Rodier type theorem for generalized principal series

In this paper, we extend Rodier's structural theorem for regular principal series to regular generalized principal series of split groups of classical types, which can be viewed as a first step to understand the internal structure of the singular case.


introduction
Following Harish-Chandra's "philosophy of cusp forms" which culminates in the Langlands classification theorems under parabolic induction both locally and globally, a longstanding local problem is to understand the decomposition structures of parabolic inductions, especially those inducing from supercuspidal representations which are the so-called generalized principal series. Among those parabolic inductions, there are two extreme cases, namely unitary parabolic inductions and regular generalized principal series of which many mathematicians have devoted their efforts to describe the corresponding internal structures. To be more precise, • Tempered parabolic induction: Knapp-Stein R-group theory and its explicit structures (Bruhat, Harish-Chandra, Knapp-Stein, Jacquet, Casselman, Howe, Silberger, Winarsky, Keys, Goldberg etc). • Principal series: a. Muller's irreducibility criterion for principal series; b. Rodier's structural theorem for regular principal series. So one might ask the following natural questions: Q1 ∶ What is the irreducibility criterion for generalized principal series in terms of Muller?
Q2 ∶ What is the story of Rodier's structural theorem for regular generalized principal series?
The first question in principal should be doable after Muller's work, but we have not seen any literature and will write down the details separately (cf. [Luo18a]). As for the second question, without the far-reaching Langlands-Shahidi theory built up by Shahidi in the early 1990s (cf. [Sha90]), it seems that one cannot push Rodier's theorem further to the regular generalized principal series if following Rodier's paper completely, especially the argument of Proposition 3. On the other hand, in order to get a glimpse of the mysterious structure theory of generalized principal series, Bernsterin and Zelevinsky have developed a novel method to attack the general linear group (cf. [BZ77,Zel80]), and then Jantzen, Tadic etc took over the mission and they initiated the study of the structure theory of parabolic induction of (relative) low rank groups, so far all rank 2 groups have been tested (cf. [ST93,Mui98,Kim96,Kon01,Han06,HM10,Mat10,Sch14,Luo18b,Luo18c]), and relative low rank groups can also be dealt with due to Moeglin-Tadić's profound work on the classification of discrete series of classical groups (cf. [Moe,MT02]). But it is still quite luxury to design a reasonable conjecture of the internal structure of parabolic induction. Even in the GL n case, to my best knowledge, there is no explicit formula describing the cardinality of the constituents of generalized principal series. To continue the mission, currently, a doable step is to generalize Rodier's theorem to generalized principal series which is what the paper is written for. To end the introduction, we first would like to point out that the argument of Proposition 3 in [Rod81] is inaccurate it seems (recently we find that this has been pointed out in [MW87, II 1.2 Remarque (i)]), but we have not yet found an intrinsic argument, only a raw argument is provided and it only works for classical types. Denote by to be the generalized principal series inducing from the parabolic subgroup P = M N to G with σ a unitary supercuspidal representation of M and ν an unramified character of M . Second, we want to mention that the novelties of the paper are the following two key observations: As those two key observations are gradually realized during the revision of earlier versions of the paper, so we want to preserve the developing trace for the readers. At last, we give the outline of the paper. In Section 2, some necessary notions of representation theory are introduced. In Section 3, we will first state and prove the Rodier type structure theorem for regular generalized principal series which roughly speaking is a parametrization of the associated Jordan-Holder series, then prove an expected formula of the cardinality of the constituents of parabolic induction for split groups of classical types. Section 4 is about the parametrization/characterization of the constituents of genericness/discreteness/temperedness of generalized principal series, while the structure of relative Weyl groups will be discussed in the appendix.

preliminaries
Let G be a connected split reductive group defined over a non-archimedean field F of characteristic 0. For our purpose, it is no harm to assume that the center Z G of G is compact. Denote by − F the absolute value, by w the uniformizer and by q the cardinality of the residue field of F . Fix a Borel subgroup B = T U of G with T the split torus and U a maximal unipotent subgroup of G, and let P = M N be a standard parabolic subgroup of G with M the Levi subgroup and N the unipotent radical.
Let X(M ) F be the group of F -rational characters of M , and set Next, let Φ be the root system of G with respect to T , and ∆ be the set of simple roots determined by U . For α ∈ Φ, we denote by α ∨ the associated coroot, and by w α the associated reflection in the Weyl group W of T in G with as ω w.α = wω α w −1 . Note that W M in general is larger than the one generated by those relative reflections, for example, where W M ≃ Z 2Z, while there are no relative reflections preserving M . For our purpose, we define the "small" relative Weyl group W 0 M ⊂ W M to be the one generated by those relative reflections, i.e.
For more details about the structure theory of W M , please refer to the Appendix. The relative walls in the "small" character vector spaces The relative Weyl chambers are the connected components of the set on which the "small" relative Weyl group W 0 M acts. An observation is that W 0 M acts simply transitively on the set of relative Weyl chambers, which follows from the fact that Φ 0 M is a root system, may not be irreducible. Denote by ∆ 0 M the relative simple roots of Φ 0 M . We denote by C + M the relative dominant Weyl chamber in 0 a ⋆ M determined by P . Recall that the canonical pairing Parabolic induction and Jacquet module: For P = M N a parabolic subgroup of G and an admissible representation (σ, V σ ) (resp. (π, V π )) of M (resp. G), we have the following normalized parabolic induction of P to G which is a representation of G , ∀n ∈ N, m ∈ M and g ∈ G} with δ P stands for the modulus character of P , i.e., denote by n the Lie algebra of N , Given an irreducible unitary admissible representation σ of M and ν ∈ a ⋆ M , let I(ν, σ) be the representation of G induced from σ and ν as follows: We say a character θ of U is generic if the restriction of θ to X α (F ) is non-trivial for each simple root α ∈ ∆. Then the Whittaker function space W θ of G with respect to θ is the space of smooth complex functions f on G satisfying, for u ∈ U and g ∈ G, In what follows, we recall Casselman's square-integrability/temperedness criterion. For our purpose, we only state it under the condition that the inducing datum σ is supercuspidal, i.e. π ∈ JH(I(ν, σ)) with σ supercuspidal, here JH(−) means the set of Jordan-Holder constituents. Let with all the coefficients x α > 0 (resp. x α ≥ 0). For an admissible representation (τ, V τ ) of M , we define the set Exp(τ ) of exponents of τ as follows: Square-integrability/Temperedness: Keep the notation as above, π ∈ JH(I(ν, σ)) is square-integrable (resp. tempered) if and only if for each standard parabolic subgroup Q = LV associated to P = M N , i.e. L and M are conjugate,

Decomposition of regular generalized principal series
An irreducible supercuspidal representation τ of M is called regular in G if the only element w ∈ W M such that τ w ≃ τ is the identity element.
As in [Sha90, Section 3], we assume that the generic character θ and w G 0 are compatible. Denote by θ M the generic character θ of U restricting to N which is compatible with w M 0 , and by S the set of relative coroots α ∨ of M such that • Ind Mα M (σ ⊗ ν) is reducible, where M α is the relative rank one subgroup of G generated by the relative root α, and Remark 1.
• The regular condition implies that there is no relative rank-one reducibility at s = 0, i.e. ν(H α ∨ (x)) = 1. • The Langlands-Shahidi theory tells us that there exists only one reducibility point occurring at s = 0, ± 1 2 or ±1 for relative rank-one parabolic inductions inducing from generic supercuspidal representations.
• For Type A split groups, reducibility does not occur at s = ± 1 2 . In what follows, we always assume that the representation σ ⊗ ν of M is regular, θ M -generic and supercuspidal, and G is of classical types, i.e. Types A n , B n , C n and D n . For such inducing data, the corresponding generalized principal series I(ν, σ) has the following properties.
Here M α is the relative rank-one subgroup of G generated by α.
Proof. The proof follows from the same argument as in Rodier's paper for principal series. We sketch the argument as follows.
• The Jacquet functor is exact.
(ii) follows from (i) and the Frobenius reciprocity property, i.e.
(iii) follows from Lemma (A.1) for G of types A n , B n and C n , i.e.
In view of Lemma (A.1), it is equivalent to show , which is equivalent to say, as α is W -conjugate to a simple root in Φ, where t is equal to the number of odd a i s. The same argument as above works if the generalized principal series, for L der ↞ SL a1 × ⋯ × SL a k × SO n0 with a 1 > ⋯ > a k odd, Ind G L (σ) is always irreducible for regular σ. In such case, we have W L ≃ (Z 2Z) (k−1) and W 0 L = 1. Constructing an auxiliaryG such that the relative Weyl group WG L of L inG is isomorphic to (Z 2Z) k as follows: Then the same argument as before shows that, as σ is regular in G, is of length at most 2. Whence the claim (⋆) holds.
Remark 2. For (iii), it is a well-known result for general connected reductive groups G ([Sil79, Theorem 5.4.3.7]). But the Jacquet module argument is much more simple and intuitive. On the other hand, they can deepen our understanding of these two arguments, and shed light on the understanding of general cases.
Before turning to the analysis of the Jordan-Holder constituents of I(ν, σ). We first recall a simple observation. For any two Weyl elements w, w ′ ∈ W M , one has the following isomorphisms, via Frobenius reciprocity, This says that, in some sense, there exist enough intertwining operators in order to distinguish the Jordan-Holder constituents of I(ν, σ) from each other. Note that Φ 0 M is a root system which may not be irreducible. Decomposing Φ 0 M and ∆ 0 M into irreducible ones as follows: M,i , and w, w ′ ∈ W 0 M,i . Suppose the relative Weyl chambers wC + M and w ′ C + M share the same wall Ker α ∨ . Let A be a base element of

Recall that in the Appendix
where the direct sum is indexed by w ′′ ∈ W 0 M such that w ′′ C + M and wC + M (resp. w ′ C + M ) are on the same side of the wall Ker α ∨ .
Proof. There is no harm to assume ∆ 0 M,i ⊂ ∆ M , and assume w = 1 and α ∨ > 0, which in turn says that α is a relative simple root. Under this setting, the argument is almost the same as in [Rod81, Lemma 1]. For the convenience of readers, we sketch the main ideas as follows. Let us first look at the relative rank-one case.

])
Hom G (W θ , I(ν, σ)) ≃ Hom M (Ind M M∩U (θ M ), σ ⊗ ν). • Standard module theorem says that if the unique Langlands quotient of a standard module is generic, then the induced representation is irreducible. (Please refer to [HM07]) An observation is that w ′ = w α if α ∨ ∈ S ∪ (−S) which follows from the fact that W 0 M acts simply transitively on the relative Weyl chambers. Therefore the second step is to use the induction by stage property of A, i.e. and In general, for w, w ′ two Weyl elements of W 0 M , let A be a base element of where Y is the set of w ′′ ∈ W 0 M for which there exists an element α ∨ ∈ S such that the chambers wC + M and w ′′ C + M are on the same side of the wall Ker α ∨ , and the chambers wC + M and w ′ C + M are separated by the wall.
Proof. It reduces to consider w, w ′ ∈ W 0 M,i based on the product property of W 0 M . Then it follows from the same argument as in [Rod81, Proposition 2] by induction on the length of galleries connecting two Weyl chambers and Lemma 3.2.
(ii) Given Y defined as in the above, As in general W M ≠ W 0 M , the following new Lemma is needed to claim a similar result as in the regular principal series case.
Remark 3. Indeed, Lemma 3.1 (iii) can be proved for general G via Casselman-Tadic's Jacquet module argument with the help of the above Lemma.
Corollary 3.6. For the unique irreducible subrepresentation π of I(ν, σ) w , we have where the sum is over those w ′′ ∈ W 0 M such that wC + M and w ′′ C + M are on the same side of all walls Ker α ∨ , for α ∨ ∈ S.
Theorem 3.7. keep the notions as before. There exists a bijection between the connected components Ker α ∨ and the set of subquotients π Γ in JH(I(ν, σ)) with the following property, for w ∈ W 0 M , Remark 4. In view of the arguments, Theorem 3.7 is indeed true for any connected reductive group G.
To say more, we need the following lemma.
Lemma 3.8. The set S is linearly independent.
Proof. To show that the set S is linearly independent, it is sufficient to show that the set S forms a Dynkin diagram. As the unitary central characters of the parabolic induction data of relative rank-one groups take care of themselves, a case-by-case argument is given as follows. Let us first recall the following facts/theorems. In view of (i,ii), for groups of classical types, it reduces to prove the Lemma for the equal partition case, so it can be, in some sense, viewed as the Borel subgroup case, i.e. Type A n : The derived Levi subgroup M der = SL m × ⋯ × SL m . Arguing by contradiction, i.e. the set S does not form a Dynkin diagram. Without loss of generality, the non-Dynkin diagram cases are as follows. • Case 1: e 2 = s ± 1, e i0 = s ± 1 and e j0 = s ± 2 or s.
As W M ≃ S m , so the regular condition implies that no two values e k and e l are equal. Then Case 2 cannot happen as 3 > 2. For Case 1, the regular condition says that e j0 ≠ s, hence a simple check says e i0 − e j0 = 3 > 1. Thus for the in-between coroots {e it − e it+1 } and {e jt+1 − e jt }, we have e it − e jt = 2(t + 1) + 1 by induction. Contradiction.
As W M ≃ S m ⋊ (Z 2Z) m , so the regular condition implies that no two values of ±e k and ±e l are equal, especially s ≠ 0. So Case 4 and Case 7 cannot happen as e i = 0 or e 1 = 0, Case 5 cannot occur as e 1 = ±e 2 , and also Case 3 cannot happen as s = 1 2 or 1 by the Langlands-Shahidi theory. It is easy to see that Case 2 cannot happen as 3 > 2. For Case 1, the regular condition says that e j0 ≠ ±s, hence a simple check says e i0 ± e j0 = 3 or 2s − 1 ≠ 1. Thus for the in-between coroots {e it ± e it+1 } and {e jt+1 ± e jt }, we have e it ± e jt = 2(t + 1) + 1 or 2s − 1 ≠ 1 by induction. Contradiction. For Case 6, for the in-between coroots {e it+1 ± e it } and {e jt+1 ± e jt }, the regular condition says that e i1 ≠ ±s, e j1 ≠ ±s and s ≠ ±1 2, a simple check shows that e i1 ± e j1 = 4 or 2s . Therefore by induction, we know that e it ± e jt = 2(t + 1) or 2s ≠ 1. Whence the Lemma hold.
As W M ≃ S m ⋊ (Z 2Z) m if m even, or S m ⋊ (Z 2Z) m−1 if m odd. If m is even, then it follows from the same argument as above. So we assume that m is odd, then the regular condition implies that no two values e k and e l are equal or of opposite signs. The only difference with the above cases is that s = 0 may be a reducible point. A case-by-case check, except Case 7, shows that two values e k and e l will be equal or of opposite signs if a relative rank-one reducibility point at s = 0 occurs in the above cases. Whence the Lemma hold. For Case 7, then the block associated to the root 2e 1 must be of even dimension, so is e i which contradicts with the regular condition.
Remark 5. From the above arguments, the generic condition is unnecessary except Case 3. Though it is not a Dynkin diagram, it is linearly independent. So the Lemma holds for all generalized principal series, i.e. without genericity assumption.
Remark 6. From the above argument, it is easy to see that the above Lemma holds for quasi-split groups of Types A n , B n and D n as well.
Corollary 3.9. Let l be the rank of the center of M . Then the length of I(ν, σ) is equal to 2 #S , which is at most 2 l .

Genericity-Discreteness-Temperedness
Among the connected components which index the constituents of I(ν, , σ), there exists a distinguished one which plays a key role in what follows.