊥ w are unchanged, hence Y w = Y ww1 . (4) If we change w to w 2 w for some w 2 ∈ W P , then we may lift w 2 toẅ 2 ∈ L, and right multiplication byẅ 2 induces an isomorphism L/Q w The above discussion implies that any function f ∈ F [w] must be supported on Y w (k) (as a function on (L/Q w )(k) under (3.3)). In view of the eigen property under B ψ , we get the following description of F w .
3.2.5. Lemma. For any w ∈ W , let [w] be its (W P , W Q ) coset in W . Then there is a canonical isomorphism where the right side is the space of eigenfunctions on Y w (k) under the left translation of B ψ (k) with eigencharacter χ. The isomorphism is given by 3.3. Spectrally meager varieties. In examining when the space F w is zero we arrive at the following notion.

Definition.
(1) Let H be a connected reductive group over k with Borel subgroup B H . Let Y be a scheme of finite type over k with an H-action. We say that Y is spectrally meager if for any geometric point y ∈ Y (k) the stabilizer Stab BH (y) contains a nontrivial torus.
(2) If Y is a spectrally meager H-scheme, let S(Y ) be the collection of (nontrivial) subtori of T H,k (the universal Cartan of H) given by the images of Stab BH (y) • → T H for all y ∈ Y (k).

Remark.
(1) The definition of spectrally meager H-scheme does not depend on the choice of the Borel subgroup B H . It is therefore intrinsic to the H-scheme Y .
(2) If Y is a spectrally meager H-scheme, the collection S(Y ) of subtori is finite. Indeed, we may partition Y into finitely many locally closed connected B H -stable subschemes {Y α } such that the torus part of the stabilizer of B H on each point of Y α has the same dimension, then each Y α contributes a single torus in S(Y ). (3) The terminology "spectrally meager" may be justified as follows.
Let χ : T H (k) → Q × ℓ be a character with the property that χ| S(k) = 1 for any S ∈ S(Y ) defined over k, then the H(k)-module Fun(Y (k)) does not contain any simple constituent of the principal series representation Ind H(k) BH (k) (χ). The same property holds after any finite base change k ′ /k. On the other hand, making the obvious definition over C, if Y is an affine H-variety over C which is spectrally meager, then O(Y ) as an algebraic H-module contains the irreducible H-module V λ with highest weight λ only if λ ∈ X * (S) ⊥ for some S ∈ S(Y ), i.e., λ lies in the union of finitely many proper sublattices in X * (T ).

3.3.3.
Corollary (of Lemma 3.2.5). Let w ∈ W . If Y w is spectrally meager as an L ψ -scheme, and χ is nontrivial on S(k) for any torus S ∈ S(Y w ) that is defined over k, then F [w] = 0.
Proof. For any y ∈ Y w (k), the stabilizer Stab B ψ (y) maps to a nontrivial torus S ∈ S(Y w ). Since χ| S(k) = 1, all (B ψ (k), χ)-eigenfunctions on Y w (k) must vanish at y. We conclude that F w = 0 by Lemma 3.2.5.

Euphotic automorphic data.
3.4.1. Definition. A pre-euphotic automorphic datum (P ∞ , ψ, χ, Q 0 ) is called a euphotic automorphic datum if it satisfies the following conditions: (1) For w ∈ Ω, any Borel subgroup B ψ of L ψ acts on Y w with an open orbit with finite stabilizers.
(2) For any w ∈ W − W P ΩW Q , the L ψ -scheme Y w is spectrally meager.
(3) Let K χ be the Kummer local system on T ψ attached to χ (see [Yun14,Appendix A.3.5]). Then for any S ∈ ∪ w / ∈WPΩWQ S(Y w ) (this is a subtorus of T ψ,k over k), the restriction K χ | S is a nontrivial local system. We call the euphotic automorphic datum (P ∞ , ψ, χ, Q 0 ) strict if moreover the following holds: (1) For any w ∈ Ω and any x ∈ (L/Q w )(k) outside the open B ψ -orbit let S x be the image of Stab B ψ (x) • → T ψ,k and assume that K χ | Sx is a nontrivial local system (in particular, S x is a nontrivial torus). (2) The stabilizer on the open B ψ -orbit is ZG.

Remark.
(1) The conditions for a pre-euphotic automorphic datum to be a euphotic automorphic datum can be checked after base changing the situation to k. (2) The open B ψ -orbit condition in Definition 3.4.1 is saying that Y w is a spherical L ψ -variety. This implies that B ψ has finitely many orbits on Y w by [Bri86]. (3) As w varies in W , there are only finitely many different L ψ -equivariant isomorphism types of the schemes Y w . Indeed, there are only finitely many possibilities for Q w and V w . Therefore the union ∪ w / ∈WPΩWQ S(Y w ) is a finite set. 3.4.3. Proposition. Let (P ∞ , ψ, χ, Q 0 ) be a euphotic automorphic datum. Then (1) The space F is finite-dimensional and consists of cusp forms. In particular, any automorphic representation π satisfying the conditions in §3.2.1 is cuspidal. (2) For any finite field extension k ′ /k, consider the similarly defined space F k ′ using the base change automorphic data (P ∞ , ψ, χ, Q 0 ). Proof.
(1) For any w ∈ W − W P ΩW Q , the assumptions in Corollary 3.3.3 are satisfied (for any S ∈ S(Y w ) defined over k, χ| S(k) is nontrivial if and only if K χ | S k is nontrivial). Therefore F [w] = 0. If w ∈ Ω, the space F [w] = Fun(Y w (k)) (B ψ (k),χ) has finite dimension because there are finitely many B ψ -orbits on Y w over k (see Remark 3.4.2(2)), hence finitely many rational orbits as well. we conclude that F is finite-dimensional and stable under the spherical Hecke operators at all places x / ∈ {0, ∞}. By [Laf18, Lemme 8.24], F consists of cusp forms.
(2) The universal bound for dim F k ′ comes from bounding the number of B ψ (k ′ )-orbits on Y w (k ′ ) for w ∈ Ω. The number of such orbits are bounded by x #π 0 (Stab B ψ (x)) where x runs over a set of representatives of the finite set Y w (k)/B ψ (k).
3.4.4. Remark. One may generalize the notion of a (pre-)euphotic automorphic datum by adding a character η of L Q (k) (or a rank one character local system K η on L Q ). We leave it to the reader to modify the third condition in Definition 3.4.1 in this situation (which should involve K χ and K η ).

Hecke eigencategory and local systems
Unless otherwise stated, in this subsection all ℓ-adic sheaves are over the relevant spaces base-changed to k.

4.1.
Automorphic sheaves. Let (P ∞ , ψ, χ, Q 0 ) be a euphotic automorphic datum. 4.1.1. A category of automorphic sheaves. We denote by Bun := Bun G (Q 0 , P ++ ∞ ) the moduli stack of G-bundles on P 1 with level structure Q 0 at 0 and P ++ ∞ at ∞. It carries an action of B ψ ⋉ V P by changing the level structure at ∞. The character ψ k • ψ : V P = P + ∞ /P ++ ∞ → Q * ℓ determines an Artin-Schreier sheaf AS ψ on V P . Similarly we get a Kummer sheaf on T ψ whose pullback along B ψ → T ψ we denote by K χ .
Let D(ψ, χ) be the derived category of Q ℓ -complexes on Bun k with (B ψ,k ⋉V P,k , K χ ⊠AS ψ )-equivariant structures. In [Yun14], a more elaborate notion of geometric automorphic data is defined, including the data of a character sheaf on the center ZG. In our case we simply take the trivial local system on ZG. For the details we refer to [Yun14, Section 2.6].

Lemma.
(1) The category P(ψ, χ) has finitely many simple objects up to isomorphism. (2) Let S be a scheme of finite type over k. Any simple perverse sheaf in P(S, ψ, χ) is of the form F S ⊠ A, where F S is a simple perverse sheaf on S k , and A ∈ P(ψ, χ) is a simple object.
Proof. All stacks in the proof are understood to be over k; we omit the base change (−) k from the notations.
(1) Stratify Bun into locally closed substacks Bun [w] indexed by [w] ∈ W P \ W /W Q using the Birkhoff decomposition (3.1) and (3.2). By the discussion in §3.2.2, the restriction of any A ∈ D(ψ, χ) to Bun [w] can be identified with an object . By the genericity condition on the Kummer sheaf K χ (see Definition 3.4.1), for any w ∈ W − W P ΩW Q , any geometric point y ∈ Y w , the restriction of K χ to Stab B ψ (y) is nontrivial. Therefore A [w] = 0. This implies that any object A ∈ D(ψ, χ) has vanishing stalks and costalks outside the open strata ⊔ w∈Ω Bun [w] (one for each connected component of Bun). For By Remark 3.4.2(2), Y w has finitely many B ψ -orbits. This implies that Perv (B ψ ,Kχ) (Y w ) has finitely many simple objects. Therefore the same is true for P(ψ, χ).
(2) The above argument shows that the restriction map along the embedding id S × j : Let B ∈ Perv (B ψ ,Kχ) (S × Y w ) be a simple object, for some w ∈ Ω. Let Y w = ⊔ α∈Σ Z α be the stratification into B ψ -orbits. Then B is the middle extension of a local system B 0 on a locally closed B ψ -stable substack of S × Y w , which is necessarily of the form S ′ × Z α for some α ∈ Σ and S ′ ⊂ S locally closed irreducible. Choose a point z ∈ Z α and let Γ α be the stabilizer of B ψ at z. Then via restriction to S ′ × {z}, (B ψ , K χ )equivariant locally systems on S ′ × Z α are the same as (Γ α , K χ |Γ α )-equivariant local systems on S ′ , with the trivial action of -equivariant local systems on S ′ (with the trivial Γ α -action) are the same as local systems on S ′ with an action of the group Ξ such that Q × ℓ ⊂ Ξ acts by scaling on the local system. Thus any irreducible (B ψ , K χ )-equivariant local system B 0 on S ′ × Z α must be of the form F 0 ⊠ ρ for an irreducible local system F 0 on S ′ and an irreducible representation ρ of Ξ on which Q × ℓ acts by scaling. View ρ as a (B ψ , K χ )-equivariant local system A 0 on Z α . We have B 0 ∼ = F 0 ⊠ A 0 . Let F S and A be the middle extensions of F 0 and A 0 respectively to S ′ and Z α , then B ∼ = F S ⊠ A.

Geometric Hecke operators.
We briefly review the construction of geometric Hecke operators. For details we refer to [Yun14,§4.2]. First consider the case G is split. Let Hk be the Hecke correspondence This perverse sheaf can be "spreadout" over Hk (still denoted IC V ) such that its * -restriction to every fiber of − → h is isomorphic to IC V . The geometric Hecke operator is the functor The formation of T V is additive in V . More generally, for any finite set I, any scheme S, V ∈ Rep( G I ), there is a functor defined using the version of the Hecke stack that modifies the bundle simultaneously at a collection of points indexed by I. These functors have factorization structures: for

Moreover, for any surjection
When G is quasi-split, the only modification to the above discussions is that G m should be replaced with the µ e -covering G m over which G is split.
(1) The functor T V [1] is exact for the perverse t-structures.
(2) For any simple perverse sheaf A ∈ P(ψ, χ) and Here A ′ runs over simple objects in P(ψ, χ) and E(V ) A,A ′ is a semisimple local system on G m,k .
(3) For simple perverse sheaves A, A ′ ∈ P(ψ, χ) and V ∈ Rep( G) there is a canonical isomorphism More generally, for any finite set I, V ∈ Rep( G I ) and any simple perverse sheaf A ∈ P(ψ, χ), Here (5) Note that µ e acts on G by pinned automorphisms and on G m,k by deck transformations. For ζ ∈ µ e , V ∈ Rep( G I ), let V ζ be the representation given by the composition G I ζ I − → G I → GL(V ); let ζ I also denote the diagonal action of ζ on G m,k I . Then there is an isomorphism functorial in A, A ′ and V , and compatible with the group structure of µ e : Proof. (1)(2) We first show a weak version of (1): suppose A is supported on the neutral component, i.e., it is a clean extension from Bun [1] , then T V (A) [1] is perverse for any V ∈ Rep( G). The proof is similar to the argument in [Yun16, third paragraph in the proof of Theorem 3.8], and we only give a sketch. The key point being that the map − → h : [1] Hk → G m × Bun is ind-affine, where [1] Hk ⊂ Hk is the preimage of Bun [1] under ← − h . This boils down to the fact that Bun [1] ⊂ Bun 0 (the neutral component of Bun) is the non-vanishing locus of a section of a certain determinant line bundle, which pulls back to an ample line bundle on the affine Grassmannian. Therefore [1] Hk ⊂ Hk is the nonvanishing locus of a section of a line bundle relative ample with respect to − → h , hence the ind-affineness of − → h | [1] Hk. Then we prove a weak version of (2): locally constant along G m follows from the ULA property of both ← − h * A and IC V with respect to the projection to G m . For details we refer to the argument in [Yun14,Lemma 4.4.6].
Now we prove (1) in general. Note that the same argument for the weaker version may fail for A supported on other components of Bun: it may happen for some ω ∈ Ω that the boundary of Bun [ω] has codimension > 1, so it cannot be the non-vanishing locus of a section of a line bundle.
For ω ∈ Ω, let Bun ω be the corresponding component of Bun (so that . The weak version of (1) that is already proven says that: Finally (1) together with the weak version of (2) implies the full version of (2).
(3) Let σ be the involution on Hk that switches the two G-bundles. Then there is a well-known isomorphism between IC V and σ * D− → h (IC V ∨ ) (both on Hk, where D− → h denotes the relative Verdier duality with respect to the map − → h ). This follows from the similar statement for the Satake category. From this and standard sheaf-theoretic functor manipulations we obtain an adjunction as complexes on G m × Bun, functorial in A, A ′ ∈ D(ψ, χ). In view of the decomposition in (2), we get (4.2).
(4) Same argument as above shows that A ′ a local system on G I m upon shifting by |I|. Now the factorization structure of T I V allows us to conclude that each Acting on both sides by T Gm,V1 again, the left side becomes T I V (A) by the factorization isomorphism (4.1), and the right side becomes with both A ′ and A ′′ run through simple objects in P(ψ, χ). We conclude The case of general I follows by an iteration of the same argument.

4.2.
Eigen local systems. Next we will extract L G-local systems from the category P(ψ, χ).
4.2.1. Let P ss (ψ, χ) ⊂ P(ψ, χ) be the subcategory of semisimple objects. Let Loc( G m,k ) be the tensor category of Q ℓ -local systems (of finite rank) over G m,k . Let C be the full subcategory of Loc( G m,k ) we see that C is stable under tensor product. Proposition 4.1.4 (3) shows that C is closed under duality. Therefore C is a semisimple rigid tensor subcategory of Loc( G m,k ), hence neutral Tannakian.

4.2.2.
Theorem. Let (P ∞ , ψ, χ, Q 0 ) be a euphotic automorphic datum. Then there are finitely many semisimple µ e -equivariant G-local systems {E σ } σ∈Σ over G m,k (for some index set Σ), and a decomposition (2) For any A ∈ P σ and V ∈ Rep( G), there is an isomorphism functorial in V and A Here the direct sum is over all irreducible local systems E over G m,k , E σ (V ) ∈ Loc( G m,k ) is the (semisimple) local system on G m,k associated to E σ and V , and Moreover, there is a version of the above isomorphism for any finite set I and V ∈ Rep( G I ), and these isomorphisms are compatible with the factorization structures.
We recall that a G-local system on G m,k is called semisimple if the Zariski closure of the image of After a finite extension of k, we may assume that each simple object A in P(ψ, χ) has the property Fr * A ∼ = A, where Fr : Bun → Bun is the Frobenius morphism with respect to k. In particular, each summand P σ in Theorem 4.2.2 is stable under Fr * . Fix a Weil structure Fr * A ∼ = A for each simple A ∈ P(ψ, χ). Applying Remark A.4.3 to the Γ = Fr Z -equivariant structure on the factorizable Rep( G)-module structure on P ss (ψ, χ) with coefficients in C (where Fr acts trivially on Rep( G) and on P ss (ψ, χ) by the fixed Weil structures, and it acts by Fr * on C), we conclude that each E σ viewed as a tensor functor Rep( G) → C carries a Fr-equivariant structure. In other words, E σ carries a Weil structure (depending on the choice of Weil structures of simple objects in P(ψ, χ)).

Remark.
(1) A µ e -equivariant G-local system on G m,k is the same thing as a L G = G ⋊ µ elocal system on G m,k , such that the induced µ e -cover of G m,k is G m,k . Therefore, after choosing a base point x ∈ G m,k , the E σ in the above theorem is the same data as continuous homomorphisms gives an extension of ρ σ to the Weil group of G m so it is a Langlands parameter in the usual sense for the quasi-split group G over F = k(t).
(2) The upshot of the above theorem is that, we not only can extract a L G-local system E σ from each indecomposable summand P σ of the category P ss (ψ, χ), but there is a residual action of Rep(Aut(E σ )) on P σ . The Rep(Aut(E σ ))-module category P σ may be viewed as a secondary invariant attached to the automorphic datum in question that is not covered by the usual Langlands parameter E σ . The relationship between this secondary invariant and global L-packets deserves further study. A closely related phenomenon is discussed in [FW08] under the name fractional Hecke eigensheaves.
Fixing a point y ∈ O(k) identifies P(ψ, χ) with Rep(ZG, Q ℓ ) and hence we obtain the desired decomposition and uniqueness of the simple perverse sheaf A σ . More explicitly A σ = j ! F σ is the perverse sheaf whose restriction to O corresponds to the character σ. For any H ∈ D(S, ψ, χ) we may decompose its restriction to S × y as follows according to the action of ZG. For H ∈ P(S, ψ, χ) (or any shifted perverse sheaf) we obtain a decomposition cf. [Yun14]. Note that the automorphism group of any point of Bun G (Q 0 , B ψ P + ∞ ) contains ZG. Therefore we may speak of the subcategory D(ψ, χ) σ on which ZG acts through σ. By [Yun14, §4.4.1] the geometric Hecke operator T V sends D(ψ, χ) σ to D(G m , ψ, χ) σ and by Proposition 4.1.4 T V (A σ ) [1] is perverse. Therefore we have 4.3. Local monodromy and rigidity. In this subsection we assume that G is split, so that L G = G. Let (P ∞ , ψ, χ, Q 0 ) be a euphotic automorphic datum. By Theorem 4.2.2, we have a G-local system E σ over G m,k for each indecomposable summand P σ of P ss (ψ, χ). Let ρ σ : π 1 (G m,k , η) → G(Q ℓ ) be the geometric monodromy representation attached to E σ .
Proposition. For any local system E σ attached to the euphotic automorphic datum the local monodromy ρ σ | IF 0 (I F0 is the inertia group at 0) is tame, and maps a topological generator of I tame

F0
into the unipotent class of G which corresponds to the two-sided cell c Q of W aff containing the longest element of W Q under Lusztig's bijection (4.6).
Proof. It suffices to prove the case where G is simply-connected. The proof is almost the same as in the epipelagic case, cf. [Yun16, §4.11-4.18], replacing P 0 in loc.cit. by Q 0 . The only thing that needs to be adapted in our situation is [Yun16,Lemma 4.12]. Here the analogous statement should be: consider the action of D Q0 (G((t))/Q 0 ) on D(ψ, χ), then any perverse sheaf K ∈ Perv Q0 (G((t))/Q 0 ) acts as a t-exact endo-functor of D(ψ, χ). Consider the Hecke correspondence Hk 0 that classifies modifications of Bun at 0, with two maps [1] ). Since Bun [1] is the preimage of the open stratum [pt/L] ⊂ Bun G (P 0 , P ∞ ), we may identify fibers of Proposition. Under the assumptions in the beginning of §4.3, assume further • The restriction ρ σ | IF ∞ is as predicted in §2.4.2.
• The image of ρ σ does not lie in any proper Levi subgroup of G (equivalently, Aut(E σ ) is finite).
Then E σ is cohomologically rigid in the sense that Here j : G m ֒→ X is the open embedding, and Ad(E σ ) is the adjoint local system attached to E σ .
Proof. We have the exact sequence By the second assumption, dim g ρσ(π1(G m,k )) = 0. Let u ∈ G be a unipotent element in the class corresponding to the longest element w Q,0 of W Q . By Proposition 4.3.2, dim g ρσ which is guaranteed by the condition (1) in Definition 3.4.1.
5. An example in type G 2 From this section on, we will give several families of examples of (strict) euphotic automorphic data.
5.1. The rigid connection from [Jak20]. The motivating example for our construction of rigid automorphic data is a certain rigid irregular G 2 -connection discovered by the first-named author. By [Jak20, Theorem 1.1.] there is a rigid irregular connection E on G m,C with differential Galois group G 2 and with the following local data. At z = 0 the connection is regular singular and has subregular unipotent monodromy. On the punctured formal disc D • at z = ∞ the connection E is isomorphic to where by (λ, λ −1 ) we denote a regular singular formal connection of rank two with monodromy λ and λ −1 and similarly for (−1). The formal connection El(z 2 , α, (λ, λ −1 )) is an elementary connection in the sense of [Sab08,§2]. It is the direct image of a formal exponential connection twisted by a regular singular connection along a twofold covering of the formal disc.
The local data of this connection dictates our guess for the local representations in §3.2.1. The parahoric subgroup Q 0 should correspond to the unipotent conjugacy class of the connection at z = 0 as in §4.3, (1).
The choice of the character ψ corresponds to the occurence of the formal exponential connection, an additive parameter, and the character χ reflects the multiplicative parameter at z = ∞ of the formal connection, given by the regular singular connection (λ, λ −1 ). Note that in addition the formal connection at z = ∞ becomes diagonalisable after pullback to a two-fold cover.

5.2.
Constructing the automorphic form. Assume G is split of type G 2 and denote by ∆ = {α 1 , α 2 } the simple roots of G 2 where α 1 is the long root. Consider the parahoric P ∞ with L ∼ = SO 4 with roots α 2 and the highest root η. We have V = Sym 3 (St) ⊗ St ′ , where St is the standard representation of the short root SL 2 ֒→ SO 4 , and St ′ is the standard representation of the long root SL 2 . In this case m = 2 and V ∼ = V * . We may identify V * with the space of bihomogeneous polynomials in two sets of variables (x, y) and (u, v) that are cubic in (x, y) and linear in (u, v). Then take ψ = x 3 u + y 3 v. We have L ψ ∼ = G m ⋊ µ 2 : the projection L ψ → PGL 2 to the short root factor is an isomorphism onto the normalizer of a maximal torus A in PGL 2 ; the other projection L ψ → PGL 2 onto the long root factor has image N PGL2 (A) with kernel µ 3 . We then have Proof. Note that for dimension reasons any point in L/Q outside the open B ψ -orbit will have a positive dimensional stabilizer. Therefore since L ψ is a torus this immediately implies that (P ∞ , ψ, χ, Q 0 ) is strict if it is euphotic.
For w = id we will prove that if Y w = ∅ then Y w is finite. The first step is to single out the cases in which Y w is empty.
Suppose w is given such that all weights of V ⊥ w lie in a half-space in X * (A) R not containing 0. For ℓ −1 ψ ∈ V ⊥ w we can then find a torus T ′ such that 0 ∈ T ′ . ℓ −1 ψ and since the orbit of ψ is closed this implies ψ = 0, a contradiction. In this case we therefore get Y w = ∅. Let β = α 1 + 3α 2 and suppose w is given such that −α 1 and β (resp. α 1 and −β) are not weights of V ⊥ w . In this case every v ∈ V ⊥ w is a reducible polynomial contradicting the irreducibility of ψ. Again this implies Y w = ∅.
These observations determine a region U ⊂ X * (A) R such that Y w = ∅ if wx Q / ∈ U in the following way. Recall that for example −β is a weight of V ⊥ w if and only if This is equivalent to −β, wx Q − x P ≤ 1 2 − 1 5 and noting that β(x P ) = 1/2 this is furthermore equivalent to β, wx Q ≥ 1 5 .

Combining all cases in which
a union of two strips in the plane. It remains to prove that Y w is finite whenever wx Q ∈ U. By W L -symmetry it suffices to consider just one quadrant, e.g. the one defined by η, x > 0 and α 2 , x > 0. This leaves us with the case where wx Q lies in the above quadrant and additionally satisifes α 1 + 2α 2 , wx Q < 0 and the case where w = s α1+α2 is a simple reflection across the hyperplane perpendicular to α 1 + α 2 . Note that Y w ′ ⊂ Y w whenever w ′ is in the first case and w = s α1+α2 , so actually it suffices to prove that Y w is finite in the second case.
Let C ⊂ P 1 (x:y) × P 1 (u:v) = L/Q w be the curve defined by ψ = x 3 u + y 3 v = 0. If ℓ −1 ψ ∈ V * w then ψ(ℓ mod Q w ) = 0. Moreover in this case the projection π : C → P 1 onto the second factor (which is a finite map of degree 3) is ramified in ℓ mod Q w . Thus Y w is contained in the ramification locus of π which is finite.
Proof. This is immediate from the proof of Proposition 3.4.3. In this case the space of functions F is one-dimensional and the statement follows. Corollary 4.2.5 implies the following geometric version of the above statement.

5.2.3.
Corollary. There is a Hecke eigensheaf A π on Bun G2 (Q 0 , P ++ ∞ ) with semisimple eigen G 2 -local system E π . Under the assumptions in §4.3 the local system E π is cohomologically rigid.
6.1.1. In this section and the next, we assume that G over F is split and simply-connected. We consider the special case where P ∞ = G(O ∞ ). The reductive quotient of P ∞ over k is G; by abuse of notation we will also denote G by G, T by T , etc.
In this case, the grading on g is trivial, and ψ ∈ g = g(−1). Extending k if necessary, we may assume ψ ∈ Lie T . Then L ψ = G ψ is a Levi subgroup of G. We will use P ψ to denote a parabolic subgroup of G containing G ψ as a Levi subgroup. Note that only the associate class of P ψ is well-defined.
Recall that Q denotes another parabolic subgroup of G, the level at 0, chosen in such a way that any Borel subgroup B ψ ⊂ G ψ acts on the partial flag variety G/Q with an open almost free orbit. This is equivalent to requiring that G/P ψ × G/Q is a spherical G-variety and where Φ G is the set of roots of G and L Q is the Levi quotient of Q.
6.1.2. Stembridge [Ste03] has classified pairs of parabolic subgroups (P ψ , Q) such that G/P ψ × G/Q is G-spherical. In type A and C this was preceded by work of Magyar-Weymann-Zelevinsky, see [MWZ99] and [MWZ00]. In this classification, the following are the ones that satisfy the dimension equality (6.1).
There are no examples of exceptional types.
6.1.3. Notation. In the sequel we will concentrate on the case where G is one of the groups SL(V ), Spin(V ) or Sp(V ), for some finite-dimensional vector space V over k (char(k) = 2) equipped with a quadratic form in the case G = Spin(V ) or a symplectic form in the case G = Sp(V ). For d ≥ 1, write P d ⊂ G for the stabilizer of a d-dimensional subspace, isotropic in the case outside type A. Similarly, for 1 ≤ d < d ′ , let P d,d ′ denote the stabilizer of a d-dimensional subspace inside a d ′ -dimensional subspace, both being isotropic outside type A.
For parabolic subgroups P ′ and P ′′ of G, we write (P ψ , Q) ∼ (P ′ , P ′′ ) to denote that P ψ is conjugate to P ′ and Q is conjugate to P ′′ .
Below we often base change to k without changing the notation. (1) Any n, (P ψ , Q) ∼ (P n , P n ) (Siegel parabolic);
6.5. Type D n , [Ste03, Corollary 1.3.D.]. Let G = Spin(2n) and let G be the split form of G over F . Note that there are two conjugacy classes of n-dimensional isotropic subspaces (permuted by O(2n)) whose stabilizers we simply denote by P n (two conjugacy classes of maximal parabolics of G). The action of B ψ on G/Q has an open orbit with finite stabilizers if we have one of the following.
6.5.1. Theorem. Assume ψ and Q are in any of the above cases. Recall that ψ ∈ Lie T so that T ψ = T .
(2) All cases in type A and D with χ as in (1) are strict euphotic automorphic data. In type B and C no case is strict.
The proof of Theorem 6.5.1 is carried out in the following section.
6.5.2. Remark. The condition on χ may be described more explicitly in each type.
Type A n . The character χ on the diagonal torus T of SL(V ) is given by a collection of characters χ 1 , . . . , χ n of k × modulo simultaneous multiplication by the same character of k × . Any maximal Levi subgroup is isomorphic to S(GL a × GL b ) with a + b = n and a, b > 0. Its center is the subtorus given by the image of the embedding (a, b), and z b ′ appears a times, z −a ′ appears b times. We therefore require for any non-empty subset I ⊂ {1, . . . , n} of cardinality a and with non-empty complement J of cardinality b that ( i∈I gcd(a,b) .
Types B n , C n , D n . Identify T ∼ = G n m in the usual way, and write χ = (χ 1 , · · · , χ n ). The maximal Levi subgroups are of the form GL a × G ′ where G ′ is a classical group of rank n − a of the same type as G (in the case of type D n , a = n − 1). The connected centers of maximal Levi subgroups are the images of maps G m → G n m , z → (ϕ 1 (z), · · · , ϕ n (z)) where ϕ i (z) is either 1 or z or z −1 . Therefore the condition on χ is that, for any disjoint subsets I J ⊂ {1, 2, · · · , n} such that I ∪ J = ∅, we have i∈I Here, when I or J is empty, the corresponding product is 1.
6.6. Stabilizers on Hessenberg varieties. Recall that V ⊥ w = α,wxQ <1 g α which we will denote by g w . Then By definition these spaces are Hessenberg varieties as defined for example in [DMPS92]. The subvector space g w of g is automatically a Hessenberg space, i.e. it is stable under the adjoint action of the parabolic subgroup Q w and it contains its Lie algebra q w . For classical groups, Hessenberg varieties may be described concretely in terms of (isotropic) flags. Let V be a finite-dimensional vector space over k and let G be SL(V ), Spin(V ) or Sp(V ) where we endow V with a symmetric bilinear (resp. symplectic) form −, − . Then ψ ∈ g is an anti-self adjoint endomorphism of the vector space V and the condition ψ ∈ g g w may be translated into the condition that if 0 ⊂ F 1 ⊂ · · · ⊂ F r ⊂ V is the flag corresponding to g we have ψ(F i ) ⊂ F h(i) for a non-decreasing function h satisfying h(i) ≥ i associated to the space g w , cf. [Tym06,§2].
In this section we study the stabilizers of a Borel subgroup B ψ of G ψ acting on the Hessenberg variety Y w . The goal is prove that for w = id the Hessenberg variety Y w is a spectrally meager G ψ -variety. 6.6.1. Lemma. Let Q ⊂ G be a parabolic subgroup with corresponding Lie algebra q and U ⊂ g a subspace containing q and which is stable under the adjoint action of Q. Assume that there is a parabolic subgroup P of G with Lie algebra p such that Q ⊂ P and U ⊂ p. Let ψ ∈ U be semisimple and let G ψ be the centralizer of ψ in G. Then the G ψ -variety Y ψ (Q, U ) = {g ∈ G/Q | ψ ∈ g U } is spectrally meager.
Proof. Let B ψ ⊂ G ψ be a Borel subgroup. Denote by e ∈ G the identity element. It suffices to prove that the stabilizer of eQ contains a non-trivial torus. Indeed, if gQ ∈ Y ψ (Q, U ) we have ψ ∈ g U ⊂ g p and the assumptions of the Lemma are satisfied for g Q, g U and g P . Since Stab B ψ (gQ) = B ψ ∩ g Q = Stab B ψ (e · g Q) we may conclude.
We will argue in two steps. Let M be the Levi quotient of P and m its Lie algebra. Consider the map induced by the projection G/Q → G/P whose fiber above eP is identified with M/Q M where Q M is the image of Q in M . Since ψ is semisimple and ψ ∈ p, the stabilizer P ψ of ψ in P is a parabolic subgroup of G ψ and its Levi quotient P ψ /U ψ coincides with the stabilizer Mψ of the image of ψ in m.
Write Y ′ = π −1 (eP ) ⊂ Y ψ (Q, U ). The parabolic P ψ acts on Y ′ and the action factors through Mψ. Since Y ′ ⊂ M/Q M we know that the center Z(M ) stabilizes every point of Y ′ . Thus H := G ψ ∩ Q contains a group of the form Z · U ψ where Z is a torus surjecting onto Z(M ). Let B ψ ⊂ G ψ be a Borel subgroup. The second step is to analyze the action of H on G ψ /B ψ . Using the Bruhat decomposition we write where W ψ is the Weyl group of Mψ. Since H acts on each cell we only need to consider one such cell P ψ wB ψ /B ψ . Note that P ψ wB ψ /B ψ ∼ = P ψ /(P ψ ∩ w B ψ ) and Stab H (pwB ψ ) = Stab H (pP ψ ∩ w B ψ ). Let H = H/U ψ . This acts on Mψ and we have that is surjective (for the action ofH on Mψ/B). HereB denotes the image of P ψ ∩ w B in Mψ. Finally, since Mψ/B is the full flag variety of Mψ every point is stabilized by the center Z(M ) ⊂H of M . 6.6.2. Corollary. Let Q, U and ψ be as in the previous Lemma. Assume that there is a maximal parabolic P with Lie algebra p such that Q ′ = Q ∩ P is a parabolic subgroup of G and ψ ∈ p. Then Y ψ (Q, U ) is spectrally meager for the G ψ -action.
For any point y ∈ Y ψ (Q ′ , U ′ ) denote byȳ the image under this morphism. We have that Stab B ψ (y) ⊂ Stab B ψ (ȳ), so it suffices to prove the claim for eQ ′ ∈ Y ψ (Q ′ , U ′ ). This immediately follows from the previous Lemma 6.6.1. 6.6.3. Remark. In particular for flag varieties of classical groups to prove that Y w is spectrally meager it suffices to prove that for any (isotropic) flag F • ∈ Y w we can refine it to a flag such that one of the (isotropic) subspaces in the resulting flag is ψ-stable. From the proof of Lemma 6.6.1 it follows that in this case the stabilizer of any point in Y w contains the center of the Levi subgroup of the maximal parabolic subgroup stabilizing the ψ-stable space.
Denote by Φ the set of roots of G and let ∆ = {α 1 , . . . , α r } be the positive simple roots. By finite Weyl group symmetry we may and will assume that wx Q is dominant. 6.6.4. Corollary. Assume that there is a simple root α i such that α, wx Q ≥ 1. Then Y w is spectrally meager.
Proof. Let P be the maximal proper parabolic subgroup of G containing Q w but not the root subgroup corresponding to α i . If g w ⊂ p = Lie P , then g αi ⊂ g w and α i , wx Q < 1. Vice versa if α i , wx Q ≥ 1, then g w ⊂ p. Therefore Lemma 6.6.1 proves the claim.
This shows that we can restrict ourselves to the study of Y w for w such that 0 ≤ α i , wx Q < 1 for all simple roots α i ∈ ∆.

Detailed analysis of stabilizers
In the following we will carry out a case-by-case analysis of the cases with an almost free open orbit listed in §6. We will show that any flag in any Hessenberg variety Y w for w = id may be refined to a flag s.th. one of the spaces appearing in it is ψ-stable. This will prove the first part of Theorem 6.5.1. Strictness in types A and D is proved in Section 7.5. 7.0.1. Remark. Let V be a vector space over k. In types B, C and D we will denote the bilinear form by −, − . We will often make use of the fact that g is the Lie algebra of endomorphisms of V which are anti-selfadjoint with respect to the given bilinear form. In particular the non-zero eigenvalues of ψ occur in pairs x, −x and the eigenspaces of x and −x are dual with respect to the given bilinear form. For any non-zero x the eigenspace V (x) is isotropic and orthogonal to all other eigenspaces apart from V (−x). In types B,C and D we will always denote the eigenvalues by ±x (and ±y in a single case in type D).
Since ψ is semisimple we can decompose any vector v according to the eigenvalues of ψ and if x is an eigenvalue, v x is the summand of v in the x-eigenspace. When working with flags F • of a vector space V we will denote flags and the condition on them imposed by ψ in the following way By this we mean that ψ(F k ) ⊂ F l and the indices denote the dimensions of the spaces.
7.1. Type A n−1 . Let V be a finite-dimensional k-vector space and G = SL(V ). We identify X * (S) ∼ = {k = (k i ) ∈ Z n | n i=1 k i = 0} such that for the standard basis e * 1 , . . . , e * n of Z n we have α ∨ i = e * i − e * i+1 for the simple roots α i . We write wx Q = (x 1 , . . . , x n ) in coordinates for that basis. Recall that The parabolic Q w is determined by the roots that evaluate non-positively on wx Q . The space g w contains the root space g α for any root α if and only if α, wx Q < 1. Let Q ′ be the opposite parabolic to Q, so that −x Q = x Q ′ . To describe the Hessenberg varieties explicitly in terms of flags we will instead work with the variety Y −w defined by the point −wx Q . This will only switch around the type of Q.
Similarly we define g −w so that g α ⊂ g −w if and only if and only if α, wx Q > −1. Write α ij = j−1 k=i α k for and α ji = −α ij for i < j. We have α ji , wx Q = x j − x i and g αji ⊂ g −w if and only if α ji , wx Q > −1 or in other words if and only if x i − x j < 1. In terms of ψ and a flag 0 ⊂ F k ⊂ · · · ⊂ F r ⊂ V the condition g αji ⊂ g −w is satisfied if and only if ψ(F i ) ⊂ F j . We can therefore read off the condition imposed on any flag F • ∈ Y w from the coordinates of wx Q . The number of different entries in wx Q determines the number of different spaces in the flag. The numbers of entries that coincide determine the dimensions of the associated graded spaces. We label the associated graded by the entries of wx Q . For example if n = 5 and wx Q = (x 1 , x 2 , x 2 , x 3 , x 4 ) we have and ψ is allowed to map F 1 to any space whose associated graded is labelled by a number y such that x 1 − y < 1. The Weyl group acts on X * (T) ⊗ Q by permutation of the coordinates and the coroot lattice by integer translations with vectors (k 1 , . . . , k n ) such that n i=1 k i = 0. The arguments in each case work for any parabolic subgroup Q corresponding to the partition λ Q . Therefore we only argue for one of these. 7.1.1. Case (1), 6.2. The barycenter of Q 0 is x Q = ( n−1 2n , − 1 2n , . . . , − 1 2n ). If wx Q = x Q we find that wx Q has at least four different entries. Mod Z only one entry may be congruent to n−1 2n , all others are congruent to − 1 2n . Four of them being pairwise different forces two consecutive entries to have a difference of 1. Therefore the corresponding space in the flag has to be ψ-stable. 7.1.2. Case (2), 6.2. The barycenter is x Q = ( n−1 2n , n−3 2n , . . . , 3−n 2n , 1−n 2n ). Let wx Q = (x 1 , . . . , x n ) be dominant and not equal to x Q . Assume that there is an index i 0 such that x i0 − x i0+1 ≥ 2/n. Then we have Since any flag F • ∈ Y w is a full flag this implies that ψ(F 1 ) ⊂ F j for some j ≤ n − 1. Let v ∈ F 1 \ {0} and write v = v x + v y according to the eigenvalues of ψ. Assume the x-eigenspace V (x) is the one-dimensional eigenspace.
If v x = 0, F 1 is ψ-stable and we are done. If We may therefore assume that x i −x i+1 ≤ 1/n. Write x i = q i +k i with q i ∈ { n−1 2n , n−3 2n , . . . , 3−n 2n , 1−n 2n } and k i ∈ Z. We have n − 1 n we have the following three cases implies that q i = 1−n 2n and q i+1 = n−1 2n and vice versa in the second case. In particular, since the q i are pairwise different, only one of the first two cases may appear and it can only happen for a single index. This implies that the integer vector (k 1 , . . . , k n ) has only two distinct entries, but this contradicts the assumption n i=1 k i = 0. Therefore the only case in which . . , − m 2n ) with both entries occuring m-times. If wx Q = x Q and wx Q = (x 1 , . . . , x n ) is dominant, then it has at least four different entries. The condition 0 ≤ α i , wx Q ≤ 1/2 implies that any flag in Y w contains a part of the form , so for any w = w x + w y + w z ∈ F 2 we get w x + w y ∈ W and xw x + yw y ∈ W , i.e. w x , w y ∈ W , proving the claim.
7.1.4. Case (4), 6.2. Here we have with entries occurring m-times, m − 1 times and one time in this order. We may assume that for x Q = wx Q = (x 1 , . . . , x n ) we have 0 ≤ x i − x i+1 ≤ 1/3, because otherwise we can insert a space of the form F + ψ(F ). In addition we know that wx Q must have at least five different entries, say y 1 , . . . y 5 and their classes mod Z need to be ordered as follows may only appear once. Write y 1 = m+1 3n + k 1 and so on. The condition 0 ≤ y i − y i+1 ≤ 1/3 implies that k 1 = k 2 = k 3 and k 4 = k 5 = k 3 − 1. Since all other entries are congruent to m+1 3n or − m−1 3n mod Z we conclude that the integer vector by which we translate has the shape (k, . . . , k, k − 1, . . . , k − 1), a contradiction to the sum of these entries vanishing.
7.1.5. Case (5) & (6), 6.2. The arguments in these cases are the same as in the previous two. 7.1.6. Case (7), 6.2. The barycenter is x Q = (1/3, 1/3, 0, 0, −1/3, −1/3). We may assume that for wx Q = (x 1 , . . . , x 6 ) we have 0 ≤ x i − x i+1 ≤ 1/3. Otherwise there is an index i such that x i − x i+2 = 1 and ψ(F i ) ⊂ F i+1 . In that case since ψ has only two eigenvalues we may insert the stable space F i + ψ(F i ). Now assume that all entries of wx Q = (x 1 , . . . , x 6 ) are pairwise different. In that case the classes of the entries must be ordered as follows because otherwise we will find two successive entries whose difference is at least 2/3. If x 1 = 2/3 + k 1 and so on where k i ∈ Z then we find that k 1 = k 2 = · · · = k 6 . This contradicts the condition 6 i=1 k i = 0. We are therefore reduced to the following situation. For wx Q = x Q we always have exactly four different proper subspaces and ψ maps as follows We are left to consider two cases, either dim(F 1 ) = 1 or dim(F 1 ) = 2.
In the first case we assume that F 1 is not already stable. Choose a non-zero vector v ∈ F 1 and extend it to a basis v, w of F 2 . Write v = v x + v y according to the eigenvalues of ψ (where the y-eigenspace V (y) has dimension 2) and similarly for w. The space F 2 + ψ(F 2 ) is spanned by v x , w x , v y and w y . If v y and w y are linearly independent, V (y) ⊂ F 2 + ψ(F 2 ) ⊂ F 4 and hence F 4 is stable. If they are linearly dependent, F 2 + ψ(F 1 ) is ψ-stable and we may insert that.
In the second case choose a basis v, w of F 2 . Then with the same notation as before F 2 + ψF 2 contains v x , w x , v y and w y . If v y and w y are linearly dependent F 2 contains a non-zero eigenvector and we may insert the line spanned by it. If not, F 2 + ψ(F 2 ) contains the whole y-eigenspace V (y) and we are done. 7.1.7. Case (8), 6.2. The barycenter is x Q = (1/6, 1/6, 1/6, 1/6, −1/3, −1/3). For wx Q = x Q we will find at least four different entries x 1 , . . . , x 4 . We may assume as before that 0 ≤ x i − x i+1 ≤ 1/2, because otherwise a space in the flag will already be ψ-stable. Therefore wx Q will have exactly four different entries, say y 1 , y 2 , y 3 , y 4 and the classes mod Z will be ordered either as (−1/3, 1/6, −1/3, 1/6) or (1/6, −1/3, 1/6, −1/3). The associated graded spaces that are labelled by −1/3 have to be one-dimensional and cannot occur consecutively. Therefore we get the following list of possible flags that we need to consider: The indices indicate the dimensions of the spaces and the endomorphism ψ always maps a space to the next one. In the first three cases we may insert the spaces The last two cases are similar to each other. We will present the argument only for the fourth case. First note that we may assume that F 4 + ψ(F 4 ) = F 5 since otherwise F 4 is already ψ-stable. Thus We may assume that they are all non-zero because otherwise F 1 + ψ(F 1 ) is ψ-stable. Since F 4 ∩ ψ(F 4 ) = 0 there is a vector w ∈ F 4 such that ψ(w) ∈ F 4 . Thus F 4 + ψ(F 4 ) contains w x , w y , w z . If any pair (w x , v x ), (w y , v y ) or (w z , v z ) is linearly independent, the corresponding eigenspace lies in F 4 + ψ(F 4 ) and this would imply that F 4 + ψ(F 4 ) is ψ-stable.
We therefore assume that the above pairs are all linearly dependent. Since w = 0 we may additionally assume that w x = 0 and hence we can write v x = λw x for some non-zero λ. Since F 4 ⊃ F 1 + ψ(F 1 ) it contains (x − y)v x + (z − y)v z and also (x − y)w x + (z − y)w z . In particular λw z − v z ∈ F 4 and therefore if λw z − v z = 0 we get v z ∈ F 4 and F 1 + ψ(F 1 ) + ψ 2 (F 1 ) ∈ F 4 . If λw z − v z = 0 we can do the same for v y and find that actually v = λw. Since w ∈ F 4 ∩ ψ(F 4 ) is arbitrary this implies dim(F 4 ∩ ψ(F 4 )) = 1 and hence ker(ψ) ∩ F 4 = 0, i.e. say z = 0 and F 4 contains a 0-eigenvector u 0 . Again if u 0 and v 0 are linearly independent, then V (0) ⊂ F 4 + ψ(F 4 ) and if they are dependent, then F 1 + ψ(F 1 ) + ψ 2 (F 1 ) ⊂ F 4 , so we are done. 7.2. Type B n . Let V be a finite dimensional k-vector space of odd dimension equipped with a nondegenerate symmetric bilinear form −, − and G = Spin(V ). As in type A we use the evident identification X * (T) ∼ = Z n to write wx Q ∈ X * (T) in coordinates. A (partial) flag in type B stabilized by a parabolic Q w can be thought of as a flag of isotropic spaces together with their complements The condition that ψ ∈ g w imposes the same conditions as in type A for the flag In addition, mapping from a space F i to F ⊥ j is determined by the value of the root e i + e j (or e i for j = i) and hence for wx Q = (x 1 , . . . , x n ) we may label the flag above as follows Again ψ is allowed to map F i to any space whose associated graded is labelled by y such that x i − y < 1.
The finite Weyl group acts on X * (T) ⊗ Q by arbitrary permutations and sign changes of the coordinates. We may translate using the coroot lattice, i.e. by integer vectors whose coordinates sum to an even number. We can therefore produce a finite list of possible cases for wx Q (using the condition that 0 ≤ α i , wx Q < 1 for all simple roots). 7.2.1. Case (1), 6.3, (P ψ , Q) ∼ (P n , P n ). In this case the barycenter is x Q = (1/2, . . . , 1/2). Therefore wx Q will have coordinates in 1/2 + Z. The condition 0 ≤ α i , wx Q < 1 immediately implies that all coordinates of wx Q have to be equal to 1/2, i.e. wx Q = x Q is the only possibility and there is nothing to prove. 7.2.2. Case (2), 6.3. This case is similar to Case (1). Since x Q = (1/2, 1/2), there is no other possibility than wx Q = x Q . 7.2.3. Case (3), 6.3. We have x Q = (1/2, 0) and the only non-trivial possibility for wx Q is wx Q = (1, 1/2). In this case the flag is and we may insert F 1 + ψ(F 1 ).

7.2.4.
Case (4), 6.3. We have x Q = (1/2, 1/2, 0). The possible non-trivial cases for wx Q are (i) wx Q = (1, 1/2, 1/2), (ii) wx Q = (2, 3/2, 1/2). In case (i) the corresponding flag is Choose a non-zero vector v ∈ F 1 . Denote the non-zero eigenvalues of ψ by x and −x and write v = v x + v 0 + v −x according to the eigenspace decomposition of V . We get that v, ψ 2 v = 0 and from that it follows that v x , v −x = 0. Since P ψ ∼ P 1 , dim V (x) = dim V (−x) = 1 and either v x = 0 or v −x = 0. Assume that v −x = 0. In that case W = F 1 + ψF 1 contains v = v 0 + v x and v x . This space therefore has a basis of eigenvectors and is ψ-stable. Since W ⊂ F 3 it is automatically isotropic and we may insert this space into the given flag.
7.3. Type C n . Let V be a finite dimensional k-vector space of even dimension equipped with a symplectic form −, − and G = Sp(V ). As before we may write wx Q in coordinates using the identification X * (T) ∼ = Z n . Similar to type B we consider flags and label them the same way, but without the middle step. The endomorphism ψ is allowed to map F i to any space whose associated graded is labelled by y such that x i − y < 1.
The finite Weyl group acts by arbitrary permutations and sign changes on the coordinates of wx Q and the coroot lattice by arbitrary integer vector translations. 7.3.1. Case (1), 6.4. The barycenter is x Q = (1/4, . . . , 1/4). Therefore any coordinate of wx Q = (x 1 , . . . , x n ) lies in 1/4 + Z or 3/4 + Z. Therefore x i − x j = 0 or x i − x j = 1/2 for any i ≤ j. By the condition 0 ≤ α n , wx Q < 1 we conclude that x n = 1/4. Therefore if all coordinates agree, wx Q = x Q . For flags of the form since the difference of any two distinct coordinates is 1/2. Since P ψ ∼ P n , the space W = F + ψ(F ) is stable and can be inserted into the flag. We can use this argument whenever we have at least three distinct coordinates. In the case that we only have two distinct coordinates the flag is of the form . Now since P ψ ∼ P n is the Siegel parabolic, ψ has eigenvalues say x and −x and it follows that ψ 2 is scalar multiplication by x 2 . Therefore − v ′ , ψ 2 (w ′ ) = −x 2 v ′ , w ′ = 0 and we find that W is isotropic. 7.3.2. Case (2), 6.4. We have x Q = (1/3, 1/3, 0). The possible non-trivial cases for wx Q are (i) wx Q = (2, 1/3, 0), We claim that W = F 2 + ψ(F 1 ) is isotropic and that if neither F 1 nor F 2 are ψ-stable then F 2 + ψ(F 1 ) is ψ-stable. Because ψ(F 1 ) ⊂ F 2 it suffices to prove that ψv, ψv ′ = 0 for any v, v ′ ∈ F 1 . This is clear since To prove W is stable, let v ∈ F 1 be non-zero. We have i.e. v x = 0 or v −x = 0. Assume we have v −x = 0 and let w = w x + w 0 + w −x = F 2 be any vector. Then If v x = 0, then F 1 is ψ-stable, so we get that w −x = 0 and F 2 + ψ(F 1 ) contains v x , v 0 and w 0 . If v 0 and w 0 are linearly dependent, then V (x) ⊂ F 2 and by what we said above no vector in F 2 has a component in the (−x)-eigenspace. This implies that F 2 is ψ-stable. We may therefore assume that v 0 and w 0 are linearly independent and hence F 2 + ψ(F 1 ) is spanned by eigenvectors. In case (ii) the flags are of the form Choose a basis v, w for F 2 . Since dim V (x) = 1 there are scalars λ, µ such that λv x + µw x = 0 and not both of them are zero. Let u = λv + µw. Since v, w are a basis, u = 0. If u −x = 0 then u 0 = 0 and hence F 2 contains the eigenline spanned by u 0 . If u −x = 0 then for any u ′ ∈ F 2 we have In this case F 2 + ψ(F 2 ) is ψ-stable and isotropic.
The same argument works for ψ 2 v, ψ 2 w = 0 and thus W is isotropic. If F 2 is not ψ-stable, then In case (iv) the flags are and we may take F 1 + ψ(F 1 ) + ψ 2 (F 1 ) which is isotropic with the same argument as in case (iii). In case (v) consider For case (vi) we get full flags and we may simply insert F 1 + ψ(F 1 ) + ψ 2 (F 1 ).
Since also v, ψv = 0 we find that v x , v −x = 0. This implies that W is isotropic.
In case (ii) we have flags As before we find that v x , v −x = 0 and hence ψv, ψ 2 v = 0. Thus ψ(F 2 ) ⊂ F ⊥ 2 and we may insert F 1 + ψ(F 1 ) + ψ 2 (F 1 ). In case (iii) we consider This works the same as case (ii).
In case (iv) we have We have F 2 ∩ ψ(F 2 ) = 0 and there is a v ∈ F 2 such that ψ(v) ∈ F 2 . Therefore F 2 is spanned by two vectors of the form w x + w 0 and u −x + u 0 and F 2 + ψ(F 2 ) is ψ-stable.
7.4. Type D n . Let V be a finite dimensional k-vector space of even dimension equipped with a nondegenerate symmetric bilinear form −, − and G = Spin(V ). Using the identification X * (T) ∼ = Z n we write wx Q in coordinates. A full flag in type D n is the data of a flag of isotropic spaces with two Lagrangian subspaces F n and F ′ n such that F n ∩ F ′ n = F n−1 . The labelling is done as before. We denote the non-zero eigenvalues of ψ by ±x (and ±y in the case n = 4). The finite Weyl group acts by arbitrary permutations and even sign changes on the coordinates of wx Q . The coroot lattice acts by translation with integer vectors whose coordinates sum to an even integer. 7.4.1. Case (1), 6.5. We have x Q = (1/2, 1/4, 0, 0). Because ψ has only two eigenvalues we may eliminate cases in the orbit of x Q for which ψ maps a space to the next one. This leaves us with the following possibilities.
In case (ii) we have flags Then ψ(F 1 ) ⊂ F 2 and we may take F 1 + ψ(F 1 ). If F 2 + ψ(F 1 ) = F 3 , then F 3 ⊂ F 2 + ψ(F 2 ) and this space is isotropic by the same argument as before. Therefore this space is a Lagrangian subspace containing F 3 and as such it is either F 4 or F ′ 4 and we are done. 7.4.2. Case (2), 6.5. We have x Q = (1/4, 1/4, 1/4, 1/4) and the possible non-trivial cases are (i) wx Q = (3/4, 3/4, 1/4, 1/4), (ii) wx Q = (5/4, 3/4, 1/4, −1/4). In case (i) we consider flags Choose a basis u, v ∈ F 2 . We first want to prove that v ′ x = 0 for all v ′ ∈ F 2 . Since dim V (x) = 1 there are scalars λ x , µ x which are not both zero such that µ x u x + λ x v x = 0. Define w = λ x v + µ x u. Then w = 0 and since w, ψ 2 w = 0 we find that w y , w −y = 0. This implies that w y = 0 or w −y = 0. Without loss of generality we may assume that w y = 0, i.e. w = w 0 + w −y + w −x . Now ψw, ψ 2 w ∈ F ⊥ 2 implies that w −x ∈ F ⊥ 2 and w −y ∈ F ⊥ 2 . If w −x = 0 then if also w −y = 0 the space F 2 contains w 0 = 0 and in particular the line spanned by it and we may insert that. So either w −x = 0 or w −y = 0 and we may assume that w −x = 0 (otherwise we end up eliminating the y-component in In the second step we will prove that also v ′ y = 0 for all v ′ ∈ F 2 . As before there are scalars λ x , µ −x such that The same argument as before shows that w ′ −y ∈ F ⊥ 2 and that we may assume it's non-zero. This proves that v ′ y = 0 for all v ′ ∈ F 2 . A simple calculation now shows that W = F 2 + ψ(F 2 ) + ψ 2 (F 2 ) is ψ-stable and isotropic. If neither F 2 nor F 2 + ψ(F 2 ) are already stable, then W is either F 4 or some other Lagrangian and we may insert that space.

The last term vanishes because for any
since the pairing is symmetric.
Cases In case (i) we consider flags We claim that W = F 2 + ψ(F 2 ) + ψ 2 (F 2 ) is isotropic. Note that we may assume ψ to be injective on F 2 , because otherwise we can insert its kernel. It suffices to show that ψv, ψ 2 w = 0 and that ψ 2 v, ψ 2 w = 0 for all v, w ∈ F 2 . We have ψv, ψ 2 w = ψv, ψ 2 w + ψv, x 2 w 0 = x 2 ψv, w = 0 and similarly we get ψ 2 v, ψ 2 w = 0. This implies that the dimension of W is at most 5 and hence we have In the first case F 2 contains a non-zero vector w such that ψw ∈ F 2 and hence it contains 2xw x + xw 0 and −2xw −x − xw 0 . We may assume that w x , w 0 , w −x = 0 because otherwise F 2 contains an eigenvector. Now F 2 is spanned by 2xw x + xw 0 and −2xw −x − xw 0 and hence F 2 + ψ(F 2 ) is ψ-stable.
In case (ii) we consider flags of the form If this space itself is not already stable, then F 4 + ψ(F 4 ) + ψ 2 (F 4 ) is Lagrangian and we may use it to refine the flag. In case (iii) we may simply insert the space F 1 + ψ(F 1 ) + ψ 2 (F 1 ). Case (iv) is similar to case (ii), we may insert the space F 4 + ψ(F 4 ) + ψ 2 (F 4 ).
In case (v) we have flags We may insert F 2 + ψ(F 2 ) if it is equal to F 4 . Otherwise F 2 ∩ ψ(F 2 ) = 0 and there is a vector w ∈ F 2 such that ψ(w) ∈ F 2 . We then find that F 2 + ψ(F 2 ) contains w x , w −x and w 0 . If any of these vanishes, F 2 contains an eigenvector. Otherwise F 2 + ψ(F 2 ) is generated by these three vectors.
Cases (vi) − (ix) are easy, simply use the spaces F + ψ(F ) + ψ 2 (F ) where F always denotes the first-non zero step of the flag. 7.5. Stabilizers on the unit coset. In the previous section we proved the first part of Theorem 6.5.1. In the following we analyze the unit coset in more detail. We prove strictness in types A and D.
7.5.1. Remark. The covering Spin(V ) → SO(V ) induces an isomorphism on flag varieties. The stabilizers only differ by the kernel of the covering. Therefore in types B and D we may (and will) work with SO(V ) in the following. 7.6. Fix a maximal torus T and a Borel B containing it. We may assume Q ⊃ B and ψ ∈ t. Then G ψ ⊃ T , and we may assume T ⊂ B ψ ⊂ B. Let P ψ = BG ψ ; this is a standard parabolic subgroup of G containing G ψ as a Levi subgroup. Let N, N ψ , N ψ be the unipotent radicals of B, P ψ and B ψ respectively.
Let W Q be the Weyl group of the Levi subgroup of Q containing T . The double cosets B\G/Q are parametrized by W/W Q . Let w ∈ W , and letẇ be any lift of w to N G (T ). Then we can identify the B-orbit ofẇQ/Q with N/N ∩ w Q (write w Q = Ad(ẇ)Q). The left translation action of B ψ = N ψ T on BẇQ/Q becomes left translation of N ψ on N/N ∩ w Q and the action of T by conjugation. Using the left N ψ -action, every B ψ orbit on N/N ∩ w Q intersects N ψ /N ψ ∩ w Q. On N ψ /N ψ ∩ w Q, there is the residual action of B ψ ∩ w Q by conjugation. Therefore, it suffices to show that the stabilizers of the action of B ψ ∩ w Q on N ψ /N ψ ∩ w Q by conjugation contain nontrivial tori, except in one case.
Let W ψ ⊂ W be the Weyl group of G ψ (with respect to T ). 7.6.1. Lemma. Let w 0 be the longest element in W with respect to the simple reflections defined by B. If w is not in the double coset W ψ w 0 W Q , then the action of T on N ψ /N ψ ∩ w Q has positive dimensional stabilizers.
Proof. Let N − Q be the unipotent radical of the parabolic of G opposite to Q and containing T . The inclusion To verify the claim, it suffices to show that the set of roots Φ(N ψ ∩ w N − Q ) does not span X * (T) Q rationally, unless w ∈ W ψ w 0 W Q . This can be checked case by case.
For example, consider the case where G = Sp(V ) and P ψ and Q are both Siegel parabolic subgroups. Let {e 1 , · · · , e n , e −n , · · · , e −1 } be a symplectic basis of V ; let T be the diagonal torus with respect to this basis. The above claim is equivalent to the following statement: let L, L ′ be two Lagrangian subspaces spanned by part of the basis. Let N L be the unipotent radical of the stabilizer of L, and similarly define N L ′ . Then as long as L = L ′ , the action of T on L ∩ L ′ has positive dimensional stabilizers.
This latter statement can be proved as follows. Identify N L with Sym 2 (L) using the symplectic form. Then N L ∩ N L ′ ⊂ N L is identified with the subspace Sym 2 (L ∩ L ′ ) of Sym 2 (L). If L = L ′ , let T 1 be the subtorus of T corresponding the basis elements e ±i such that e ±i / ∈ L ∩ L ′ . Then T 1 acts trivially on Sym 2 (L ∩ L ′ ) ∼ = N L ∩ N L ′ . 7.6.2. Remark. More precisely let U denote the rational span of Φ(N ψ ∩ w N − Q ). If U = X * (T ) Q then the stabilizer contains α∈Φ(G)∩U ker(α). The set Φ(G) ∩ U is the root system of a Levi subgroup of G and α∈Φ(G)∩U ker(α) is its center. 7.7. It remains to treat the case w ∈ W ψ w 0 W Q . Write w ∈ vw 0 W Q for some v ∈ W ψ . We need to consider the stabilizers of the conjugation action of B ψ ∩ vw0 Q on N ψ /N ψ ∩ vw0 Q. Conjugating by a lifting of v in N G ψ (T ), we may as well consider the action of v −1 B ψ ∩ w0 Q on N ψ /N ψ ∩ w0 Q. Let Q ′ = ( w0 Q) − be the opposite parabolic of w0 Q and let M = Q ′ ∩ w0 Q (common Levi of Q ′ and w0 Q).
we reduce to considering the action of B H on N ψ ∩ N Q ′ by conjugation (the whole H acts on N ψ ∩ N Q ′ by conjugation). We remark that after this reduction, the roles played by P ψ and Q ′ are symmetric.
Below we describe case-by-case the action of H on N ψ ∩ N Q ′ in linear algebra terms. In the following, V n , V ′ n always denote an n-dimensional space; when n = 1, we use G m (V 1 ), G m (V ′ 1 ) to denote the onedimensional torus that acts on V 1 , V ′ 1 by scaling. In all cases N ψ ∩N Q ′ is a vector group, and we describe it as a representation of H. The Borel subgroup B H of H acts on N ψ ∩ N Q ′ with an open orbit with finite stabilizer. In the following we analyze the orbits of B H acting on N ψ ∩ N Q ′ more precisely. Recall that by P d we denote the stabilizer of a d-dimensional subspace V d and by P d,d ′ the stabilizer of a flag 0 ⊂ We list the possible conjugacy classes of Q and the corresponding H-representation N ψ ∩ N Q ′ . 7.8. Type A. (P ψ , Q) ∼ (P m , P m,m+1 ) or (P m , P 1,m+1 ) or (P m+1 , P m,2m ) or (P m+1 , P m,m+1 ). Then 2 ) or its dual. Case (1) is clear for dimension reasons: any non-open orbit in N ψ ∩ N Q ′ has dimension less than that of T , hence has positive dimensional subtorus in the stabilizers. In addition it's easy to check that the stabilizer on the open orbit is just the center of SL n and that outside the open orbit the stabilizer always contains the center of a maximal Levi subgroup.
We reduce the cases (2a)(2b)(3a)(3b) to Case (1). We give the argument only for (2a). Let a : (a ⊕ b)) contains a Borel subgroup of the target. Therefore Stab BH (a, b) contains a nontrivial torus if a ⊕ b is not surjective (equivalently not an isomorphism). In the remaining case, we may assume a ⊕ b : ( (1) is not strict as it may have several relevant orbits. In Case (2) the stabilizer on the open orbit is µ 2 = {±id}, so it is not strict.
We consider Case (3). We identify If v is isotropic or x = 0 it's easy to see that the stabilizer of (v, x) in B H = B 3 × G m × G m contains a torus. Therefore assume that v is anisotropic and x = 0. Denote by , the symmetric bilinear form on The open orbit is given by those (v, x) where v / ∈ ℓ ⊥ and x = 0. It is easily verified that in this case the stabilizer is µ 2 = {±id} and hence this case is also not strict. 7.10. Type C. ( We consider Case (1). We may identify Sym 2 (V n ) with the space of quadratic forms on V * n . Let L = ker(q) ⊂ V * n , and P L ⊂ GL(V n ) be the parabolic subgroup stabilizing L. Then we have a natural map Stab BH (q) ⊂ B H ∩ P L → GL(L) whose image is a Borel subgroup of GL(L). Therefore if L = 0, Stab BH (q) contains a nontrivial torus. Now suppose L = 0, i.e., q is nondegenerate. We equip V n with the quadratic form induced from q, and still denote it by q. Let B H be the stabilizer of a complete flag F = (0 ⊂ V 1 ⊂ · · · ⊂ V n−1 ⊂ V n ). Consider the relative position of the flag F ⊥ = (0 ⊂ V ⊥ n−1 ⊂ · · · V ⊥ 1 ⊂ V n ) ((−) ⊥ is taken under the quadratic form q). If F and F ⊥ are not opposite, consider the first i ≥ 1 such that V i ∩ V ⊥ i = 0 (i.e., the first i such that q| Vi is degenerate), in which case ker(q|V i ) is 1-dimensional. Then Stab BH (q) → G m (ker(q|V i )) is surjective, hence Stab BH (q) contains a nontrivial torus. If F and F ⊥ are opposite, then q is the in the open B H -orbit of Sym 2 (V n ). The intersection of the stabilizers of F and F ⊥ is a maximal torus T of GL(V n ) and the stabilizer on the open orbit is the 2-torsion T [2] ∼ = µ n 2 . We consider Case (2). An element (λ, µ, X) ∈ H acts on a vector (v, x, y) as (λXv, λ 2 x, λµy). Let 7.11. Type D.
. We consider Case (1). An element (λ, µ, X) ∈ H acts on a vector (u, v, x) via (u, v, x) → (λXu, µXv, λµx). Let B 2 ∈ GL(V 2 ) be a Borel subgroup stabilizing a line ℓ ⊂ V 2 and B H = G m × G m × B 2 . If x = 0 the stabilizer of (u, v, x) contains a torus coming from the G m -factors. If u = 0 then the condition on X is equivalent to asking that it stabilizes the line spanned by v and Stab BH (0, v, w) contains the intersection of B 2 with the Borel subgroup stabilizing v . The same happens for v = 0 and we may therefore assume that u = 0, v = 0 and x = 0. The open orbit is given by those vectors (u, v, x) for which x = 0, u / ∈ ℓ, v / ∈ ℓ and u and v are linearly independent (i.e. X is contained in the intersection of three pairwise different Borel subgroups). It's easy to check that the stabilizer on the open orbit is µ 2 ∼ = {±id}. One may check explicitly that outside the open orbit the stabilizers contain tori which are contained in centers of maximal Levi subgroups. For x = 0 one finds a torus of the form Z(GL 2 ) in a Levi subgroup isomorphic to GL 2 × SO (4) and for x = 0 one obtains the center of a Levi subgroup isomorphic to G m × SO(6) (as long as we're outside the open orbit).
For Case (2) and let B H = Stab(ℓ)× Stab(F ) for a line ℓ ⊂ V 2 and a full flag F given by 0 ⊂ F 1 ⊂ F 2 ⊂ V 3 . If f is not surjective clearly Stab BH (f, ω) contains a torus. We therefore assume f is surjective and hence has a one-dimensional kernel 3 orthogonal to F with respect to the canonical pairing for V 3 and V * 3 . The open orbit is given by those (f, ω) for which F ⊥ and the above flag are opposite (i.e. X lies in a maximal torus of GL(V 3 )) and for which ω (considered as a 2-form on V * 3 ) is not contained in any of the duals of the planes 8.1.1. Setup. Let G = PGL(V ) for some vector space V over k of dimension n, and let G be the split form of G over F . Since all parahoric subgroups of G(F ∞ ) can be conjugated to be contained in G(O ∞ ) in this case, we may assume P ∞ ⊂ G(O ∞ ). For such P ∞ the corresponding Z/mZ-grading on g = ⊕ i∈Z/mZ g(i) is induced from a Z/mZ-grading on the vector space Conversely, any Z/mZ-grading on V with V i = 0 for all i ∈ Z/mZ arises from a parahoric subgroup Let Q 0 ⊂ GL(V 0 ) be a parabolic subgroup; let ψ 0 ∈ End(V 0 ) be a semisimple element. We assume • The pair (Q 0 , ψ 0 ) appears in the list of hyperspecial euphotic data of type A in §6.2.
• Let V 0 0 be the zero eigenspace of ψ 0 ; let V ′ 0 be the sum of nonzero eigenspaces. Then dim V 0 With these preliminary data, we construct Q and ψ as follows. Let Q ⊂ L be the parabolic subgroup Viewing ψ as a collection of maps V i → V i−1 for i ∈ Z/mZ, we then require it to restrict to isomorphisms and to restrict to zero on V 0 0 . Moreover we require that ψ m |V 0 = ψ 0 . 8.1.3. We check that in the above situation, B ψ acts on L/Q with an open orbit with finite stabilizers. Indeed, L/Q ∼ = GL(V 0 )/Q 0 , and L ψ ∼ = PGL(V 0 ) ψ0 (the centralizer of ψ 0 in PGL(V 0 )). Therefore we reduce to the case discussed in §6.2 for the group PGL(V 0 ). 8.1.4. Case (2). Take Q 0 to be the standard Iwahori subgroup of G(F 0 ) (i.e., Q ⊂ L is a Borel subgroup). Fix a decomposition P as a collection of maps V i → V i−1 , let it restrict to an isomorphism ℓ i ∼ → ℓ i−1 and be zero on V 0 i . 8.1.5. We check that in the above situation, B ψ acts on L/Q with an open orbit with finite stabilizers. We have L/Q = i∈Z/mZ Fl(V i ). We also have L ψ ∼ = i∈Z/mZ GL(V 0 i ) (an extra factor of G m acting on all the lines ℓ i gets cancelled after dividing by scalar matrices). Therefore The required property of the B ψ -action on L/Q follows from the same property for the B ψ,i -action on Fl(V i ), which is checked in case (2) of §6.2.
8.1.6. Remark. We expect case (2) to correspond to hypergeometric local systems with slope 1/m at ∞ and unipotent monodromy at 0. Rigid automorphic data corresponding to hypergeometric local systems are constructed in the work of Kamgarpour and Yi [KY20]. 8.2. Convention. In the exceptional cases, we always assume G is of adjoint type. We will indicate the type of P ∞ by coloring the affine Dynkin diagram of G(F ∞ ): the white nodes are simple roots of L = L P , and the black nodes are simple roots not contained in L.
When we describe L and V * P , we will use V i , V ′ i , W i , F i , etc. to indicate vector spaces of dimension i over k.  (2) L/Q = P(V ′ 3 ) or P ∨ (V ′ 3 ). In these cases B ψ = L • ψ acts on L/Q with an open orbit with stabilizer µ 2 2 .
8.5.1. Type of P ∞ : is an isomorphism onto the diagonal torus in the target with respect to the chosen bases. Same for the other two projections.
Potential choices of Q: L/Q is a partial flag variety of PGL(V 6 ) of type (3, 2, 1) (dimensions of associated graded of the partial flag, in any order). Now B ψ is a Borel of L • ψ , which projects isomorphically to a Levi of PGL(V 6 ) of type (3,3). The situation B ψ \L/Q appears as a special case of §6.2(4), from which we know that B ψ acts on L/Q with an open free orbit. 8.6. Type 2 E 6 . 8.6.1. Type of P ∞ : Choose a basis {e 1 , e 2 } for V 2 and a basis {x 1 , y 1 , x 2 , y 2 } for V 4 . Take ψ = x 1 y 1 ⊗ e 1 + x 2 y 2 ⊗ e 2 . Then the projection L • ψ → PGL(V 4 ) is an isomorphism onto the diagonal torus with respect to the basis {x 1 , y 1 , x 2 , y 2 }.
Potential choices of Q: L/Q = P(V 4 ) or P ∨ (V 4 ). It is clear that in both cases B ψ = L • ψ acts on L/Q with an open free orbit. 8.6.2. Type of P ∞ : In this case m = 4. We have L = (Spin(V 7 ) × SL(V 2 ))/∆µ 2 acting on V * P = ∆ 8 ⊗ V 2 where ∆ 8 is the 8-dimensional spin representation of Spin(V 7 ).
We have an embedding ϕ : Spin(V 7 )/P 3 ֒→ P∆ 8 , where Spin(V 7 )/P 3 classifies maximal isotropic subspaces in V 7 . Choose a splitting V 7 = W 3 ⊕ W ′ 3 ⊕ x 0 , where W 3 and W ′ 3 are maximal isotropic and paired perfectly to each other and both orthogonal to x 0 . Let {e 1 , e 2 } be a basis for  In this case m = 3. We have L ∼ = SL(V 6 ) × SL(V 3 )/(µ 2 × 1)∆µ 3 acting on V * P ∼ = ∧ 2 (V 6 ) ⊗ V 3 . Choose a basis {e 1 , e 2 , e 3 } for V 3 ; choose a splitting V 6 = W 1 ⊕ W 2 ⊕ W 3 into three 2-dimensional subspaces. Let θ i be a volume form on W i . Consider the element ψ = θ 1 ⊗e 1 +θ 2 ⊗e 2 +θ 3 ⊗e 3 . The projection L • ψ → PGL(V 6 ) is an isomorphism onto the Levi of SL(V 6 ) stabilizing the splitting V 6 = W 1 ⊕ W 2 ⊕ W 3 , and the projection L • ψ → PGL(V 3 ) has image equal to the diagonal torus with respect to the basis {e i }. Potential choices of Q: L/Q is the partial flag variety of PGL(V 6 ) with associated graded dimensions (4, 2) (in any order). The situation B ψ \L/Q appears in the example §6.2(8), from which we know that B ψ acts on L/Q with an open free orbit. 8.7.2. Type of P ∞ : and another case by symmetry. In this case m = 2. Then L is isogenous to (Spin + (V 12 ) × SL(V 2 ))/∆µ 2 acting on V * P ∼ = ∆ + 32 ⊗ V 2 , where Spin + (V 12 ) is one of the half-spin quotient of Spin(V 12 ) acting on its half-spin representation ∆ + 32 . Choose a basis {e 1 , e 2 } for V 2 . There is an embedding ϕ : Spin(V 12 )/P 6 ֒→ P∆ + 32 , where Spin(V 12 )/P 6 is the partial flag variety of one of the two families of Lagrangian subspaces in V 12 . Fix a splitting V 12 = W 6 ⊕ W ′ 6 into Lagrangians. Take  (1) L/Q = SO(V 12 )/P 3 is the partial flag variety classifying isotropic F 3 ⊂ V 12 . The situation B ψ \L/Q has been analyzed in §6.5(3).
(2) L/Q = SO(V 12 )/P 5 × P(V 2 ), where the first factor classifies isotropic F 5 ⊂ V 12 . We check the open orbit condition as follows. We first reduce to study the action of B 1 ψ (a Borel subgroup of SL(W 6 )) on Y = SO(V 12 )/P 5 . Note that Y classifies a pair of Lagrangians U 6 , U ′ 6 ⊂ V 12 such that dim(U 6 ∩ U ′ 6 ) = 5. We may assume U 6 is conjugate to W 6 . There is an open subset Y ′ ⊂ Y classifying those (U 6 , U ′ 6 ) such that U 6 is the graph of a skew-symmetric map a : W ′ 6 → W 6 . We may identify Y ′ with ∧ 2 (W 6 ) × P(W 6 ) (the choice of U ′ 6 is the same as choosing a hyperplane in U 6 , or in W ′ 6 ). The situation of B 1 ψ acting on Y ′ is essentially the same as case (1) of §7.9. (3) L/Q = SO(V 12 )/P 1,6 × P(V 2 ), where the first factor classifies isotropic F 1 ⊂ F 6 ⊂ V 12 for F 6 a Lagrangian in the same connected component of W 6 . The same argument as in the previous case reduces to the action of B 1 ψ on ∧ 2 W 6 × P ∨ (W 6 ), which is essentially the same as case (1) of §7.9. 8.7.3. Type of P ∞ : In this case m = 4. We have L isogenous to (SL(V 4 )×SL(V ′ 4 )×SL(V 2 ))/Γ acting on V * 8.8. Type E 8 . Type of P ∞ : In this case m = 5. We have L ∼ = (SL(V 5 ) × SL(V ′ 5 ))/µ 5 acting on V * P = ∧ 2 (V 5 ) ⊗ V ′ 5 , here the embedding µ 5 ֒→ SL(V 5 ) × SL(V ′ 5 ) is z → (z 2 id V5 , zid V ′ 5 ). Choose a basis {x i } 1≤i≤5 of V 5 , and a basis {e i } 1≤i≤5 of V ′ 5 . Take ψ = i∈Z/5Z x i−1 ∧ x i+1 ⊗ e i . Then the projection L • ψ → PGL(V 5 ) is an isomorphism onto the diagonal torus of PGL(V 5 ) with respect to the basis {x i }. Same for the other projection L • ψ → PGL(V ′ 5 ). Potential choices of Q: L/Q = P(V 5 ), P ∨ (V 5 ), P(V ′ 5 ) or P ∨ (V ′ 5 ). In all these cases B ψ = L • ψ acts on L/Q with an open free orbit.

Appendix A. Factorizable module categories
In this appendix we define and classify semisimple factorizable module categories over a neutral Tannakian category with coefficients. We will apply the classification result here to the category of semisimple perverse sheaves in the automorphic category D(ψ, χ) in §4. The materials presented here are an elementary case of the theory of chiral homology that does not involve the language of ∞-categories, so that we give self-contained proofs.
A.1. Notations. The notations used in the appendix differ from the ones in the main body of the paper.
Let L be an algebraically closed field of characteristic zero. All abelian categories in this subsection will be L-linear. Let Vect denote the category of finite-dimensional vector spaces over L.
Let P be a semisimple L-linear abelian category such that End P (X) = L for each simple object X ∈ P. Let Irr(P) denote the set of isomorphism classes of simple objects in P. Objects in P will be denoted X, Y, · · · .
Let (R, ⊗) and (C, ⊗) be semisimple rigid tensor category over L. Objects in R and C will be denoted V, W, · · · .
A.2. Factorizable module categories with coefficients. We say that P is a factorizable R-module category with coefficients in C, if for every finite set I there is a bi-exact functor R ⊠I × P → C ⊠I ⊠ P (Deligne's tensor product) (V, X) → V ⋆ I X, for V ∈ R ⊠I , X ∈ P with the following extra structures: (1) When I = ∅, we understand that R ⊠∅ ∼ = Vect, and the action of R ⊠∅ is the usual action of Vect on P by tensoring. (2) Any map of finite sets ϕ : I → J induces ϕ R : R ⊠I → R ⊠J sending ⊠ i∈I V i to ⊠ j∈J (⊗ i →j V i ).
Similarly it induces ϕ C : C ⊠I → C ⊠J . Then there is a functorial isomorphism ϕ R (V ) ⋆ J X ∼ = (ϕ C ⊠ id P )(V ⋆ I X) ∈ C ⊠J ⊠ P, for V ∈ R ⊠I and X ∈ P. (3) If I = I ′ ⊔ I ′′ is a partition of I then there is a functorial isomorphism V ′ ⋆ I ′ (V ′′ ⋆ I ′′ X) ∼ = (V ′ ⊠ V ′′ ) ⋆ I X for V ′ ∈ R ⊠I ′ , V ′′ ∈ R ⊠I ′′ and X ∈ P. Here on the left side, when V ′ acts on V ′′ ⋆ I ′′ X ∈ C ⊠I ′′ ⊠ P, it only acts on the P-factor. These structures have to satisfy the usual compatibilities: composition of maps in (1), refinement of partitions in (2), and the compatibility of (1) and (2) for maps ϕ ′ ⊔ ϕ ′′ : I ′ ⊔ I ′′ → J ′ ⊔ J ′′ . We do not spell out the details.
For X ′ ∈ C ⊠I ⊠ P and Y ∈ P, let Hom(X ′ , Y ) and Hom(Y, X ′ ) denote the inner homs taking values in C ⊠I . For example, Hom(Y, X ′ ) is characterized by having an isomorphism Hom C ⊠I (C, Hom(Y, X ′ )) ∼ = Hom C ⊠I ⊠P (C ⊠ Y, X ′ ) functorial in C ∈ C ⊠I . Then the axioms imply that for X, Y ∈ P, V ∈ R ⊠I , there is a functorial isomorphism (A.1) Hom(V ⋆ I X, Y ) ∼ = Hom(X, V ∨ ⋆ I Y ) ∈ C ⊠I .
factorizable R-module category with coefficients in C. Then there is a well-defined finite set of (isomorphism classes of ) tensor functors {σ : R → C} σ∈Σ and a unique decomposition such that the factorizable R-module structure on P σ with coefficients in C is inflated from an E 2 -action of Rep(Aut ⊗ (σ)) on P σ .
Proof. Choose fiber functors of R and C to identify them with Rep(H) and Rep(M ). In the decomposition of P into indecomposables (see Lemma A.3.1(2)), we apply Theorem A.4.1 to each P s to get a homomorphism ρ s : M → H, such that P s is inflated from an E 2 -action of Rep(H ρs ) on P s . Now let Σ be the set of H-conjugacy classes of {ρ s } s∈Irr(P)/∼ . A homomorphism ρ : M → H up to H-conjugacy is the same datum as a tensor functor σ : R → C, so we may identify Σ with a set of tensor functors {σ : R → C}.
For σ ∈ Σ with the corresponding ρ : M → H, Let P σ be the direct sum of P s for those ρ s conjugate to ρ under H. Note that Aut ⊗ (σ) ∼ = H ρ , so P σ is inflated from an E 2 -action of Rep(Aut ⊗ (σ)).
A.4.3. Remark. We state an equivariant version of Corollary A.4.2. Suppose both R and C are equipped with actions of a group Γ. The action of γ ∈ Γ on V ∈ R ⊠I and W ∈ C ⊠I are denoted V γ and W γ . Suppose further that the action of R on P is equipped with functorial isomorphisms V γ ⋆ I X ∼ = (V ⋆ I X) γ , ∀γ ∈ Γ compatible with the group structure on Γ and the factorization structure. Here the action of γ on the right side is only on the C ⊠I -factor. Under these assumptions, each functor σ : R → C constructed in Corollary A.4.2 is equipped with a Γ-equivariant structure. Therefore Aut ⊗ (σ) also carries an action of Γ. Moreover, the E 2 -action of Rep(Aut ⊗ (σ)) on P σ (denoted •) is equipped with a Γ-invariant structure, i.e., functorial isomorphisms U • I X ∼ = U γ • I X, ∀γ ∈ Γ, U ∈ Rep(Aut ⊗ (σ)) ⊠I , X ∈ P σ compatible with the group structure on Γ and the factorization structure.
The rest of the appendix is devoted to the proof of Theorem A.4.1.
A.5. Proof of Theorem A.4.1. First some notations. Let IndP be the category of ind-objects in P: it is equivalent to Irr(P)-graded vector spaces of possibly infinite dimension. We denote by ω : C ⊠I = Rep(M I ) → Vect the forgetful functor for various I. We also denote the forgetful functor C ⊠I ⊠ P → P by ω. Let Irr(H) and Irr(M ) denote the set of (isomorphism classes of) irreducible representations of H and M . For any V ∈ Irr(H) we have an embedding m V : V ⊗ V ∨ ∼ = End(V ) ⊂ O H as matrix coefficients. Same for M .
The proof goes in several steps.
A.5.1. The affine scheme S. For each V ∈ Irr(H), the action V ⋆ X ∈ Rep(M ) ⊠ P for various X ∈ Irr(P) only involves finitely many irreducible representations W ∈ Irr(M ). We denote this finite set by Γ V ⊂ Irr(M ). Note that Γ V always contains the trivial representation 1 R of M , for V ⋆ (V ∨ ⋆ X) contains 1 R ⊠ X as a direct summand.
We define a moduli problem as follows. For any L-algebra R, let S(R) be the set of Hopf algebra homomorphisms ϕ : O H → O M ⊗ R such that, for any V ∈ Irr(H), ϕ(m V (End(V ))) lies in the span of m W (End(W )) ⊗ R for W ∈ Γ V . Then S(R) is a subset of homomorphisms of algebraic groups M R → H R . Note that S(R) = ∅ since it contains the trivial homomorphism M R → H R (because Γ V contains 1 R ).
We claim that S is representable by an affine scheme of finite type over L. Indeed, choose a faithful V 0 ∈ Irr(H), then ϕ ∈ S(R) is determined by the restriction ϕ| mV 0 (End(V0)) : End(V 0 ) → ⊕ W ∈ΓV 0 End(W )⊗ R, which is representable by an affine space of finite dimension. This realizes S as a closed subscheme of an affine space. Moreover, S carries an action of M × H by conjugation on O M and O H .
We give generators and relations for the ring of regular functions O S . This part is inspired by [LZ18,§6]. For any f ∈ O H and any g ∈ M (L), define a function Φ f,g ∈ O S that assigns to each R-point ϕ : O H → O M ⊗ R the value Φ f,g (ϕ) = ev g ϕ(f ) ∈ R, where ev g : O M ⊗ R → R is the evaluation at g.
The functions {Φ f,g } f ∈OH ,g∈M(L) generate O S as an L-algebra. Indeed it suffices to run f through a basis of the matrix coefficients for a faithful V 0 ∈ Irr(H), and take a finite set of g i such that their images in W ∈ΓV 0 End(W ) span. We now give the relations among {Φ f,g } f ∈OH ,g∈M(L) . We claim that the relations are generated by the following: (1) For any g ∈ M (L), the assignment f → Φ f,g is L-linear.
(2) Let V ∈ Irr(H). Then for any finite L-linear combination i c i g i of elements in M (L) such that i c i g i | W = 0 for all W ∈ Γ V , then i c i Φ f,gi = 0 for all f ∈ m V (End(V )).
(3) For any f, f ′ ∈ O H and g ∈ M (L), we have Φ f f ′ ,g = Φ f,g Φ f ′ ,g . (4) For any f ∈ O H and g, g ′ ∈ M (L), we have Φ f,gg It is easy to see that these relations indeed hold in O S . To show they are all the relations, suppose we are given an assignment Φ f,g → ϕ f,g ∈ R satisfying the above relations, we show how to construct a Hopf algebra map ϕ : O H → O M ⊗ R such that Φ f,g evaluated at ϕ ∈ S(R) is ϕ f,g . For any f ∈ O H , relation (2) ensures that there exists a unique element ϕ(f ) ∈ ⊕ W ∈ΓV m W (End(W )) ⊗ R ⊂ O M ⊗ R such that ev g ϕ(f ) = ϕ f,g for any g ∈ M (L). Relation (1) says that the assignment f → ϕ(f ) gives a linear map ϕ : O H → O M ⊗ R. Relation (3) shows that ϕ is an algebra homomorphism. Relation (4) shows that ϕ is a coalgebra homomorphism. This proves the claim.
A.5.2. Construction of an action of O S on each object in P. We view O S as an algebra ind-object in Rep(H) using the conjugation action of H. For each X ∈ P we will construct a map α X : ω(O S ⋆ X) → X in IndP, compatible with the algebra structure on O S , such that the following diagram is commutative for any V ∈ Rep(H) and X ∈ P Here the top row is induced by the commutativity constraint of the action of R on P; in (ω ⊠ id C )(O S ⋆ (V ⋆ X)) we emphasize that ω is applied to the first factor of C, so the result is still an object in C ⊠ P. Moreover, all maps in IndP are compatible with the O S -actions. See [Gai05,§22].
In other words, if we define an internal Hom(X, Y ) ∈ IndRep(H) for X, Y ∈ P by the adjunction Hom P (ω(V ⋆ X), Y ) = Hom H (V, Hom(X, Y )) for all V ∈ Rep(H), then we need to construct an algebra homomorphism α ′ X : O S → End(X) in IndRep(H), that is compatible with morphisms in P. The construction of α X is analogous to V. Lafforgue's excursion operators [Laf18]. For any g ∈ M (L) we first define a map α g,X : ω(O H ⋆ X) → X as the composition Here the second step uses that (V ⊠ V ∨ ) ⋆ {1,2} X ∈ Rep(M 2 ) ⊠ P, hence (g, 1) ∈ M 2 acts on ω((V ⊠ V ∨ ) ⋆ {1,2} X). The last map ev 1 : O H → 1 R is evaluation at 1 ∈ H. The map α g,X then gives a map α ′ g,X : O H → End(X) in IndRep(H). It is supposed to be the composition Both compositions ω(V ⊠ V ∨ ⋆ X) → X are adjoint to the following map This finishes the proof of relation (4).
A.5.6. The action of Rep(H) on P is inflated from the Rep(H ρ )-action. To show this, we need to check that for any g ∈ M (L), V ∈ Rep(H) and X ∈ P, the following two endomorphisms of ω(V ⋆ X) are the same: the first one, which we denote by a g , is obtained by the action of g on the forgetful functor ω; the second one, which we denote by b g , comes by evaluating γ V,V,X at ρ(g) ∈ Hom Hρ (V, V ). Consider the map c V,g : V ⊗V ∨ → O Z given by ξ ⊗v → c V,g (h) = ξ, hρ(g)h −1 v = h −1 ξ, ρ(g)h −1 v . We are using that ρ(g) commutes with H ρ to conclude that c V,g is right invariant under H ρ hence giving a function on Z. The map c V,g is H-equivariant for the diagonal action of H on V ⊗ V ∨ . Consider the following map By adjunction it gives an endomorphism d g of ω(V ⋆ X). On the one hand, the definition of the O S -action, see (A.2), implies that d g = a g ; on the other hand, comparing δ V,g,X to the map γ V,V,X in (A.3), we see that δ V,g,X is the restriction of γ V,V,X to ρ(g) ⊗ ω((V ⊗ V ∨ ) ⋆ X), hence d g = b g . This shows a g = b g and finishes the proof of Theorem A.4.1.