Representations of unramified U(2,2) over a p-adic field I: representations of non-integral level
aa r X i v : . [ m a t h . R T ] A p r REPRESENTATIONS OF UNRAMIFIED U (2 , OVER A p -ADIC FIELD I:REPRESENTATIONS OF NON-INTEGRAL LEVEL MICHITAKA MIYAUCHI
Abstract.
Let F be a non-archimedean local field of odd residual characteristic and let G be the unramified unitary group U (2 ,
2) defined over F . In this paper, we give a classificationof the irreducible smooth representations of G of non-integral level using the Hecke algebraicmethod developed by Allen Moy for GSp (4).
Introduction
Let F be a non-archimedean local field of odd residual characteristic and let G be the unram-ified unitary group U (2 ,
2) defined over F . Although Konno [6] classified the non-supercuspidalrepresentations of G , supercuspidal representations of G has not been classified. The purposeof this paper is to classify the irreducible smooth representations of G of non-integral level.In [12] and [13], Moy gave a classification of the irreducible smooth representations of un-ramified U (2 ,
1) and
GSp (4) over F , based on the concepts of nondegenerate representationsand Hecke algebra isomorphisms. A nondegenerate representation of GSp (4) is an irreduciblerepresentation σ of an open compact subgroup K which satisfies a certain cuspidality or semisim-plicity condition. An important property of nondegenerate representations of GSp (4) is thatevery irreducible smooth representation of
GSp (4) contains some nondegenerate representation.For σ a nondegenerate representation of GSp (4), the set of equivalence classes of irreduciblerepresentations of
GSp (4) which contain σ can be identified with the set of equivalence classesof irreducible representations of a Hecke algebra H associated to σ . Moy described H as a Heckealgebra of some smaller group and thus reduced the classification of the irreducible smoothrepresentations of GSp (4) which contain σ to that of a smaller group.In this paper, we attempt to classify the irreducible smooth representations of G by thismethod. The keypoint of our classification is to construct an analog of nondegenerate represen-tations of GSp (4) for G .In [14], Moy and Prasad developed the concept of nondegenerate representations into that ofunrefined minimal K -types for reductive p -adic groups. For classical groups, Stevens [18] gavean explicit construction of unrefined minimal K -types as fundamental skew strata, based on theresults of Bushnell and Kutzko [2] and Morris [9]. However, neither of these give a classificationof the irreducible smooth representations of p -adic classical groups.Throughout this paper, we use the notion of fundamental skew strata introduced by [18] forour nondegenerate representations of G . Let F be the unramified quadratic extension over F .Then G is realized as the group of isometries of an F/F -hermitian form on 4-dimensional F -vector space V . According to [18], a skew stratum is a 4-tuple [Λ , n, r, β ]. A periodic latticefunction Λ with a certain duality induces a filtration { P Λ ,k } k ≥ on a parahoric subgroup P Λ , of G . Integers n > r ≥ β in the Lie algebra of G determine a character ψ of thegroup P Λ ,r +1 which is trivial on P Λ ,n +1 . Writing e (Λ) for the period of Λ, we refer to n/e (Λ) asthe level of the stratum. Mathematics Subject Classification.
Key words and phrases. p -adic group, unitary group, Hecke algebra. In Section 2, we prove Theorem 2.2, a rigid result on the existence of fundamental skew strata.This theorem, which is an analog of the result in [13], says that every irreducible representationof G contains some fundamental skew stratum [Λ , n, n − , β ] such that { P Λ ,n } n ≥ is the standardfiltration of P Λ , . So we can start our classification with 7 filtrations up to conjugacy.In the latter part of this paper, we give a classification of the irreducible representations of G which contain a skew stratum [Λ , n, n − , β ] whose level is not an integer. As in [18], we mayfurther assume that β is a semisimple element in Lie( G ) by replacing P Λ ,n with a nonstandardfiltration subgroup of a parahoric subgroup.In Section 3, we consider the case when the G -centralizer of β forms a maximal compact torusin G . Stevens [16] and [17] gave a method to construct irreducible supercuspidal representa-tions of p -adic classical groups from such a stratum. Applying his result, we can construct theirreducible representations of G containing such a stratum. Those are irreducibly induced fromopen and compact subgroups.In Section 4, we treat the case when the G -centralizer G ′ of β is not compact. We describethe Hecke algebra associated to such a stratum as a certain Hecke algebra of G ′ . This allows usto identify the equivalence classes of irreducible representations of G containing such a stratumwith those of irreducible representations of G ′ . The construction of Hecke algebra isomorphismshere is along the lines of that in [12] and [13]. This method is valid only if G ′ is tamely ramifiedover F . But there are no wildly ramified G ′ for our group G = U (2 , G .Moreover, we consider the condition when two fundamental skew strata occur in a commonirreducible smooth representation in G . This problem relates to the “intertwining implies con-jugacy” property. We prove that this property holds among the skew strata [Λ , n, n − , β ]with compact centralizers. In the other cases, we normalize skew strata to be appropriate tothis problem. Then the “intertwining implies conjugacy” property of normalized skew strata istrivial except only one case (case (4d)).The remaining problem that we need to consider is to classify the irreducible representationsof G of integral level. The level 0 representations have been classified by Moy-Prasad [15] andMorris [11] for more general p -adic reductive groups. In a second part of this article, we willcomplete our classification of the irreducible representations of G , by classifying those containinga fundamental skew stratum of positive integral level.The Hecke algebra isomorphisms established for G preserve the unitary structure of Heckealgebras, so we can calculate formal degrees of supercuspidal representations of G via thoseisomorphisms. We hope to return to this in the future.Part of this article is based on the author’s doctoral thesis. The author would like to thank hissupervisors, Tetsuya Takahashi and Tadashi Yamazaki, for their useful comments and patientencouragement during this work. The author also thank Kazutoshi Kariyama and Shaun Stevensfor corrections and helpful comments. The research for this paper was partially supported byEPSRC grant GR/T21714/01. 1. Preliminaries
In this section, we recall the notion of fundamental skew strata for unramified unitary groupsover a non-archimedean local field. For details in more general settings, one should consult [1],[2] and [16].1.1.
Filtrations.
Let F be a non-archimedean local field of odd residual characteristic. Let o denote the ring of integers in F , p the prime ideal in o , k = o / p the residue field, and q the number of elements in k .Let F = F [ √ ε ], ε a nonsquare element in o × , denote the unramified quadratic extension over F . We denote by o F , p F , k F the objects for F analogous to those above for F . For x in F or EPRESENTATIONS OF U (2 ,
2) OVER A p -ADIC FIELD I 3 k F , we write x for the Galois conjugate of x . Since F is unramified over F , we can (and do)select a common uniformizer ̟ of F and F .Let V be an N -dimensional F -vector space equipped with a nondegenerate hermitian form f with respect to F/F . We put A = End F ( V ) and e G = A × . Let σ denote the involution on A induced by f . We also put G = { g ∈ e G | gσ ( g ) = 1 } , the corresponding unramified unitarygroup over F , and g = { X ∈ A | X + σ ( X ) = 0 } ≃ Lie( G ).Recall from [2] (2.1) that an o F -lattice sequence in V is a function Λ from Z to the set of o F -lattices in V such that(i) Λ( i ) ⊃ Λ( i + 1), i ∈ Z ;(ii) there exists an integer e (Λ) called the o F -period of Λ such that ̟ Λ( i ) = Λ( i + e (Λ)), i ∈ Z .We say that an o F -lattice sequence Λ is strict if Λ is injective.For L an o F -lattice in V , we define its dual lattice L by L = { v ∈ V | f ( v, L ) ⊂ o F } . An o F -lattice sequence Λ in V is called self-dual if there exists an integer d (Λ) such that Λ( i ) =Λ( d (Λ) − i ), i ∈ Z .An o F -lattice sequence Λ in V induces a filtration { a n (Λ) } n ∈ Z on A by a n (Λ) = { X ∈ A | X Λ( i ) ⊂ Λ( i + n ) , i ∈ Z } , n ∈ Z . This filtration determines a sort of “valuation” map ν Λ on A by ν Λ ( x ) = sup { n ∈ Z | x ∈ a n (Λ) } , x ∈ A \{ } , with the usual understanding that ν Λ (0) = ∞ .For Λ an o F -lattice sequence in V and k ∈ Z , we define a translate Λ + k of Λ by (Λ + k )( i ) =Λ( i + k ), i ∈ Z . Then we have a n (Λ) = a n (Λ + k ), n ∈ Z . For g ∈ G and Λ a self-dual o F -latticesequence in V , we define a self-dual o F -lattice sequence g Λ by ( g Λ)( i ) = g Λ( i ), i ∈ Z . Note that e ( g Λ) = e (Λ) and d ( g Λ) = d (Λ). Remark 1.1.
Given an o F -lattice sequence Λ in V , we define its dual sequence Λ by Λ ( i ) =Λ( − i ) , i ∈ Z . Since a n (Λ ) = σ ( a n (Λ)), n ∈ Z , it follows from [2] (2.5) that Λ is self-dual ifand only if σ ( a n (Λ)) = a n (Λ), n ∈ Z .For Γ an o F -lattice in A , we define its dual by Γ ∗ = { X ∈ A | tr A/F ( X Γ) ⊂ p } , where tr A/F denotes the composition of traces tr F/F ◦ tr A/F . By [2] (2.10), if Λ is an o F -lattice sequence in V , then we have a n (Λ) ∗ = a − n (Λ).For Λ a self-dual o F -lattice sequence in V , we will write g Λ ,n = g ∩ a n (Λ), n ∈ Z . We alsoset P Λ , = G ∩ a (Λ) and P Λ ,n = G ∩ (1 + a n (Λ)), n ≥
1. Then { P Λ ,n } n ≥ is a filtration of theparahoric subgroup P Λ , of G by its open normal subgroups.We fix an additive character Ω of F with conductor p . Let ∧ denote the Pontrjagin dualand for x a real number, let [ x ] denote the greatest integer less than or equal to x . Proposition 1.2.
Let Λ be a self-dual o F -lattice sequence in V and let n, r ∈ Z satisfy n > r ≥ [ n/ ≥ . Then the map p p − induces an isomorphism P Λ ,r +1 /P Λ ,n +1 ≃ g Λ ,r +1 / g Λ ,n +1 and there exists an isomorphism g Λ , − n / g Λ , − r ≃ ( P Λ ,r +1 /P Λ ,n +1 ) ∧ ; b + g Λ , − r ψ b , where ψ b ( p ) = Ω(tr A/F ( b ( p − , p ∈ P Λ ,r +1 . Skew strata.Definition 1.3 ([2] (3.1), [16] Definition 4.5) . (i) A stratum in A is a 4-tuple [Λ , n, r, β ] consistingof an o F -lattice sequence Λ in V , integers n, r verifying n > r ≥
0, and an element β in a − n (Λ).We say that two strata [Λ , n, r, β i ], i = 1 ,
2, are equivalent if β − β ∈ a − r (Λ).(ii) A stratum [Λ , n, r, β ] in A is called skew if Λ is self-dual and β ∈ g Λ , − n . MICHITAKA MIYAUCHI
The fraction n/e (Λ) is called the level of the stratum. If n > r ≥ [ n/ , n, r, β ] parametrize the dual of P Λ ,r +1 /P Λ ,n +1 .For [Λ , n, r, β ] a stratum in A , we set y β = ̟ n/k β e (Λ) /k , where k = ( e (Λ) , n ). Then y β belongsto a (Λ) and its characteristic polynomial Φ β ( X ) lies in o F [ X ]. We define the characteristicpolynomial φ β ( X ) of this stratum to be the reduction modulo p F of Φ β ( X ). Definition 1.4 ([1] (2.3)) . A stratum [Λ , n, r, β ] in A is called fundamental if φ β ( X ) = X N .The representations of G we will consider are always assumed to be smooth and complex. Let π be a smooth representation of G and [Λ , n, r, β ] a skew stratum with n > r ≥ [ n/ π contains [Λ , n, r, β ] if the restriction of π to P Λ ,r +1 contains the character ψ β . A smoothrepresentation π of G is called of positive level if π has no non-zero P Λ , -fixed vector, for anyself-dual o F -lattice sequence Λ in V . By [18] Theorem 2.11, an irreducible smooth representationof G of positive level contains a fundamental skew stratum [Λ , n, n − , β ].For g ∈ e G and x ∈ A , we write Ad( g )( x ) = gxg − . Proposition 1.5.
Let [Λ , n, r, β ] and [Λ ′ , m, s, γ ] be skew strata in A contained in some irre-ducible smooth representation of G .(i) There exists g ∈ G such that ( β + g Λ , − r ) ∩ Ad( g )( γ + g Λ ′ , − s ) = ∅ .(ii) If [Λ , n, r, β ] is fundamental, then we have n/e (Λ) ≤ m/e (Λ ′ ) .(iii) If n/e (Λ) = m/e (Λ ′ ) , then φ β ( X ) = φ γ ( X ) .Proof. The proof is very similar to those of [4] Theorem 4.1 and Corollary 4.2. (cid:3)
By Proposition 1.5, we can define the level and the characteristic polynomial of an irreduciblesmooth representation π of G of positive level to be those of the fundamental skew stratacontained in π .Given a skew stratum [Λ , n, r, β ] in A , we define its formal intertwining I G [Λ , n, r, β ] = { g ∈ G | ( β + g Λ , − r ) ∩ Ad( g )( β + g Λ , − r ) = ∅} . If n > r ≥ [ n/ ψ β of P Λ ,r +1 in G .1.3. Semisimple strata.Definition 1.6 ([1] (1.5.5), [2] (5.1)) . A stratum [Λ , n, r, β ] in A is called simple if(i) the algebra E = F [ β ] is a field, and Λ is an o E -lattice sequence;(ii) ν Λ ( β ) = − n ;(iii) β is minimal over F in the sense of [1] (1.4.14). Remark 1.7.
The definition above is restricted for our use. With the notion of [1], our simplestratum is a simple stratum [Λ , n, r, β ] with k ( β, Λ) = − n .The following criterion of the simplicity of strata is well known. Proposition 1.8 ([7] Proposition 1.5) . Let Λ be a strict o F -lattice sequence in V of period N and n an integer coprime to N . Suppose that [Λ , n, n − , β ] is a fundamental stratum in A .Then F [ β ] is a totally ramified extension of degree N over F and [Λ , n, r, β ] is simple, for any n > r ≥ . Let [Λ , n, r, β ] be a stratum in A . We assume that there is a non-trivial F -splitting V = V ⊕ V such that(i) Λ( i ) = Λ ( i ) ⊕ Λ ( i ), i ∈ Z , where Λ j ( i ) = Λ( i ) ∩ V j , for j = 1 , βV j ⊂ V j , j = 1 , EPRESENTATIONS OF U (2 ,
2) OVER A p -ADIC FIELD I 5 For j = 1 ,
2, we write β j = β | V j . By [2] (2.9), we get a stratum [Λ j , n, r, β j ] in End F ( V j ). Recallfrom [2] (3.6) that a stratum [Λ , n, r, β ] in A is called split if(iii) ν Λ ( β ) = − n and X does not divide φ β ( X );(iv) either ν Λ ( β ) > − n , or else all the following conditions hold:(a) ν Λ ( β ) = − n and X does not divide φ β ( X ),(b) ( φ β ( X ) , φ β ( X )) = 1. Definition 1.9 ([16] Definition 4.8, [18] Definition 2.10) . (i) (Inductive definition on the dimen-sion of V ) A stratum [Λ , n, r, β ] is called semisimple if it is simple, or else it is split as aboveand satisfies the following conditions:(a) [Λ , n, r, β ] is simple;(b) [Λ , n , r, β ] is semisimple or β = 0, where n = max {− ν Λ ( β ) , r + 1 } .(ii) A skew stratum in A is called split if it is split with respect to an orthogonal F -splitting V = V ⊥ V .If a skew stratum [Λ , n, r, β ] is split with respect to V = V ⊥ V , then Λ j is a self-dual o F -lattice sequence in ( V j , f | V j ) with d (Λ j ) = d (Λ), for j = 1 , Hecke algebras.
Let G be a unimodular, locally compact, totally disconnected topologicalgroup. Let J be an open compact subgroup of G and let ( σ, W ) be an irreducible smoothrepresentation of J . For g ∈ G , we write σ g for the representation of J g = g − J g defined by σ g ( x ) = σ ( gxg − ), x ∈ J g . We define the intertwining of σ in G by I G ( σ ) = { g ∈ G | Hom J ∩ J g ( σ, σ g ) = 0 } . Let ( e σ, f W ) denote the contragradient representation of ( σ, W ). The Hecke algebra H ( G//J, σ )is the set of compactly supported functions f : G → End C ( f W ) such that f ( kgk ′ ) = e σ ( k ) f ( g ) e σ ( k ′ ) , k, k ′ ∈ J, g ∈ G. Let dg denote the Haar measure on G normalized so that the volume vol( J ) of J is 1. Then H ( G//J, σ ) becomes an algebra under convolution relative to dg . Recall from [1] (4.1.1) thatthe support of H ( G//J, σ ) is the intertwining of e σ in G , that is, I G ( e σ ) = [ f ∈H ( G//J,σ ) supp( f ) . Since J is compact, there exists a J -invariant, positive definite hermitian form on f W . Thisform induces an involution X X on End C ( f W ). For f ∈ H ( G//J, σ ), we define f ∗ ∈H ( G//J, σ ) by f ∗ ( g ) = f ( g − ) , g ∈ G . Then the map ∗ : H ( G//J, σ ) → H ( G//J, σ ) is aninvolution on H ( G//J, σ ).Let Irr( G ) denote the set of equivalence classes of irreducible smooth representations of G and Irr( G ) ( J,σ ) the subset of Irr( G ) consisting of the elements those σ -isotypic components arenot zero. Let Irr H ( G//J, σ ) denote the set of equivalence classes of irreducible representationsof H ( G//J, σ ). Then, by [1] (4.2.5), there is a bijection Irr( G ) ( J,σ ) ≃ Irr H ( G//J, σ ).1.5.
Unramified hermitian forms on a 2-dimensional space.
We shall consider hermitianforms on the space V = F . Let { e , e } be an F -basis of V . Up to isometries, there are two F/F -hermitian forms f and f on V , where f ( x, y ) = x y + ̟x y , f ( x, y ) = x y + x y , for x = x e + x e , y = y e + y e in V . MICHITAKA MIYAUCHI
The space (
V, f ) is anisotropic and has the unique o F -lattice L such that L ⊃ L ⊃ ̟L .In fact, we have L = o F e ⊕ o F e . Therefore, every strict self-dual o F -lattice sequence in ( V, f )is a translate of the following sequence Λ:Λ(2 i ) = ̟ i L, Λ(2 i + 1) = ̟ i +1 L , i ∈ Z . (1.1)Hence every strict self-dual o F -lattice sequence Λ ′ in ( V, f ) satisfies e (Λ ′ ) = 2 and d (Λ ′ ) is odd.Next, we consider the space ( V, f ). Its anisotropic part is trivial. Set N = o F e ⊕ o F e , N = o F e ⊕ p F e . Let Λ be a strict self-dual o F -lattice sequence in ( V, f ) of o F -period 2. Then there exist g ∈ G and k ∈ Z such that( g Λ + k )(2 i ) = ̟ i N , ( g Λ + k )(2 i + 1) = ̟ i N , i ∈ Z . (1.2)In particular, the integer d (Λ) should be even.As a consequence, we obtain the following: Lemma 1.10.
Let f be an unramified hermitian form on a 2-dimensional F -space V and let Λ be a strict self-dual o F -lattice sequence Λ in V with e (Λ) = 2 . Then the parity of d (Λ) isindependent of the choice of Λ . Moreover, The space ( V, f ) is anisotropic if and only if d (Λ) isodd. Fundamental strata for U (2 , U (2 ,
2) and state a rigid version of theexistence of fundamental skew strata for U (2 , U (2 ,
2) of positive level contains a fundamental skew stratum [Λ , n, n − , β ]such that Λ is a strict o F -lattice sequence.2.1. Reduction to strict lattice sequences.
From now on, we will denote by V the fourdimensional space of column vectors F . We write A = M ( F ) and e G = GL ( F ). Let e i (1 ≤ i ≤
4) be the standard basis vectors of V , and let E ij denote the element in A whose ( k, l )entry is δ ik δ jl . Set H = E + E + E + E , and define a nondegenerate hermitian form f on V by f ( x, y ) = t xHy , x, y ∈ V . The form f induces an involution σ on A by σ ( X ) = H − t XH , X ∈ A .We define G = U (2 ,
2) = { g ∈ e G | gσ ( g ) = 1 } and g = { X ∈ A | X + σ ( X ) = 0 } ≃ Lie( G ).Then g consists of matrices of the form Y Z C a √ εM N b √ ε − CD c √ ε − N − Zd √ ε − D − M − Y , C, D, M, N, Y, Z ∈ F, a, b, c, d ∈ F . We recall the structure of strict self-dual o F -lattice sequences in ( V, f ) from [10] §
1. We define o F -lattices N , N , and N in V by N = o F e ⊕ o F e ⊕ o F e ⊕ o F e ,N = o F e ⊕ o F e ⊕ o F e ⊕ p F e ,N = o F e ⊕ o F e ⊕ p F e ⊕ p F e . Then we obtain a sequence of lattices N = N ) N ) N = ̟N ) ̟N ) ̟N . EPRESENTATIONS OF U (2 ,
2) OVER A p -ADIC FIELD I 7 A self-dual o F -lattice sequence Λ in V is called standard if its image Λ( Z ) = { Λ( i ) | i ∈ Z } iscontained in the set { ̟ m N , ̟ m N , ̟ m N , ̟ m N | m ∈ Z } . By [10] Proposition 1.10, everyself-dual o F -lattice sequence is a G -conjugate of some standard sequence.Up to translation, the standard strict self-dual o F -lattice sequences correspond to the non-empty subsets S of { N , N , N } . Therefore, there are just following 7 standard strict self-dual o F -lattice sequences in V up to translation:(1) S = { N , N , N } : Λ(4 i ) = ̟ i N , Λ(4 i + 1) = ̟ i N , Λ(4 i + 2) = ̟ i N , Λ(4 i + 3) = ̟ i +1 N , i ∈ Z , (2.1)(2) S = { N , N } :Λ(3 i ) = ̟ i N , Λ(3 i + 1) = ̟ i N , Λ(3 i + 2) = ̟ i +1 N , i ∈ Z , (2.2)(3) S = { N , N } :Λ(3 i ) = ̟ i N , Λ(3 i + 1) = ̟ i N , Λ(3 i + 2) = ̟ i +1 N , i ∈ Z , (2.3)(4) S = { N , N } : Λ(2 i ) = ̟ i N , Λ(2 i + 1) = ̟ i N , i ∈ Z , (2.4)(5) S = { N } : Λ(2 i ) = ̟ i N , Λ(2 i + 1) = ̟ i +1 N , i ∈ Z , (2.5)(6) S = { N } : Λ( i ) = ̟ i N , i ∈ Z , (2.6)(7) S = { N } : Λ( i ) = ̟ i N , i ∈ Z . (2.7) Remark 2.1.
By the argument above, if Λ is a strict self-dual o F -lattice sequence in V of period2, then we have [Λ( i ) : Λ( i + 1)] = q for all i ∈ Z . Theorem 2.2.
Let π be an irreducible smooth representation of G of positive level. Then π con-tains a fundamental skew stratum [Λ , n, n − , β ] which satisfies one of the following conditions:(i) Λ is a standard strict self-dual o F -lattice sequence and ( e (Λ) , n ) = 1 ,(ii) Λ is the strict o F -lattice sequence in (2.5) and n is even.Proof. Recall from [5] § C -sequence in V is a self-dual o F -lattice sequence Λ in V which satisfies C (i) Λ(2 i + 1) ) Λ(2 i + 2), i ∈ Z , C (ii) e (Λ) is even and d (Λ) is odd.It follows from [5] Proposition 3.1.1 that an irreducible smooth representation π of G of positivelevel contains a fundamental skew stratum [Λ , n, n − , β ] such that Λ is a C -sequence and( e (Λ) , n ) = 2. After conjugation by some element in G , we may assume Λ is a standard C -sequence.We claim that it suffices to find an o F -lattice sequence Λ ′ and an integer n ′ verifying one ofthe conditions in the theorem, n/e (Λ) = n ′ /e (Λ ′ ) and a n +1 (Λ) ⊃ a n ′ +1 (Λ ′ ). If this is the case,then π contains a P Λ ′ ,n ′ +1 -fixed vector and some skew stratum [Λ ′ , n ′ , n ′ − , β ′ ]. Proposition 1.5(iii) says that it is fundamental.For L a set of o F -lattices, we write L = { L | L ∈ L} . Note that Λ(2 Z ) and Λ(2 Z + 1) areclosed under the multiplication by elements in F × , and Λ(2 Z ) = Λ(2 Z + 1) . MICHITAKA MIYAUCHI (a) Suppose that Λ(2 Z ) = Λ(2 Z + 1). Then we have Λ(2 i ) = Λ(2 i + 1), i ∈ Z . Define an o F -lattice sequence Λ ′ by Λ ′ ( i ) = Λ(2 i ), i ∈ Z . Then Λ ′ is a strict standard self-dual o F -latticesequence such that e (Λ ′ ) = e (Λ) / a k (Λ ′ ) = a k − (Λ) = a k (Λ), k ∈ Z . Putting n ′ = n/ a n ′ +1 (Λ ′ ) = a n +1 (Λ).(b) Suppose that Λ(2 Z ) ∩ Λ(2 Z + 1) = ∅ . Then Λ is strict and Λ(2 Z ) ∩ Λ(2 Z ) = ∅ . Since N = N and N = ̟ − N , the set Λ(2 Z ) is either { ̟ m N | m ∈ Z } or { ̟ m N | m ∈ Z } ,and hence Λ( Z ) = { ̟ m N , ̟ m N | m ∈ Z } . Replacing Λ with some translate, we may assumeΛ is the o F -lattice sequence in (2.5), and [Λ , n, n − , β ] satisfies the condition (ii).(c) Suppose that Λ(2 Z ) = Λ(2 Z + 1) and Λ(2 Z ) ∩ Λ(2 Z + 1) = ∅ . Then we see that Λ( Z )contains both N and ̟N , and Λ(2 Z ) contains N or N .Suppose that Λ(2 Z ) (and hence Λ(2 Z + 1)) contains N . Replacing Λ by some translate,we may assume Λ(0) = Λ(1) = N . Since N and ̟N belong to Λ( Z ), we have Λ(2) = N and therefore Λ(3) should be either N or ̟N . If Λ(3) = N , then Λ(3) = Λ(4) = N since N ∈ Λ(2 Z + 1) = Λ(2 Z ). This contradicts to the condition C (i). We therefore haveΛ(0) = Λ(1) = N , Λ(2) = N , Λ(3) = ̟N (2.8)and e (Λ) = 4.Similarly, if Λ(2 Z ) contains N , then we may assume that Λ satisfies e (Λ) = 4 andΛ(1) = N , Λ(2) = Λ(3) = N , Λ(4) = ̟N . (2.9)In both cases, we have n = 4 m + 2, for m ≥
0. Let Λ ′ be the sequence in (2.4) and let n ′ = 2 m + 1. Then it is easy to check that a n +1 (Λ) ⊃ a n ′ +1 (Λ ′ ). This complete the proof. (cid:3) Remark 2.3.
Replacing Λ by some translate, an irreducible smooth representation π of G ofpositive level contains a fundamental skew stratum [Λ , n, n − , β ] such that Λ and n satisfy oneof the conditions listed below:Λ e (Λ) d (Λ) ( n, e (Λ)) e (Λ) / ( n, e (Λ))(2.1) 4 0 1 4(2.2) 3 0 1 3(2.3) 3 − − − − Characteristic polynomials.
Let Λ be a strict o F -lattice sequence in V and n an integercoprime to e (Λ). Let [Λ , n, n − , β ] be a stratum in A . We write e = e (Λ). Note that φ β ( X )coincides with the characteristic polynomial of y β + a (Λ) in the k F -algebra a (Λ) / a (Λ).As in [4] §
2, we can form a Z /e Z -graded k F -vector space Λ = L i ∈ Z /e Z Λ( i ), where Λ( i ) =Λ( i ) / Λ( i + 1). There is a natural isomorphism a k (Λ) / a k +1 (Λ) ≃ M i ∈ Z /e Z Hom k F (Λ( i ) , Λ( i + k )); x ( x i ) i ∈ Z /e Z , given by x i : v + Λ( i + 1) xv + Λ( i + k + 1) , x ∈ a k (Λ) , v ∈ Λ( i ) . For i ∈ Z /e Z , let β i and y i denote the image of β and y β in Hom k F (Λ( i ) , Λ( i − n )) andEnd k F (Λ( i )), respectively. If we write φ i ( X ) for the characteristic polynomial of y i , then wehave φ β ( X ) = Q i ∈ Z /e Z φ i ( X ). Choose j ∈ Z /e Z so that n j = dim k F (Λ( j )) is minimal. Since y i = β i − ( e − n · · · β i − n β i , we have φ β ( X ) = φ j ( X ) e X m , where m = dim F ( V ) − en j . EPRESENTATIONS OF U (2 ,
2) OVER A p -ADIC FIELD I 9 Suppose that [Λ , n, n − , β ] is skew. Then φ β (( − e X ) = φ β ( X ) since σ ( y β ) = ( − e y β . Wetherefore have φ β ( X ) = ( X − a ) , a ∈ k , if e = 4 , ( X − a √ ε ) X, a ∈ k , if e = 3 , ( X + aX + b ) , a, b ∈ k , if e = 2 ,X + a √ εX + bX + c √ εX + d, a, b, c, d ∈ k , if e = 1 . Every quadratic polynomial in k [ X ] is reducible in k F [ X ]. So, if e = 2, then there arefollowing three cases(a) φ β ( X ) = ( X − λ ) ( X − λ ) , λ ∈ k F , λ = λ ;(b) φ β ( X ) = ( X − u ) ( X − v ) , u, v ∈ k , u = v ;(c) φ β ( X ) = ( X − u ) , u ∈ k .3. Representations associated to maximal tori
The remaining part of this paper is devoted to a classification of the irreducible smoothrepresentations of G of non-integral level. In this section, we classify irreducible representationsof G which contain a semisimple skew stratum [Λ , n, n − , β ] such that the G -centralizer G β of β is compact. We also discuss the “intertwining implies conjugacy” property of such strata.3.1. Construction of supercuspidal representations.
We first recall the construction ofsupercuspidal representations from [16] and [17].Let [Λ , n, n − , β ] be a semisimple skew stratum in A . We write G β for the G -centralizer of β . By [16] Theorem 4.6, we get I G [Λ , n, [ n/ , β ] = P Λ , [( n +1) / G β P Λ , [( n +1) / . (3.1)Suppose that G β is compact. Then G β lies in P Λ , , and hence I G [Λ , n, [ n/ , β ] = G β P Λ , [( n +1) / .This implies that the intertwining of ( P Λ , [ n/ , ψ β ) coincides with the normalizer of ( P Λ , [ n/ , ψ β ).So we obtain the following: Proposition 3.1.
Let [Λ , n, n − , β ] be a semisimple skew stratum such that G β is compact.Set J = G β P Λ , [( n +1) / . Then the map ρ Ind GJ ρ gives a bijection between Irr( J ) ( P Λ , [ n/ ,ψ β ) and Irr( G ) ( P Λ , [ n/ ,ψ β ) .Proof. For ρ ∈ Irr( J ) ( P Λ , [ n/ ,ψ β ) , the restriction of ρ to P Λ , [ n/ is a multiple of ψ β since J normalizes ψ β . If g ∈ G intertwines ρ , then g intertwines ψ β , and hence g lies in J . Thus, Ind GJ ρ is irreducible and supercuspidal.The surjectivity of this map follows from Frobenius reciprocity. Let ρ i ∈ Irr( J ) ( P Λ , [ n/ ,ψ β ) , i = 1 ,
2. Suppose that Ind GJ ρ is isomorphic to Ind GJ ρ . Then there is g ∈ G which intertwines ρ and ρ . Restricting these to P Λ , [ n/ , we get g ∈ J , and ρ ≃ ρ g ≃ ρ . This implies theinjectivity. (cid:3) We shall give a description of Irr( J ) ( P Λ , [ n/ ,ψ β ) when G β is a maximal torus in G . In thiscase, the group G β is abelian. Put H = ( G β ∩ P Λ , ) P Λ , [ n/ and J = ( G β ∩ P Λ , ) P Λ , [( n +1) / .Let σ ∈ Irr( J ) ( P Λ , [ n/ ,ψ β ) . Since H /P Λ , [ n/ ≃ ( G β ∩ P Λ , ) / ( G β ∩ P Λ , [ n/ ), σ containssome extension θ of ψ β to H . Applying the proof of [17] Proposition 4.1, we see that thereexists a unique irreducible representation η θ of J containing θ . Moreover, the restriction of η θ to H is the [ J : H ] / -multiple of θ . The group J/J ≃ G β /G β ∩ P Λ , is cyclic with ordercoprime to [ J : H ] / since [ J : H ] / is a power of q . Hence η θ can extend to an irreduciblerepresentation κ θ of J . It is easy to observe that Irr( J ) ( J ,η θ ) = { χ ⊗ κ θ | χ ∈ ( G β /G β ∩ P Λ , ) ∧ } ,and χ ⊗ κ θ ≃ χ ′ ⊗ κ θ if and only if χ = χ ′ , for χ, χ ′ ∈ ( G β /G β ∩ P Λ , ) ∧ . In particular, therestriction of every element in Irr( J ) ( J ,η θ ) to H is a multiple of θ . We conclude the following: Proposition 3.2.
With the notation as above, the set
Irr( J ) ( P Λ , [ n/ ,ψ β ) is a disjoint union S θ Irr( J ) ( H ,θ ) , where θ runs over the extensions of ψ β to H . We have Irr( J ) ( H ,θ ) = { χ ⊗ κ θ | χ ∈ ( G β /G β ∩ P Λ , ) ∧ } . Representations of level n/ . In this section, we fix a positive integer n coprime to 4and the o F -lattice sequence Λ in (2.1). By Remark 2.3, an irreducible smooth representation of G of level n/ , n, [ n/ , β ].Let Λ ′ be a strict self-dual o F -lattice sequence in V of period 4 and [Λ ′ , n, n − , β ] a fun-damental skew stratum. Proposition 1.8 says that this is a skew simple stratum and E β is atotally ramified extension of degree 4 over F , where E β = F [ β ]. Propositions 3.1 and 3.2 give aclassification of the irreducible smooth representations of G which contain [Λ ′ , n, [ n/ , β ].We give a necessary and sufficient condition when two such strata occur in a common irre-ducible representation of G . Theorem 3.3.
Let Λ be the o F -lattice sequence in (2.1) and let n be a positive integer coprimeto . Let [Λ , n, [ n/ , β ] and [Λ , n, [ n/ , γ ] be fundamental skew strata. Suppose that there isan irreducible smooth representation of G containing [Λ , n, [ n/ , β ] and [Λ , n, [ n/ , γ ] . Then ( P Λ , [ n/ , ψ γ ) is a P Λ , -conjugate of ( P Λ , [ n/ , ψ β ) .Proof. By Proposition 1.5 (i), there is g ∈ G such that ( β + g Λ , − [ n/ ) ∩ Ad( g )( γ + g Λ , − [ n/ ) = ∅ .Take δ ∈ ( β + g Λ , − [ n/ ) ∩ Ad( g )( γ + g Λ , − [ n/ ). We obtain fundamental skew strata [Λ , n, n − , δ ]and [ g Λ , n, n − , δ ]. We see that Λ and g Λ are strict o E δ -lattice sequence in the one dimension E δ -space V , and hence Λ is a translate of g Λ. Since d ( g Λ) = d (Λ), we get Λ = g Λ and g ∈ P Λ , .This completes the proof. (cid:3) Representations of level n/ . We fix a positive integer n coprime to 3. By Remark 2.3,an irreducible smooth representation of G of level n/ , n, [ n/ , β ] such that Λ is the o F -lattice sequence in (2.2) or (2.3).Let Λ be any strict self-dual o F -lattice sequence in V with e (Λ) = 3 and let [Λ , n, [ n/ , β ]be a fundamental skew stratum. As seen in § φ β ( X ) = ( X − a √ ε ) X , for a ∈ k × .Using Hensel’s Lemma, we can lift this to Φ β ( X ) = f a ( X ) f ( X ) where f a ( X ), f ( X ) are monic, f a ( X ) mod p F = ( X − a √ ε ) and f ( X ) mod p F = X . Applying the argument in the proof of[18] Theorem 4.4, if we put V a = ker f a ( y β ) and V = ker f ( y β ), then [Λ , n, [ n/ , β ] is splitwith respect to V = V a ⊥ V . As usual, for b ∈ { a, } , we write β b = β | V b and Λ b ( i ) = Λ( i ) ∩ V b , i ∈ Z . Lemma 3.4.
The form ( V , f | V ) represents 1 if and only if d (Λ) is even.Proof. Suppose that d (Λ) = 2 k , k ∈ Z . Then the dual lattice of Λ ( k ) in ( V , f | V ) is itself.Since dim F V = 1, we see that the form f | V represents 1.If d (Λ) = 2 k + 1, k ∈ Z , then the dual of Λ ( k + 2) in ( V , f | V ) is ̟ − Λ ( k + 2). The form f | V represents ̟ , and does not represent 1 since F is unramified over F . (cid:3) Proposition 3.5.
Let Λ and Λ ′ be the o F -lattice sequence defined by (2.2) and (2.3), respectively.Let [Λ , n, [ n/ , β ] and [Λ ′ , n, [ n/ , γ ] be fundamental skew strata. Then, there are no irreduciblesmooth representations of G which contain [Λ , n, [ n/ , β ] and [Λ , n, [ n/ , γ ] .Proof. Suppose that there exists an irreducible smooth representation of G containing [Λ , n, [ n/ , β ]and [Λ , n, [ n/ , γ ]. By Proposition 1.5 (i), we can take δ ∈ ( β + g Λ , − [ n/ ) ∩ (Ad( g ) γ + g g Λ ′ , − [ n/ ),for some g ∈ G .Applying the above construction of a splitting to fundamental skew strata [Λ , n, n − , δ ] and[ g Λ ′ , n, n − , δ ], it follows from Lemma 3.4 that d (Λ) ≡ d ( g Λ ′ ) (mod 2), which contradicts thechoice of Λ and Λ ′ . This completes the proof. (cid:3) EPRESENTATIONS OF U (2 ,
2) OVER A p -ADIC FIELD I 11 Due to [8] Lemma 3.6, Λ a is a strict o F -lattice sequence of period 3 in the 3-dimensional F -space V a , and by Proposition 1.8, [Λ a , n, n − , β a ] is simple. Let E a = F [ β a ]. Since Λ a is astrict o E a -lattice sequence in the one-dimensional E a -space V a , the integer d (Λ) determines Λ a uniquely. Similarly, Λ is an o F -lattice sequence in the one-dimensional F -space V of period 3,so it is also determined by d (Λ). We therefore see that given such a stratum [Λ , n, n − , β ], wecan reconstruct Λ from d (Λ).We get the following theorem, imitating the proof of Theorem 3.3. Theorem 3.6.
Let Λ be the o F -lattice sequence in (2.2) or (2.3), and let n be a positive in-teger coprime to . Let [Λ , n, [ n/ , β ] and [Λ , n, [ n/ , γ ] be fundamental skew strata. Supposethat there exists an irreducible smooth representation of G which contains [Λ , n, [ n/ , β ] and [Λ , n, [ n/ , γ ] . Then ( P Λ , [ n/ , ψ γ ) is a P Λ , -conjugate of ( P Λ , [ n/ , ψ β ) . Let [Λ , n, [ n/ , β ] be a skew stratum as above. The restriction [Λ , n, n − , β ] is not semisimpleonly if β = 0 since φ β ( X ) = X . Even in this case, β is scalar and does not matter incomputing the intertwining. Hence we can classify the irreducible smooth representations of G which contain [Λ , n, [ n/ , β ], using Propositions 3.1 and 3.2.3.4. Representations of half-integral level.
We fix a positive odd integer n . Let π be anirreducible smooth representation of G of level n/ X − a ) ( X − b ) , for a ∈ k × , b ∈ k , a = b. (3.2)Then by Remark 2.3, π contains a fundamental skew stratum [Λ , n, [ n/ , β ] such that Λ is thesequence in (2.4) or (2.5) and φ β ( X ) has the form (3.2).Let Λ be any strict self-dual o F -lattice sequence in V with e (Λ) = 2 and let [Λ , n, n − , β ]be a fundamental skew stratum such that φ β ( X ) has the form (3.2). As in § φ β ( X ) to Φ β ( X ) = f a ( X ) f b ( X ) where, for c ∈ { a, b } , f c ( X ) is monic and f c ( X ) mod p F = ( X − c ) . As in the proof of [18] Theorem 4.4, if we put V a = ker f a ( y β ) and V b = ker f b ( y β ), then [Λ , n, [ n/ , β ] is split with respect to V = V a ⊥ V b .For c ∈ { a, b } , we write β c = β | V c and Λ c ( i ) = Λ( i ) ∩ V c , i ∈ Z . It follows from [8] Lemma3.6 that Λ a is a strict o F -lattice sequence in V a of period 2, and hence [Λ a ( i ) : Λ a ( i + 1)] = q for all i ∈ Z . By Remark 2.1, Λ b is also strict. Lemma 3.7. (i) If d (Λ) is even, then ( V a , f | V a ) and ( V b , f | V b ) are isotropic.(ii) If d (Λ) is odd, then ( V a , f | V a ) and ( V b , f | V b ) are anisotropic.Proof. The assertion follows from Lemma 1.10. (cid:3)
Proposition 3.8.
Let Λ and Λ ′ be the o F -lattice sequences in (2.4) and (2.5), respectively. Let [Λ , n, [ n/ , β ] and [Λ ′ , n, [ n/ , γ ] be fundamental skew strata. Suppose φ β ( X ) = φ γ ( X ) = ( X − a ) ( X − b ) , for a, b ∈ k such that a = b . Then there are no irreducible smooth representationsof G which contain [Λ , n, [ n/ , β ] and [Λ , n, [ n/ , γ ] .Proof. The assertion follows from Lemma 3.7 as in the proof of Proposition 3.5. (cid:3)
For c ∈ { a, b } , we put E c = F [ β c ]. Proposition 1.8 says that [Λ a , n, n − , β a ] is simple andΛ a is a strict o E a -lattice sequence in the one-dimensional E a -space V a . An analog holds for[Λ b , n, n − , β b ] if b = 0. In this case, d (Λ) determines Λ.Suppose that d (Λ) is odd. Lemma 3.7 implies the uniqueness of Λ c in the anisotropic space( V c , f | V c ) up to translation. Hence d (Λ) determines Λ. Theorem 3.9.
Let Λ be the o F -lattice sequence in (2.4) or (2.5), and let n a positive odd integer.Let [Λ , n, [ n/ , β ] and [Λ , n, [ n/ , γ ] be skew strata such that φ β ( X ) = φ γ ( X ) = ( X − a ) ( X − b ) , a = b . Assume that ab = 0 or Λ is defined by (2.5). Suppose that there exists an irreduciblesmooth representation of G containing [Λ , n, [ n/ , β ] and [Λ , n, [ n/ , γ ] . Then ( P Λ , [ n/ , ψ γ ) is a P Λ , -conjugate of ( P Λ , [ n/ , ψ β ) . Proof.
This is exactly as in the proof of Theorem 3.3. (cid:3)
We shall describe the set Irr( G ) ( P Λ , [ n/ ,ψ β ) under the assumption of Theorem 3.9. If ab = 0,then we can apply Propositions 3.1 and 3.2.Suppose that Λ is defined by (2.5) and b = 0. We write G c = G ∩ End F ( V c ), for c ∈ { a, } .Let G β a denote the G a -centralizer of β a and I the formal intertwining of [Λ , n, [ n/ , β ] in G .Since G coincides with P Λ , , I is just the normalizer of ( P Λ , [ n/ ∩ G , ψ β | P Λ , [ n/ ∩ G ) in G . By [16] Theorem 4.6 and Corollary 4.14, we have I G [Λ , n, [ n/ , β ] = G ′ P Λ , [ n/ , (3.3)where G ′ = G β a × I . Therefore, we get a natural isomorphism of Hecke algebras H ( G//P Λ , [ n/ , ψ β ) ≃ H ( G ′ //P ′ , ψ ′ ) , (3.4)where P ′ = G ′ ∩ P Λ , [ n/ and ψ ′ = ψ β | P ′ . Via this isomorphism, we can identify Irr( G ) ( P Λ , [ n/ ,ψ β ) with Irr( G ′ ) ( P ′ ,ψ ′ ) . Moreover, every irreducible representation of G containing [Λ , n, [ n/ , β ] issupercuspidal since G ′ is compact.4. Hecke algebra isomorphisms
In this section, we complete our classification of the irreducible smooth representations of G ofnon-integral level. The remaining representations we need to consider are those of half-integrallevel. We fix a positive integer m and put n = 2 m −
1. By Proposition 1.5, the set of equivalenceclasses of irreducible smooth representations of G of level n/ o F -lattice sequence of period 2. Asclaimed in § , n, n − , β ] hasone of the following forms:(4a) ( X − λ ) ( X − λ ) , λ ∈ k F , λ = λ ;(4b) ( X − u ) ( X − v ) , u ∈ k × , v ∈ k , u = v ;(4c) ( X − u ) , u ∈ k × .We have classified the representations of case (4b) in § X − u ) X , u ∈ k × and Λ is defined by (2.4).4.1. Normalization of β . Let Λ and Λ ′ be the o F -lattice sequences in (2.4) and (2.5), re-spectively. By Remark 2.3, an irreducible smooth representation of G of level n/ , n, n − , β ] or [Λ ′ , n, n − , β ′ ].The filtrations { a k (Λ) } k ∈ Z and { a k (Λ ′ ) } k ∈ Z are given by the following: a (Λ) = o F o F o F o F o F o F o F o F p F p F o F o F p F p F o F o F , a (Λ) = p F p F o F o F p F p F o F o F p F p F p F p F p F p F p F p F , (4.1) a (Λ ′ ) = o F o F o F p − F p F o F o F o F p F o F o F o F p F p F p F o F , a (Λ ′ ) = p F o F o F o F p F p F p F o F p F p F p F o F p F p F p F p F . (4.2)Up to equivalence classes of skew strata, we can take β ∈ g Λ , − n and β ′ ∈ g Λ ′ , − n to be β = ̟ − m C a √ ε b √ ε − C̟D ̟c √ ε ̟d √ ε − ̟D , C, D ∈ o F , a, b, c, d ∈ o ; (4.3) EPRESENTATIONS OF U (2 ,
2) OVER A p -ADIC FIELD I 13 β ′ = ̟ − m Z C ̟M − C̟D − Z − ̟D − ̟M , C, D, M, Z ∈ o F . (4.4) Lemma 4.1.
Let [Λ ′ , n, n − , β ′ ] be a fundamental skew stratum whose characteristic polynomialis of the form (4a) or (4c). Suppose that an irreducible smooth representation π of G contains [Λ ′ , n, n − , β ′ ] . Then there exists a fundamental skew stratum [Λ , n, n − , β ] which occurs in π .Proof. Take β ′ as in (4.4). By assumption, after conjugation by an element in P Λ ′ , , we mayassume y β is upper triangular. By replacing β with a P Λ ′ , -conjugate again, we may assume that Z ∈ p F . Then we have β ′ + a − n (Λ ′ ) ⊂ a − n (Λ). Thus the assertion follows from the standardargument. (cid:3) By Lemma 4.1, it suffices to consider irreducible smooth representations of G containing afundamental skew stratum [Λ , n, n − , β ] such that Λ is the o F -lattice sequences in (2.4) and φ β ( X ) has the form (4a), (4c) or (4d). We shall normalize β as in [13] Case (5.1b). Set X = (cid:18) C a √ εb √ ε − C (cid:19) , Z = (cid:18) D c √ εd √ ε − D (cid:19) , S = (cid:18) (cid:19) and abbreviate β = ̟ − m (cid:18) X̟Z (cid:19) . Then we have y β = (cid:18) XZ ZX (cid:19) . For Y ∈ GL ( o F ), we define an element g = g ( Y ) in P Λ , by g = (cid:18) Y S t Y − S (cid:19) . Without loss, we can replace [Λ , n, n − , β ] by [Λ , n, n − , Ad ( g )( β )]. Observe that Ad ( g )( β ) = ̟ − m (cid:18) Y XS t Y S̟S t Y − SZY − (cid:19) , and Ad ( g )( y β ) = (cid:18) Ad ( Y )( XZ ) 00 Ad ( S t Y − S )( ZX ) (cid:19) . We will confuse elements in k F with those in o F .Case (4a): By Hensel’s lemma, we may assume that the characteristic polynomial of XZ has theform ( X − λ )( X − λ ), for λ ∈ o F such that λ λ (mod p F ). Replacing β by a P Λ , -conjugate,we may assume XZ = (cid:18) λ λ (cid:19) . The spaces
F e ⊕ F e and F e ⊕ F e are the kernels of y β − λ and y β − λ , respectively, so thatthese spaces are stable under the action of β . Thus X and Z are diagonal. After conjugationby a diagonal matrix in P Λ , , we may assume X = (cid:18) − (cid:19) , Z = (cid:18) λ − λ (cid:19) . (4.5) Case (4c): After P Λ , -conjugation, we may assume XZ is upper triangular modulo p F . After g ( Y )-conjugation by some upper triangular Y ∈ GL N ( o F ), we may also assume X is antidiagonalor upper triangular modulo p F .If X is antidiagonal, then Z is also antidiagonal hence XZ is scalar. We can take X to beupper triangular by replacing β with a P Λ , -conjugate. So we can always assume X is uppertriangular. Then Z must be upper triangular as well. Replacing β by some P Λ , -conjugate, wecan assume X = (cid:18) − (cid:19) , Z = (cid:18) u c √ ε − u (cid:19) , u ∈ o × , c ∈ o F . (4.6)Case (4d): As in the case (4a), we may assume XZ = (cid:18) u v (cid:19) , u ∈ o × , v ∈ p , so that X and Z are both anti-diagonal. Replacing β by a P Λ , -conjugate, we may assume that ̟ m β is one of the following elements: √ ε ̟d √ ε , √ ε √ ε
00 0 0 0 ̟d √ ε , √ ε ̟ √ ε ̟d √ ε , where d = uε − ∈ o × . Remark 4.2.
In case (4a), we can swap λ for λ .4.2. Case (4c).
Let Λ ′ be the o F -lattice sequence defined by (2.4). We fix a positive integer m and α ∈ o × . Put n ′ = 2 m −
1. For c ∈ o , set β ( c ) = ̟ − m − ̟α ̟c √ ε − ̟α ∈ g Λ ′ , − n ′ . (4.7)Define a self-dual o F -lattice sequence Λ in V with e (Λ) = 8 and d (Λ) = 2 byΛ(0) = Λ(1) = Λ(2) = N , Λ(3) = N , Λ(4) = Λ(5) = Λ(6) = N , Λ(7) = ̟N , Λ( i + 8 k ) = ̟ k Λ( i ) , ≤ i ≤ , k ∈ Z . (4.8)Then the induced filtration { a n (Λ) } n ∈ Z is given by a (Λ) = o F o F o F o F p F o F o F o F p F p F o F o F p F p F p F o F , a (Λ) = p F o F o F o F p F p F o F o F p F p F p F o F p F p F p F p F , a (Λ) = a (Λ) = p F p F o F o F p F p F o F o F p F p F p F p F p F p F p F p F , a (Λ) = p F p F o F o F p F p F p F o F p F p F p F p F p F p F p F p F , a (Λ) = p F p F p F o F p F p F p F p F p F p F p F p F p F p F p F p F , a (Λ) = a (Λ) = p F p F p F p F p F p F p F p F p F p F p F p F p F p F p F p F . (4.9)Put n = 8 m − β = β (0). EPRESENTATIONS OF U (2 ,
2) OVER A p -ADIC FIELD I 15 Lemma 4.3.
Suppose that an irreducible smooth representation π of G contains a skew stratum [Λ ′ , n ′ , n ′ − , β ( c )] , for some c ∈ o . Then π contains [Λ , n, n − , β ] .Proof. The assertion follows from the inclusion β ( c ) + a − n ′ (Λ ′ ) ⊂ β + a − n (Λ). (cid:3) Remark 4.4.
Let π be an irreducible smooth representation of G . By Lemma 4.3 and theargument in § π is of level n/ X − α ) (mod p F ) if andonly if π contains [Λ , n, n − , β ].We are going to classify the irreducible smooth representations of G of level n/ X − α ) (mod p F ). Let E = F [ β ] and let B denote the A -centralizerof β . Then E and B are σ -stable, and the algebra E is a totally ramified extension of degree 2over F with uniformizer ̟ E = ̟ m β √ ε . Put E = { x ∈ E | σ ( x ) = x } and G ′ = G ∩ B . Then E is the quadratic unramified extension over E and G ′ is the unramified unitary group over E corresponding to the involutive algebra ( B, σ ). Note that ̟ E ∈ E .Observe that β Λ( i ) = Λ( i − n ), i ∈ Z . Then Λ is an o E -lattice sequence in V of o E -period 4since o E = o F [ ̟ E ]. Since β ∈ F × , the map s : A → B defined by s ( X ) = ( X + βXβ − ) / , X ∈ A (4.10)is a ( B, B )-bimodule projection. Let B ⊥ denote the orthogonal complement of B in A withrespect to the pairing induced by tr A/F . We have B ⊥ = ker s and A = B ⊕ B ⊥ . Note that theset B ⊥ is also σ -stable.For k ∈ Z , we abbreviate a k = a k (Λ), a ′ k = a k ∩ B and a ⊥ k = a k (Λ) ∩ B ⊥ . Proposition 4.5.
For all k ∈ Z , we have a k = a ′ k ⊕ a ⊥ k .Proof. Since β normalizes a k , we have s ( a k ) = a ′ k . The proposition follows from the equation X = s ( X ) + ( X − s ( X )), X ∈ A . (cid:3) As in [13] (4.16), we define a σ -stable o F -lattice in A by J = a ′ n ⊕ a ⊥ [ n/ (4.11)and an open compact subgroup J = (1 + J ) ∩ G of G . Then we get P Λ ,n ⊂ J ⊂ P Λ , [ n/ andhence the quotient J/P Λ ,n +1 is abelian. As usual, there is an isomorphism g Λ , − n / g ∩ J ∗ ≃ ( J/P Λ ,n +1 ) ∧ ; b + g ∩ J ∗ Ψ b , given by Ψ b ( x ) = Ω(tr A/F ( b ( x − , x ∈ J. (4.12)By Proposition 4.5 and [2] (2.10), we have J ∗ = a ′ − n ⊕ a ⊥− [ n/ . (4.13) Remark 4.6.
In general, we need to consider the lattice a ′ n ⊕ a ⊥ [( n +1) / . But in this case, itfollows from (4.9) that a ⊥ [ n/ = a ⊥ [( n +1) / .We also abbreviate g k = a k ∩ g , g ′ k = a ′ k ∩ g and g ⊥ k = a ⊥ k ∩ g , for k ∈ Z . For X ∈ A , we writead( β )( X ) = βX − Xβ . Since β ∈ g ′− n , the map ad( β ) induces a quotient mapad( β ) : g ⊥ k / g ⊥ k +1 → g ⊥ k − n / g ⊥ k − n +1 , k ∈ Z . Lemma 4.7.
For k ∈ Z , the map ad( β ) : g ⊥ k / g ⊥ k +1 ≃ g ⊥ k − n / g ⊥ k − n +1 is an isomorphism.Proof. By the periodicity of { a k } k ∈ Z , it suffices to prove that ad( β ) induces an injection, for all k ∈ Z . Let X ∈ g ⊥ k satisfy ad( β )( X ) ∈ g ⊥ k − n +1 . Since β − ∈ a n , we have X = X − s ( X ) = − ad( β )( X ) β − / ∈ g ⊥ k +1 , as required. (cid:3) Proposition 4.8 ([13] Lemma 4.4) . Suppose that an element γ ∈ β + g Λ , − n lies in g ∩ B modulo g Λ ,k − n for some integer k ≥ . Then, there exists p ∈ P Λ ,k such that Ad( p )( γ ) ∈ g ∩ B .Proof. Exactly the same as the proof of [13] Lemma 4.4. (cid:3)
As an immediate corollary of the proof we have
Corollary 4.9 ([13] Corollary 4.5) . Ad( J )( β + g ′ − n ) = β + g ∩ J ∗ . Proposition 4.10 ([13] Theorem 4.1) . Let π be an irreducible smooth representation of G . Then π contains [Λ , n, n − , β ] if and only if the restriction of π to J contains Ψ β .Proof. Since Ψ β is an extension of ψ β to J , π contains [Λ , n, n − , β ] if π contains Ψ β .Suppose that π contains [Λ , n, n − , β ]. Then we can find γ ∈ β + g Λ , − n so that a skewstratum [Λ , n, [ n/ , γ ] occurs in π . By Proposition 4.8, after conjugation by some element in P Λ , , we may assume γ lies in β + g ′ − n . This implies that the restriction of ψ γ to J is Ψ β . Thiscompletes the proof. (cid:3) We write Ψ = Ψ β , J ′ = J ∩ G ′ and Ψ ′ = Ψ | J ′ . Theorem 4.11.
With the notation as above, there is a ∗ -isomorphism η : H ( G ′ //J ′ , Ψ ′ ) ≃H ( G//J, Ψ) which preserves support, that is, η satisfies supp( η ( f )) = J supp( f ) J , for f ∈H ( G ′ //J ′ , Ψ ′ ) . Remark 4.12.
The isomorphism η allows us to identify Irr( G ′ ) ( J ′ , Ψ ′ ) with Irr( G ) ( J, Ψ) . Since Ψ ′ can extend to a character of G ′ , we obtain a bijection from Irr( G ′ ) ( J ′ , to Irr( G ) ( J, Ψ) , where 1denotes the trivial representation of J ′ . Since the centers of G and G ′ are compact, the supportpreservation of η implies that this map preserves the supercuspidality of representations.We commence the proof of Theorem 4.11. Proposition 4.13. I G (Ψ) = J G ′ J .Proof. Note that an element g in G lies in I G (Ψ) if and only if Ad( g )( β + g ∩ J ∗ ) ∩ ( β + g ∩ J ∗ ) = ∅ . Clearly, we have J I G (Ψ) J = I G (Ψ), G ′ ⊂ I G (Ψ) and hence J G ′ J ⊂ I G (Ψ).Let g ∈ I G (Ψ). Then Corollary 4.9 says that there exists an element k in J gJ such thatAd( k )( β + g ′ − n ) ∩ ( β + g ′ − n ) = ∅ . Then there are x, y in g ′ − n such that ad( β )( k ) = kx − yk .Projecting this equation on B ⊥ , we have ad( β )( k ⊥ ) = k ⊥ x − yk ⊥ , where k ⊥ denote the B ⊥ -component of k . Suppose k ⊥ ∈ a l , for some l ∈ Z . Then we have ad( β )( k ⊥ ) ∈ a l − n +1 , and byapplying the proof of Lemma 4.7, k ⊥ ∈ a l +1 . So we conclude k ⊥ = 0 and hence k ∈ G ′ . Thiscompletes the proof. (cid:3) For x in A , we denote by x ′ its B -component and by x ⊥ its B ⊥ -component. Lemma 4.14.
For g ∈ J G ′ J , we have ν Λ ( g ⊥ ) ≥ ν Λ ( g ′ ) + [ n/
2] + 1 .Proof.
Put k = ν Λ ( g ). Then, for any element y in J gJ , we have y ≡ g (mod a k +[ n/ ), so that y ≡ g ⊥ (mod B + a k +[ n/ ). Therefore if g ∈ J G ′ J , then g ⊥ ∈ a ⊥ k +[ n/ . Now the lemmafollows immediately. (cid:3) Proposition 4.15.
For g ∈ G ′ , we have J gJ ∩ G ′ = J ′ gJ ′ .Proof. Since βσ ( β ) = − β ∈ F × , the map Ad( β ) gives an automorphism on G of order 2 and J is a pro- p subgroup of G closed under Ad( β ). The assertion follows from [16] Lemma 2.1. (cid:3) We can identify M ( E ) with B as follows: For a, b, c, d, x, y, z, w ∈ F , X = (cid:18) a + ̟ E x b + ̟ E yc + ̟ E z d + ̟ E w (cid:19) a − b √ ε − x √ ε y − c √ ε d − εz − w √ ε̟αx √ ε − ̟αy a b √ ε − ̟αεz − ̟αw √ ε c √ ε d . EPRESENTATIONS OF U (2 ,
2) OVER A p -ADIC FIELD I 17 The involution σ on M ( E ) induced by this identification maps X to σ ( X ) = (cid:18) d + ̟ E w b + ̟ E yc + ̟ E z a + ̟ E x (cid:19) . So the group G ′ is isomorphic to the unramified U (1 ,
1) over E .We write B ′ = G ′ ∩ P Λ , . Since the o E -lattice sequence Λ satisfies Λ(0) ) Λ(3) ) ̟ E Λ(0),the group B ′ is the Iwahori subgroup of G ′ and J ′ = G ′ ∩ (1 + ̟ m − E a (Λ)) is its (4 m − s = (cid:18) (cid:19) , s = (cid:18) ̟ − E ̟ E (cid:19) ∈ G ′ ,S = { s , s } , and W ′ = h S i . Then we have a Bruhat decomposition G ′ = B ′ W ′ B ′ . Lemma 4.16.
Let t be a non negative integer. Then we have(i) [ J ′ ( s s ) t J ′ : J ′ ] = [ J ′ ( s s ) t J ′ : J ′ ] = q t , [ J ′ ( s s ) t s J ′ : J ′ ] = [ J ′ ( s s ) t s J ′ : J ′ ] = q t +1 .(ii) [ J ( s s ) t J : J ] = [ J ( s s ) t J : J ] = q t , [ J ( s s ) t s J : J ] = q t +1 , [ J ( s s ) t s J : J ] = q t +3 .Proof. For g in G ′ , we have [ J ′ gJ ′ : J ′ ] = [ J ′ : J ′ ∩ gJ ′ g − ] = [ g ′ n : g ′ n ∩ g g ′ n g − ] and[ J gJ : J ] = [ J : J ∩ gJ g − ] = [ g ∩ J : g ∩ J ∩ g J g − ]= [ g ′ n : g ′ n ∩ g g ′ n g − ] · [ g ⊥ [ n/ : g ⊥ [ n/ ∩ g g ⊥ [ n/ g − ] . These can be directly calculated for w ∈ W ′ by (4.9). (cid:3) For µ ∈ E and ν ∈ E × , we define u ( µ ) = (cid:18) µ (cid:19) , u ( µ ) = (cid:18) µ (cid:19) , and h ( ν ) = (cid:18) ν σ ( ν ) − (cid:19) . For g ∈ G ′ , let e g denote the element in H ( G ′ //J ′ , Ψ ′ ) such that e g ( g ) = 1 and supp( e g ) = J ′ gJ ′ . Theorem 4.17.
The algebra H ( G ′ //J ′ , Ψ ′ ) is generated by the elements e g , g ∈ B ′ ∪ S . Theseelements satisfy the following relations:(i) e k = Ψ( k ) e , k ∈ J ′ ,(ii) e k ∗ e k ′ = e kk ′ , k, k ′ ∈ B ′ ,(iii) e s ∗ e k = e sks ∗ e s , s ∈ S , k ∈ sB ′ s ∩ B ′ ,(iv) e s ∗ e s = [ J ′ s J ′ : J ′ ] P x ∈ o E / p E e u ( ̟ m − E x √ ε ) , e s ∗ e s = [ J ′ s J ′ : J ′ ] P x ∈ o E / p E e u ( ̟ m − E x √ ε ) ,(v) For µ ∈ o × E √ ε , e s ∗ e u ( µ ) ∗ e s = [ J ′ s J ′ : J ′ ] e u ( µ − ) ∗ e s ∗ e h ( µ ) ∗ e u ( µ − ) ,e s ∗ e u ( ̟ E µ ) ∗ e s = [ J ′ s J ′ : J ′ ] e u ( ̟ E µ − ) ∗ e s ∗ e h ( − µ − ) ∗ e u ( ̟ E µ − ) . The above relations are a defining set for this algebra.Proof.
The proof is very similar to the proof of [3] Chapter 3 Theorem 2.1. (cid:3)
For g ∈ G ′ , let f g denote the element in H ( G//J,
Ψ) such that f g ( g ) = 1 and supp( f g ) = J gJ . Theorem 4.18.
The algebra H ( G//J, Ψ) is generated by f g , g ∈ S ∪ B ′ and satisfies the followingrelations:(i) f k = Ψ( k ) f , k ∈ J ′ ,(ii) f k ∗ f k ′ = f kk ′ , k, k ′ ∈ B ′ ,(iii) f s ∗ f k = f sks ∗ f s , s ∈ S , k ∈ sB ′ s ∩ B ′ ,(iv) f s ∗ f s = [ J s J : J ] P x ∈ o E / p E f u ( ̟ m − E x √ ε ) , f s ∗ f s = [ J s J : J ] P x ∈ o E / p E f u ( ̟ m − E x √ ε ) ,(v) For µ ∈ o × E √ ε , f s ∗ f u ( µ ) ∗ f s = [ J s J : J ] f u ( µ − ) ∗ f s ∗ f h ( µ ) ∗ f u ( µ − ) ,f s ∗ f u ( ̟ E µ ) ∗ f s = q f u ( ̟ E µ − ) ∗ f s ∗ f h ( − µ − ) ∗ f u ( ̟ E µ − ) . Proof.
Recall that if x, y ∈ G ′ satisfy [ J xJ : J ][ J yJ : J ] = [ J xyJ : J ], then J xJ yJ = J xyJ and f x ∗ f y = f xy .By Proposition 4.13, H ( G//J,
Ψ) is linearly spanned by f g , g ∈ G ′ . For g ∈ G ′ , we write g = b wb where b , b ∈ B ′ and w ∈ W ′ . Then we have f g = f b ∗ f w ∗ f b because B ′ normalizes J . Let w = s i s i · · · s i l be a minimal expression for w with s i j ∈ S . It follows from Lemma 4.16that f w = f s i ∗ f s i ∗ · · · ∗ f s il . Therefore H ( G//J,
Ψ) is generated by f g , g ∈ B ′ ∪ S .Relations (i), (ii), and (iii) are obvious. Since [ J s J : J ] = [ J ′ s J ′ : J ′ ], we can take a commonsystem of representatives for J/J ∩ sJ s and J ′ /J ′ ∩ sJ ′ s . Then the proof of relations on s isalso obvious.We shall give the proof of relations on s only in the case when m = 2 k + 1, k ≥
0, and omitthe easy modification when m is even.We will abbreviate s = s . By the identification M ( E ) = B , we have s = ̟ − α − ε − − ε − ε − ̟αε . (4.14)We can take a common system of representatives for J/J ∩ sJ s and J ∩ sJ s \ J as x ( a, b, c ) = ̟ k +1 a − ̟ m cε ̟ k α − b
00 0 1 0 ̟ k +1 b ̟ k +1 a + ̟ m cε , (4.15)where a, b ∈ o √ ε/ p √ ε, c ∈ o / p . Note that x ( a, b, c ) lies in the kernel of Ψ.(iv) For g ∈ G , let δ g denote the unit point mass at g . Then we have f s = X x ∈ J ∩ sJs \ J Ψ − ( x ) f ∗ δ sx = X y ∈ J/J ∩ sJs Ψ − ( y ) δ ys ∗ f , so that f s ∗ f s = X x ∈ J ∩ sJs \ J, y ∈ J/J ∩ sJs Ψ − ( xy ) f ∗ δ sxys ∗ f = [ J sJ : J ] X x ∈ J ∩ sJs \ J Ψ − ( x ) f ∗ δ sxs ∗ f . We remark that for g ∈ G , f ∗ δ g ∗ f = (cid:26) [ J gJ : J ] − f g , if g ∈ I G (Ψ) , , if g I G (Ψ) . The B -component of sx ( a, b, c ) s is sx (0 , , c ) s and lies in a . Therefore, if sx ( a, b, c ) s ∈ I G (Ψ) = J G ′ J , then by Lemma 4.14, sx ( a, b, s ∈ a [ n/ and hence a ≡ b ≡ EPRESENTATIONS OF U (2 ,
2) OVER A p -ADIC FIELD I 19 So we obtain f s ∗ f s = [ J sJ : J ] X c ∈ o / p f ∗ δ sx (0 , ,c ) s ∗ f = [ J sJ : J ] X c ∈ o / p f ∗ δ u ( ̟ − E ̟ m c √ ε ) ∗ f = [ J sJ : J ] X x ∈ o E / p E f u ( ̟ m − E x √ ε ) . (v) Let µ ∈ o × E √ ε . Put u = u ( ̟ E µ ) ∈ B ′ . Then f u = f ∗ δ u = δ u ∗ f . Since u normalizesthe pair ( J, Ψ), we have f s ∗ f u ∗ f s = X x ∈ J ∩ sJs \ J, y ∈ J/J ∩ sJs Ψ − ( xy ) f ∗ δ sxuys ∗ f = [ J sJ : J ] X x ∈ J ∩ sJs \ J Ψ − ( x ) f ∗ δ sxus ∗ f . The B -component of sx ( a, b, c ) us , which is identified with u ( ̟ − E µ + ̟ m ̟ − E c √ ε ), lies in a − .Suppose sx ( a, b, c ) us ∈ I G (Ψ) = J G ′ J . Then by Lemma 4.14, we have sx ( a, b, s ∈ a k andhence b ≡
0. So we obtain f s ∗ f u ∗ f s = [ J sJ : J ] X a ∈ o √ ε/ p √ ε, c ∈ o / p f ∗ δ sx ( a, ,c ) us ∗ f . For the moment, we fix a and c . We put ν = µ + ̟ m ̟ − E c √ ε , v = u ( ̟ E ν − ), h = h ( − ν − ),and x = x ( a, , sx ( a, , c ) us = sxssu ( ̟ E ν ) s = sxsvshv = [ sxs, v ] vshv ( hv ) − xhv .Since hv ∈ B ′ , we have ( hv ) − xhv ∈ J and Ψ(( hv ) − xhv ) = Ψ( x ) = 1. Since sxs ∈ P Λ , k +1 and v ∈ P Λ , , we obtain [ sxs, v ] ∈ P Λ , [ n/ . If we write sxs = 1 + y and v = 1 + z , then wehave [ sxs, v ] = (1 + y )(1 + z )(1 − y )(1 − z ) ≡ yzy + yzyz (mod B ⊥ ). So the B -componentof [ sxs, v ] lies in 1 + a k +5 = 1 + a n +1 . This implies that [ sxs, v ] ∈ J and Ψ([ sxs, v ]) = 1.We therefore have f ∗ δ sx ( a, ,c ) us ∗ f = f ∗ δ vshv ∗ f = [ J sJ : J ] − f v ∗ f s ∗ f h ∗ f v = [ J sJ : J ] − f u ( ̟ E µ − ) ∗ f s ∗ f h ( − µ − ) ∗ f u ( ̟ E µ − ) , for any a ∈ o √ ε/ p √ ε and c ∈ o / p . So we conclude that f s ∗ f u ∗ f s = q f u ( ̟ E µ − ) ∗ f s ∗ f h ( − µ − ) ∗ f u ( ̟ E µ − ) , as required. (cid:3) We return to the proof of Theorem 4.11. By Theorems 4.17 and 4.18, there is an algebrahomomorphism η : H ( G ′ //J ′ , Ψ) → H ( G//J,
Ψ) induced by η ( e s ) = f s , η ( e s ) = q − f s , η ( e b ) = f b , b ∈ B ′ . Note that, for g ∈ G ′ , η maps e g to (vol( J ′ gJ ′ ) / vol( J gJ )) / f g . Then, it follows from Proposi-tion 4.13 that η is surjective. Proposition 4.15 implies that η is injective. The ∗ -preservation of η is obvious since e ∗ g = e g − and f ∗ g = f g − , for g ∈ G ′ . Case (4d).
Let Λ ′ denote the o F -lattice sequence in (2.4). We fix an element d ∈ o × anda positive odd integer n ′ = 2 m −
1. For b, c ∈ o , we set β ( b, c ) = ̟ − m √ ε b √ ε ̟c √ ε ̟d √ ε ∈ g Λ ′ , − n ′ . (4.16)In this section, we consider the irreducible smooth representations of G which contain a skewstratum [Λ ′ , n ′ , n ′ − , β ], where β equals to β (0 , β (0 , β (1 , o F -lattice sequence Λ in V with e (Λ) = 8 and d (Λ) = 3 byΛ(0) = Λ(1) = Λ(2) = Λ(3) = N , Λ(4) = N , Λ(5) = Λ(6) = N , Λ(7) = ̟N , Λ( i + 8 k ) = ̟ k Λ( i ) , ≤ i ≤ , k ∈ Z . (4.17)The filtration { a k (Λ) } k ∈ Z is given by the following: a (Λ) = o F o F o F o F p F o F o F o F p F p F o F o F p F p F p F o F , a (Λ) = p F o F o F o F p F p F o F o F p F p F p F o F p F p F p F p F , a (Λ) = p F p F o F o F p F p F o F o F p F p F p F p F p F p F p F p F , a (Λ) = p F p F o F o F p F p F p F o F p F p F p F p F p F p F p F p F , a (Λ) = p F p F p F o F p F p F p F p F p F p F p F p F p F p F p F p F , a (Λ) = p F p F p F p F p F p F p F p F p F p F p F p F p F p F p F p F , a (Λ) = p F p F p F p F p F p F p F p F p F p F p F p F p F p F p F p F , a (Λ) = p F p F p F p F p F p F p F p F p F p F p F p F p F p F p F p F . (4.18)If we write n = 8 m − β = β (0 , G ) ( P Λ ′ ,n ′ ,ψ β (0 , ) ∪ Irr( G ) ( P Λ ′ ,n ′ ,ψ β ) ⊂ Irr( G ) ( P Λ ,n ,ψ β ) . (4.19)Take λ ∈ o F such that λλ = d and put t = λ − ̟ ̟λ . Then t is a similitude on ( V, f ) and hence t acts on the set of skew strata as well. Note that t Λ ′ is a translate of Λ ′ . As a conjugate of (4.19) by t , we getIrr( G ) ( P Λ ′ ,n ′ ,ψ β (1 , ) ∪ Irr( G ) ( P Λ ′ ,n ′ ,ψ β ) ⊂ Irr( G ) ( P t Λ ,n ,ψ β ) . (4.20)Since Irr( G ) ( P t Λ ,n ,ψ β ) is the t -conjugate of Irr( G ) ( P Λ ,n ,ψ β ) , there is a bijection Irr( G ) ( P Λ ,n ,ψ β ) ≃ Irr( G ) ( P t Λ ,n ,ψ β ) ; π π t . We shall concentrate on classifying the irreducible smooth representa-tions of G containing [Λ , n, n − , β ]. Remark 4.19.
We note that Irr( G ) ( P t Λ ,n ,ψ β ) ∩ Irr( G ) ( P Λ ,n ,ψ β ) ⊃ Irr( G ) ( P Λ ′ ,n ′ ,ψ β ) . EPRESENTATIONS OF U (2 ,
2) OVER A p -ADIC FIELD I 21 Define an orthogonal F -splitting V = V ⊥ V by V = F e ⊕ F e and V = F e ⊕ F e . Thenthe skew stratum [Λ , n, n − , β ] is split with respect to V = V ⊥ V . Put β j = β | V j , for j = 1 , β = 0 and E = F [ β ] is a totally ramified extension of degree 2 over F . We denote by B the A -centralizer of β and by B ⊥ its orthogonal complement with respect to the pairing inducedby tr A/F . Then B and B ⊥ are σ -stable. For i, j ∈ { , } , we write A ij = Hom F ( V j , V i ). Since β ∈ F × · V , the map s : A → E ; X ( X + β Xβ − ) / , X ∈ A is an ( E , E )-bimodule projection. So we get B = A ⊕ E , B ⊥ = A ⊕ A ⊕ ker s , A = B ⊕ B ⊥ . (4.21)For brevity, we write a k = a k (Λ), a ′ k = a k ∩ B , a ⊥ k = a k ∩ B ⊥ , g ′ k = g Λ ,k ∩ B and g ⊥ k = g Λ ,k ∩ B ⊥ ,for k ∈ Z . Proposition 4.20.
For k ∈ Z , we have a k = a ′ k ⊕ a ⊥ k .Proof. By [2] (2.9), we have a k = L ≤ i,j ≤ a k ∩ A ij and a k ∩ A ii = a k (Λ i ), for i = 1 ,
2. Since β normalizes a k (Λ ), we have a k (Λ ) = ( a k ∩ E ) ⊕ ( a k ∩ ker s ). Now the proposition followsfrom (4.21). (cid:3) Remark 4.21.
Since [( n + 1) / ≡ a ⊥ [( n +1) / = a ⊥ [ n/ .As in § σ -stable o F -lattice J in A by (4.11) and an open compact subgroup J of G by J = G ∩ (1 + J ). For b ∈ g ∩ J ∗ , we denote by Ψ b the character of J defined by (4.12). Lemma 4.22.
For k ∈ Z , the map ad( β ) induces an isomorphism g ⊥ k / g ⊥ k +1 ≃ g ⊥ k − n / g ⊥ k − n +1 .Proof. Since { a k } k ∈ Z is periodic, it is sufficient to prove that the induced map is injective, forall k ∈ Z . For X in A , we use the notation X = (cid:18) X X X X (cid:19) , X ij ∈ A ij . Let X ∈ g ⊥ k satisfy ad( β )( X ) ∈ g ⊥ k − n +1 . Then we obtainad( β )( X ) = (cid:18) − X β β X ad( β )( X ) (cid:19) ∈ g ⊥ k − n +1 and hence X β , β X , ad( β )( X ) ∈ a k − n +1 . Since β − ∈ a n , we have X , X ∈ a k +1 and X = X − s ( X ) = − ad( β )( X ) β − / ∈ a k +1 . This completes the proof. (cid:3) We write G ′ = G ∩ B , J ′ = G ′ ∩ J , Ψ = Ψ β and Ψ ′ = Ψ | J . Then, by Lemma 4.22, we get theanalogues of Propositions 4.8, 4.10, 4.13, and Corollary 4.9. Theorem 4.23.
With the notation as above, there exists a support-preserving, ∗ -isomorphism η : H ( G ′ //J ′ , Ψ ′ ) ≃ H ( G//J, Ψ) . Remark 4.24.
Since the centers of G and G ′ are compact, the same remark as in the case (4c)holds, that is, η induces a bijection from Irr( G ′ ) ( J ′ , to Irr( G ) ( J, Ψ) , which preserves supercusp-idality of representations. Proposition 4.25.
For g ∈ G ′ , we have J gJ ∩ G ′ = J ′ gJ ′ .Proof. Put M = ( A ⊕ A ) × and x = 1 V − V ∈ M . Since 1 + J is a pro- p subgroup of e G andstable under the adjoint action of x , it follows from [16] Lemma 2.1 that (1 + J ) g (1 + J ) ∩ M =((1 + J ) ∩ M ) g ((1 + J ) ∩ M ). Consider the adjoint action of h = ̟d ∈ B × on M . Then Ad( h ) induces an automorphism of M of order 2 and B × is the set of fixed pointsof this automorphism. Note that (1 + J ) ∩ M is Ad ( h )-stable. Then, by [16] Lemma 2.1, we get(1 + J ) g (1 + J ) ∩ B = ((1 + J ) ∩ B ) g ((1 + J ) ∩ B ).We note that B × is σ -stable and (1 + J ) ∩ B is a σ -stable pro- p subgroup of B × . Since G ′ = B ∩ G , it follows from [16] Lemma 2.1 again that (1 + J ) g (1 + J ) ∩ G ′ = J ′ gJ ′ . Thiscompletes the proof. (cid:3) We put B ′ = P Λ , ∩ G ′ . Then B ′ is the Iwahori subgroup of G ′ and normalizes J ′ = P Λ ,n ∩ G ′ .We define elements s and s in G ′ by s = , s = ̟ − ̟ . Put S = { s , s } and W ′ = h S i . Then we have a Bruhat decomposition G ′ = B ′ W ′ B ′ . Lemma 4.26.
Let t be a non negative integer. Then(i) [ J ′ ( s s ) t J ′ : J ′ ] = [ J ′ ( s s ) t s J ′ : J ′ ] = [ J ′ ( s s ) t J ′ : J ′ ] = q t , [ J ′ ( s s ) t s J ′ : J ′ ] = q t +1) .(ii) [ J ( s s ) t J : J ] = [ J ( s s ) t J : J ] = q t , [ J ( s s ) t s J : J ] = q t +2 , [ J ( s s ) t s J : J ] = q t +4 .Proof. As in the proof of Lemma 4.16, we can directly calculate these indices using (4.18). (cid:3)
For µ ∈ F and ν ∈ F × , we set u ( µ ) = µ
00 0 1 00 0 0 1 , u ( µ ) = µ , h ( µ ) = ν ν −
00 0 0 1 . For g ∈ G ′ , let e g denote the element in H ( G ′ //J ′ , Ψ ′ ) such that e g ( g ) = 1 and supp( e g ) = J ′ gJ ′ . Theorem 4.27.
Suppose m ≥ . Then the algebra H ( G ′ //J ′ , Ψ ′ ) is generated by the elements e g , g ∈ B ′ ∪ S . These elements are subject to the following relations:(i) e k = Ψ( k ) e , k ∈ J ′ ,(ii) e k ∗ e k ′ = e kk ′ , k, k ′ ∈ B ′ ,(iii) e k ∗ e s = e s ∗ e sks , s ∈ S , k ∈ B ′ ∩ sB ′ s ,(iv) e s ∗ e s = e , e s ∗ e s = [ J ′ s J ′ : J ′ ] P x ∈ o / p e u ( ̟ m − x √ ε ) ,(v) For µ ∈ o × √ ε , e s ∗ e u ( µ ) ∗ e s = e u ( µ − ) ∗ e s ∗ e h ( µ ) ∗ e u ( µ − ) , e s ∗ e u ( ̟µ ) ∗ e s = [ J ′ s J ′ : J ′ ] e u ( ̟µ − ) ∗ e s ∗ e h ( − µ − ) ∗ e u ( ̟µ − ) .Proof. Note that s normalizes J ′ . Then the proof is very similar to that of [3] Chapter 3Theorem 2.1. (cid:3) For g ∈ G ′ , we denote by f g the element in H ( G//J,
Ψ) such that f g ( g ) = 1 and supp( f g ) = J gJ . EPRESENTATIONS OF U (2 ,
2) OVER A p -ADIC FIELD I 23 Theorem 4.28.
Suppose m ≥ . Then the algebra H ( G//J, Ψ) is generated by f g , g ∈ B ′ ∪ S and satisfies the following relations:(i) f k = Ψ( k ) f , k ∈ J ′ ,(ii) f k ∗ f k ′ = f kk ′ , k, k ′ ∈ B ′ ,(iii) f k ∗ f s = f s ∗ f sks , s ∈ S , k ∈ B ′ ∩ sB ′ s ,(iv) f s ∗ f s = [ J s J : J ] f , f s ∗ f s = [ J s J : J ] P x ∈ o / p f u ( ̟ m − x √ ε ) ,(v) For µ ∈ o × √ ε , f s ∗ f u ( µ ) ∗ f s = − qf u ( µ − ) ∗ f s ∗ f h ( µ ) ∗ f u ( µ − ) , f s ∗ f u ( ̟µ ) ∗ f s = − q f u ( ̟µ − ) ∗ f s ∗ f h ( − µ − ) ∗ f u ( ̟µ − ) .Proof. As in the proof of Theorem 4.18, it follows from Lemma 4.26 that H ( G//J,
Ψ) is generatedby f g , g ∈ S ∪ B ′ . Relations (i), (ii), (iii) are obvious. We shall prove relations (iv) and (v) on s in the case when m = 2 k + 1, k ≥
1. The other cases can be treated in a similar fashion.We will abbreviate s = s . We can choose a common system of representatives for J/J ∩ sJ s and J ∩ sJ s \ J to be x ( a, A ) = ̟ k +1 A ̟ m a √ ε − ̟ k +1 A , A ∈ o F / p F , a ∈ o / p . We note that x ( a, A ) lies in the kernel of Ψ.(iv) As in the proof of Theorem 4.18, we obtain f s ∗ f s = [ J sJ : J ] X a ∈ o / p , A ∈ o F / p F f ∗ δ sx ( a,A ) s ∗ f . Since the B -component sx ( a, s of sx ( a, A ) s lies in a , it follows from Lemma 4.14 that if sx ( a, A ) s ∈ I G (Ψ) = J G ′ J , then sx (0 , A ) s ∈ a [ n/ and hence A ≡
0. So f ∗ δ sx ( a,A ) s ∗ f = 0unless A ≡
0. Hence we have f s ∗ f s = [ J sJ : J ] X a ∈ o / p f ∗ δ sx ( a, s ∗ f = [ J sJ : J ] X a ∈ o / p f ∗ δ u ( ̟ m − a √ ε ) ∗ f = [ J sJ : J ] X a ∈ o / p f u ( ̟ m − a √ ε ) . (v) Let µ ∈ o × √ ε . Put u = u ( ̟µ ) ∈ B ′ . As in the proof of Theorem 4.18, we have f s ∗ f u ∗ f s = [ J sJ : J ] X a ∈ o / p , A ∈ o F / p F f ∗ δ sx ( a,A ) us ∗ f . Put x = x (0 , A ), ν = µ + ̟ m − a , v = u ( ̟ν − ), and h ( − ν − ). Then sx ( a, A ) us = sxssu ( ̟ν ) s = sxsvshv = [ sxs, v ] vshv ( hv ) − x ( hv ) . Since hv ∈ B ′ , we see that ( hv ) − x ( hv ) ∈ J lies in the kernel of Ψ. If we write sxs = 1 + y and v = 1 + z , then y ∈ a k − and z ∈ a . So we have [ sxs, v ] ∈ P Λ , k +5 ⊂ P Λ , [ n/ and [ sxs, v ] ≡ yzy (mod B ⊥ + a n +1 ). These observations imply that [ sxs, v ] ∈ J and Ψ([ sxs, v ]) = Ω(tr
A/F ( βyzy )) = Ω(2 d √ εν − AA ). We therefore have f s ∗ f u ∗ f s = [ J sJ : J ] X a ∈ o / p , A ∈ o F / p F Ω − (2 √ εν − AA ) f ∗ δ vshv ∗ f = − q [ J sJ : J ] X a ∈ o / p f ∗ δ vshv ∗ f = − q X a ∈ o / p f v ∗ f s ∗ f h ∗ f v = − q f u ( ̟µ − ) ∗ f s ∗ f h ( − µ − ) ∗ f u ( ̟µ − ) . (cid:3) Remark 4.29. If m = 1, the algebra H ( G ′ //J ′ , Ψ) is generated by the elements e g , g ∈ B ′ ∪ S .In this case, these elements are subject to the following relations:(i) e k = Ψ( k ) e , k ∈ J ′ ,(ii) e k ∗ e k ′ = e kk ′ , k, k ′ ∈ B ′ ,(iii) e k ∗ e s = e s ∗ e sks , s ∈ S , k ∈ B ′ ∩ sB ′ S ,(iv) e s ∗ e s = e , e s ∗ e s = ( P y ∈ o √ ε/ p √ ε, y e s ∗ e h ( − y − ) + q e )( P x ∈ o / p e u ( x √ ε ) ),(v) For µ ∈ o × √ ε , e s ∗ e u ( µ ) ∗ e s = e u ( µ − ) ∗ e s ∗ e h ( µ ) ∗ e u ( µ − ) .We can easily see that the analogue of Theorem 4.23 holds as well. We omit the details.Combining Proposition 4.25, the analog of Proposition 4.13 and Theorems 4.27, 4.28, wecomplete the proof of Theorem 4.23. Indeed, the map η ( e s ) = − q − f s , η ( e s ) = − q − f s , η ( e b ) = f b , b ∈ B ′ induces the required algebra isomorphism.4.4. Case (4a).
Let Λ denote the o F -lattice sequence in (2.4) and n = 2 m − λ ∈ o F satisfy λ λ (mod p F ). We consider the irreducible smooth representationsof G which contain a skew stratum [Λ , n, n − , β ], where β = ̟ − m − ̟λ − ̟λ ∈ g Λ , − n . (4.22)The assertions and proofs are very similar to those in § Remark 4.30.
By the argument in § G ) ( P Λ ,n ,ψ β ) is precisely the set of equivalenceclasses of irreducible smooth representations of G of level n/ X − λ ) ( X − λ ) (mod p F ).Set V − = F e ⊕ F e and V = F e ⊕ F e . Then this stratum is split with respect to V = V − ⊕ V . For j = ±
1, we write β j = β | V j , E j = F [ β j ] and Λ j ( i ) = Λ( i ) ∩ V j , i ∈ Z .Then Proposition 1.8 implies that [Λ j , n, n − , β j ], j = ±
1, are simple strata. We set A ij =Hom F ( V j , V i ), i, j = ±
1. Since β j ∈ F × · V j , we obtain an ( E j , E j )-bimodule projection s j : A jj → E j by s j ( X ) = ( X + β j Xβ − j ) / , X ∈ A jj . EPRESENTATIONS OF U (2 ,
2) OVER A p -ADIC FIELD I 25 Writing B for the A -centralizer of β and B ⊥ for its orthogonal component with respect to thepairing induced by tr A/F , we have B = E − ⊕ E , B ⊥ = ker s − ⊕ ker s ⊕ A − , ⊕ A , − , and A = B ⊕ B ⊥ . For j = ±
1, the map s j satisfies s j ( a k (Λ j )) = a k (Λ j ) ∩ E j . Combining thiswith [2] (2.9), we obtain a k (Λ) = a k (Λ) ∩ B ⊕ a k (Λ) ∩ B ⊥ , k ∈ Z . So we can define a σ -stable o F -lattice J in A by (4.11) and an open compact subgroup J = J ∩ G of G . For b ∈ g ∩ J ∗ , we denote by Ψ b the character of J defined by (4.12).As in § a k = a k (Λ), a ′ k = a k ∩ B , a ⊥ k = a k ∩ B ⊥ , etc. Lemma 4.31.
For k ∈ Z , ad( β ) induces an isomorphism g ⊥ k / g ⊥ k +1 ≃ g ⊥ k − n / g ⊥ k − n +1 .Proof. It follows from [2] (3.7) Lemma 1 that ad( β ) maps a k ∩ A − , onto a k − n ∩ A − , and a k ∩ A , − onto a k − n ∩ A , − .Let j ∈ {± } . For x ∈ a k − n ∩ ker s j , we have − xβ − j / ∈ a k ∩ ker s j and ad( β j )( − xβ − j /
2) =( x − β j xβ − j ) / x − s j ( x ) = x . This implies that ad( β j ) maps a k ∩ ker s j onto a k − n ∩ ker s j ,so that ad( β ) maps a ⊥ k onto a ⊥ k − n .By the periodicity of { a k } k ∈ Z , we conclude that ad( β ) induces an isomorphism a ⊥ k / a ⊥ k +1 ≃ a ⊥ k − n / a ⊥ k − n +1 . The lemma follows immediately from this. (cid:3) We also set G ′ = G ∩ B , J ′ = G ′ ∩ J , Ψ = Ψ β , and Ψ ′ = Ψ | J . Then, by Lemma 4.31,we get the analogues of Propositions 4.10 and 4.13, that is, Irr( G ) ( P Λ ,n ,ψ β ) = Irr( G ) ( J, Ψ) and I G (Ψ) = J G ′ J . Applying the proof of Proposition 4.25, we have J gJ ∩ G ′ = J ′ gJ ′ , for g ∈ G ′ .We shall establish a Hecke algebra isomorphism η : H ( G ′ //J ′ , Ψ ′ ) ≃ H ( G//J,
Ψ). Since G ′ isabelian, the structure of the algebra H ( G ′ //J ′ , Ψ ′ ) is obvious.We have G ′ = h ζ i B ′ , where B ′ = G ′ ∩ P Λ , and ζ = ̟ − λ −
00 0 0 11 0 0 00 ̟λ . (4.23)The group B ′ normalizes the pair ( J, Ψ) and ζ satisfies the following: Lemma 4.32.
For t ∈ Z , we have [ J ζ t J : J ] = q | t | .Proof. It follows from direct computation. (cid:3)
For g ∈ G ′ , let e g denote the element in H ( G ′ //J ′ , Ψ ′ ) such that e g ( g ) = 1 and supp( e g ) = J ′ gJ ′ , and let f g denote the element in H ( G//J,
Ψ) such that f g ( g ) = 1 and supp( f g ) = J gJ . Lemma 4.33. f ζ − ∗ f ζ = f ζ ∗ f ζ − = q f .Proof. As in the proof of Theorem 4.18, we have f ζ − ∗ f ζ = [ J ζJ : J ] X y ∈ J/J ∩ ζJζ − Ψ − ( y ) f ∗ δ ζ − yζ ∗ f . Suppose that m = 2 k . Then we can choose a system of representative for J/J ∩ ζJ ζ − to be x ( a, b, A ) = ̟ k A ̟ k a √ ε
00 0 1 0 ̟ k +1 b √ ε − ̟ k A , a, b ∈ o / p , A ∈ o F / p F . The B -component of ζ − x ( a, b, A ) ζ lies in a , so that, if ζ − x ( a, b, A ) ζ ∈ J G ′ J , then byLemma 4.14, the B ⊥ -part of ζ − x ( a, b, A ) ζ belongs to a [ n/ . We conclude that f ∗ δ ζ − x ( a,b,A ) ζ ∗ f = 0 unless a ≡ b ≡ A ≡
0. So we have f ζ − ∗ f ζ = q f , as required.The case of m = 2 k + 1, k ≥ f ζ ∗ f ζ − = q f are quite similar to this. (cid:3) Theorem 4.34.
With the notation as above, there is a ∗ -isomorphism η : H ( G ′ //J ′ , Ψ ′ ) ≃H ( G//J, Ψ) with support preservation.Proof. Since I G (Ψ) = J G ′ J , the algebra H ( G//J,
Ψ) is spanned by f g , g ∈ G ′ . Since G ′ = h ζ i B ′ ,we can write g = ζ t b where t ∈ Z , b ∈ B ′ . By Lemma 4.32, we have f g = f tζ ∗ f b if t ≥ f g = f | t | ζ − ∗ f b otherwise.Since B ′ normalizes the pair ( J, Ψ), we have f b ∗ f g = f bg = f gb = f g ∗ f b , for b ∈ B ′ and g ∈ G ′ . In particular, f b lies in the center of H ( G//J,
Ψ), for b ∈ B ′ . Therefore the map η ( e ζ ) = q − f ζ , η ( e ζ − ) = q − f ζ − , η ( e b ) = f b , b ∈ B ′ , induced the required isomorphism η : H ( G ′ //J ′ , Ψ ′ ) ≃ H ( G//J,
Ψ) by Lemma 4.33. (cid:3)
Remark 4.35.
As in Remark 4.12, the map η induces a bijection from Irr( G ′ ) ( J ′ , to Irr( G ) ( J, Ψ) .Since G ′ is abelian and not compact, every irreducible smooth representation of G which contains( J, Ψ) is not supercuspidal.
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