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It is a part of classical to answer the following question: How to decompose the representation O(n) yC[Rn] ? To do this, it is profitable to introduce a Lie algebra action n (1) sl2(C) yC[R ], specified by 0 1 e := 7→ − 1 n x2,  0 0  2 i=1 i P 1 0 n n ∂ h := 7→ + xi ,  0 −1  2 i=1 ∂xi P 0 0 1 n ∂2 f := 7→ 2 i=1 ∂x2 .  1 0  i P The most important feature of this Lie algebra action is that it commutes with the action of O(n), and the resulting extended action of O(n) × sl2(C) will help us in decomposing the action of O(n) on C[Rn]. The group O(n) acts on the space of harmonic polynomials: (2) O(n) y H[Rn] := {ϕ ∈ C[Rn] | f · ϕ =0}. According to the action of h, we have the obvious decomposition: ∞ H[Rn]= Hd[Rn], Md=0 where Hd[Rn] := {ϕ ∈ H[Rn] | ϕ is homogeneous of degree d}. The following result is well-known, and commonly called the theory of spherical harmonics. Theorem 1.1. The representation O(n) y Hd[Rn] is either irreducible or zero. Moreover, we have the following decomposition as an O(n) × sl2(C)-module: n C[Rn] ∼= Hd[Rn] ⊗ L(d + ). 2 d≥0, HMd[Rn]=06 n C Here L(d + 2 ) is the irreducible lowest weight module of sl2( ) with the lowest n weight d + 2 . More generally, we consider the action of O(n) on the direct sum of k-copies of Rn: O(n) yRn×k, (k ≥ 0, Rn×k denotes the space of n × k real matrices). LOCAL : THE BASIC THEORY 3

Again it induces a representation (3) O(n) yC[Rn×k].

Generalizing the action of O(n) × sl2(C) for k = 1, we now have a natural extension of the representation (3) to a representation n×k (4) O(n) × sp2k(C) yC[R ].

(The action of the symplectic Lie algebra sp2k(C) may be viewed as the hidden n×k symmetry of C[R ].) The symplectic Lie algebra sp2k(C) has the usual decom- position + − sp2k(C)= p ⊕ glk(C) ⊕ p , n×k and in the standard coordinates {xi,j}1≤i≤n,1≤j≤k of R , the action is described by the following: + n p 7→ span{ l=1 xlixlj}1≤i,j≤k, n n ∂ glk(C) 7→ span{Pδi,j + xli } ≤i,j≤k, 2 l=1 ∂xlj 1 − n P∂2 p 7→ span{ } ≤i,j≤k. l=1 ∂xli∂xlj 1 We now have a representation P n×k n×k − (5) O(n) ⊗ glk(C) y H[R ] := {ϕ ∈ C[R ] | X · ϕ = 0 for all X ∈ p }. Generalizing Theorem 1.1, the following theorem is one of the main results in classical invariant theory. See [Ho2] or [KV].

Theorem 1.2. As an O(n) × glk(C)-module, H[Rn×k]= π ⊗ σ M is multiplicity free, and π and σ determines each other, where π runs over a set of irreducible finite-dimensional representations of O(n), and σ runs over a set of irreducible finite-dimensional representations of glk(C). As an O(n) × sp2k(C)- module, C[Rn×k]= π ⊗ L(σ), where L(σ) is the unique irreducible quotientM of the generalized Verma module C − U(sp2k( )) ⊗U(glk (C)⊕p ) σ, and “U” indicates the universal enveloping algebra. A natural question is about the occurrence, namely, for a given irreducible rep- resentation π of O(n), we want to know whether n×k HomO(n)(C[R ], π) =6 0? Note that k C[Rn×k]= C[Rn]. Since C[Rn] contains the trivial representationO of O(n), we have n×k n×(k+1) (6) HomO(n)(C[R ], π) =06 ⇒ HomO(n)(C[R ], π) =06 . 4 BINYONGSUNANDCHEN-BOZHU

Also we have a surjective homomorphism C[Rn×n] −−−−−→restriction C[O(n)], and since π always occurs in C[O(n)], we see that n×k (7) HomO(n)(C[R ], π) =6 0 if k ≥ n. In view of (6) and (7), we define the first occurrence index n×k n(π) := min{k ≥ 0 | HomO(n)(C[R ], π) =06 }. The following result may be read off from the explicit description of the cor- respondence π ↔ σ of Theorem 1.2 (see [KV]), and it is a special case of the conservation relations which we will discuss in the next section. Theorem 1.3. For any irreducible finite dimensional representation π of O(n), we have n(π)+n(π ⊗ det) = n. Remark. The fact that n(det) = n is implicit in Weyl’s book [Wey, Chapter 2].

2. Transcending classical invariant theory In this section, we give an informal introduction to Howe’s paper of the same title [Ho3]. We also describe two fundamental principles of the theory: the Howe duality theorem and Kudla-Rallis conservation relations. We start from the setting of Section 1, where we have the representation n×k O(n) × sp2k(C) yC[R ].

The above action of the symplectic Lie algebra sp2k(C) does not integrate to a Lie group representation in general. This is to say that the space C[Rn×k] has local symmetry of the , but no global symmetry. However, if we consider the space of Schwartz functions instead of the space of polynomials, we do have the global symmetry. We shall describe this in what follows. In general, we may consider the real (definite or indefinite)

t 1p 0 1p 0 O(p, q) := g ∈ GLp+q(R) | g g = , (p, q ≥ 0).   0 −1q  0 −1q Similar to (3), we have an obvious representation (8) O(p, q) y S(R(p+q)×k), where “S” indicates the space of Schwartz functions. In the same spirit as (4) but for more subtle reasons, the representation in (8) extends to a representation R y R(p+q)×k (9) O(n) × Sp2k( ) S( ), R R where Sp2k( ) denotes the metaplecticf double cover of the symplectic group Sp2k( ). f LOCAL THETA CORRESPONDENCE: THE BASIC THEORY 5

Let Irr(O(p, q)) denote the set of isomorphism classes of irreducible represen- tations (with certain technical conditions) of O(p, q). Similar notations, such as R Irr(Sp2k( )), will be used. The following result is the Howe duality theorem for R the so-called dual pair (O(p, q), Sp2k( )), and we will discuss its general statement in thef later part of this section. Theorem 2.1. Denote (p+q)×k ωk : = S(R )

Ω: = {π ∈ Irr(O(p, q)) | HomO(p,q)(ωk, π) =06 }, ′ ′ ′ Ω : = {π ∈ Irr(Sp (R)) | Homf R (ωk, π ) =06 }. 2k Sp2k( ) f Then the relation ′ ′ {(π, π ) ∈ Irr(O(p, q)) × Irr(Sp (R)) | Hom f R (ωk, π⊗π ) =06 } 2k O(p,q)×Sp2k( ) defines a one-to-one correspondencef b ′ R Irr(O(p, q)) ⊃ Ω ↔ Ω ⊂ Irr(Sp2k( )). ′ R Moreover, for any (π, π ) ∈ Irr(O(p, q)) × Irr(Sp2k(f)), we have ′ dim Hom f R (ωfk, π⊗π ) ≤ 1. O(p,q)×Sp2k( ) We proceed to describe another fundamental principleb governing the correspon- dence of Theorem 2.1. Fix π ∈ Irr(O(p, q)). Similar to (6), we have that

(p+q)×k (p+q)×(k+1) (10) HomO(p,q)(S(R ), π) =06 ⇒ HomO(p,q)(S(R ), π) =06 . This is called Kudla’s persistence principle [Ku1]. Similar to (7), we have that

(p+q)×k (11) HomO(p,q)(S(R ), π) =6 0 if k ≥ p + q. The condition “k ≥ p + q” is called Howe’s stable range condition. In view of (10) and (11), we define the first occurrence index

(p+q)×k (12) n(π) := min{k ≥ 0 | HomO(p,q)(S(R ), π) =06 }. The following result is called conservation relations (for the group O(p, q)). It was conjectured by Kudla and Rallis [KR], and established by the authors in [SZ]. Theorem 2.2. For any π ∈ Irr(O(p, q)), we have n(π)+n(π ⊗ det) = p + q. Remark. The paper [SZ] proves a conservation relation for a (type I) G, which is valid for any irreducible smooth representation of G, and for any local field of characteristic zero. 6 BINYONGSUNANDCHEN-BOZHU

Now we come to the general statement of Howe duality conjecture [Ho1]. Let W be a finite-dimensional symplectic over R with symplectic form h , iW . Denote by τ the involution of EndR(W ) specified by

τ hx · u, viW = hu, x · viW , u,v ∈ W, x ∈ EndF(W ).

′ Let (A, A ) be a pair of τ-stable semisimple R-subalgebras of EndR(W ) which are mutual centralizers of each other. Put G := A ∩ Sp(W ) and G′ := A′ ∩ Sp(W ), which are closed subgroups of Sp(W ). Following Howe, we call the pair (G,G′) so obtained a in Sp(W ). We say that the pair (A, A′) (or the reductive dual pair (G,G′)) is irreducible of type I if A (or equivalently A′) is a simple algebra, and that it is irreducible of type II if A (or equivalently A′) is the product of two simple algebras which are exchanged by τ. A complete classification of such dual pairs was also given by Howe [Ho1]. Denote by H(W ) := W × R the attached to W , with group multiplication

(u,α)(v, β) := (u + v,α + β + hu, viW ), u,v ∈ W, α, β ∈ R.

Its is identified with R in the obvious way. Fix an arbitrary non-trivial unitary character ψ : R → C×.

Theorem 2.3. (Stone-von Neumann) Up to isomorphism, there is a unique irre- ducible unitary representation ωψ of H(W ) with the central character ψ.

Let ωψ be as in Theorem 2.3.b Set

b ωψ := {v ∈ ωψ | the map H(W ) → ωψ, g 7→ g · v is smooth}.

This is a smooth representationb of H(W ) withb the usual smooth topology. Let (G,G′) be a reductive dual pair in Sp(W ). Let G and G′ be finite fold covering groups of G and G′, respectively. Then G × G′ acts on H(W ) as group e e automorphisms via the action of G × G′ on W . Using this action, we form the semidirect product e e (G × G′) ⋉ H(W ). We say that a representation of (eG × eG′) ⋉ H(W ) is a smooth oscillator represen- tation if its restriction to H(W ) is isomorphic to ω . We say that a representation e e ψ of G × G′ is a smooth oscillator representation if it extends to a smooth oscillator representation of (G × G′) ⋉ H(W ). Basic facts about smooth oscillator represen- e e tations may be found in [Wei, Ku2]. The following ise the fundamentale result of Howe [Ho2], establishing his famous conjecture for all real reductive dual pairs. LOCAL THETA CORRESPONDENCE: THE BASIC THEORY 7

Theorem 2.4. Let ω be a smooth oscillator representation of (G × G′) ⋉ H(W ). Denote e e

Ω: = {π ∈ Irr(G) | HomGe(ω, π) =06 }, ′ ′ ′ ′ Ω : = {π ∈ Irr(eG ) | HomGe′ (ω, π ) =06 }. e Then the relation

′ ′ ′ {(π, π ) ∈ Irr(G) × Irr(G )) | HomGe×Ge′ (ω, π⊗π ) =06 } defines a one-to-one correspondencee e b

Irr(G) ⊃ Ω ↔ Ω′ ⊂ Irr(G′).

Moreover, for any (π, π′) ∈ Irr(eG) × Irr(G′), we havee e e ′ dim HomGe×Ge′ (ω, π⊗π ) ≤ 1. The correspondence specified in the aboveb theorem is called the local theta correspondence, or theta correspondence in short. R R Example 1. The groups O(p, q) and Sp2k( ) form a reductive dual pair in Sp2k(p+q)( ). ′ R When G = O(p, q) and G = Sp2k( ), Theorem 2.4 reduces to Theorem 2.1. R R R Examplee 2. The groups GLe m(f) and GLn( ) form a reductive dual pair in Sp2mn( ). The following representation is a smooth oscillator representation

m×n −1 GLm(R) × GLn(R) y S(R ), ((g, h) · f)(x) := f(g xh),

m×n m×n where g ∈ GLm(R), h ∈ GLn(R), x ∈ R , f ∈ S(R ). Suppose m ≤ n. Then the local theta correspondence yields an injective map

.((Irr(GLm(R)) ֒→ Irr(GLn(R

If m = n, then this is the bijection which sends π ∈ Irr(GLm(R)) to its contragra- dient representation π∨.

Remark. Howe made his duality conjecture for both real and p-adic local fields in [Ho1]. For the latter, the Howe duality conjecture has also been established completely, thanks to the works of Waldspurger [Wa], Minguez [Mi], Gan and Takeda [GT], and Gan and Sun [GS]. The companion statement of multiplicity one was due to Waldspurger [Wa] and Li, Sun and Tian [LST]. We refer the readers to [MVW] on basic approaches to local theta correspondence for p-adic local fields. 8 BINYONGSUNANDCHEN-BOZHU

3. The explicit correspondence For applications to automorphic forms and unitary , it is of great interest to explicitly describe the local theta correspondence, in terms of Langlands-Vogan parametrization of irreducible representations [Vo]. The first important cases concern the so-called (irreducible) compact dual pairs, namely

(G,G′) = (O(p), Sp(2n, R)), (U(p), U(r, s)), (Sp(p), O∗(2n)).

The representations of G′ arising from the correspondence are the so-called unitary lowest weight modules. The explicit correspondence is described in two landmark papers on the subjectf of unitary lowest weight modules, one by Kashiwara and Vergne [KV] and another by Enright, Howe and Wallach [EHW]. For non-compact dual pairs, the situation is considerably more complicated. (The case of compact dual pairs is almost always a stepping stone for the general case.) In the following, we list some of the most relevant works, which give com- plete description of the correspondence for specific dual pairs (or specific classes of representations). • Dual pairs (G,G′) of type I, where the “size” of G is not greater than that of G′: for a “sufficiently regular” (genuine) discrete series representation π of G, Li shows that π occurs in the correspondence, and the corresponding representation π′ of G′ is a (explicitly described) unitary representation e with nonzero cohomology. See [Li] for details. • Dual pairs of type II:f complete correspondence is given by Moeglin [Mo] (over the real), Adams-Barbasch [AB1] (over the complex) and Li-Paul- Tan-Zhu [LPTZ] (over the quaternion). • Complex dual pairs of type I: complete correspondence is given by Adams- Barbasch [AB1]. • The dual pair (O(p, q), Sp(2n, R)) in the so-called “(almost) equal rank” case: complete correspondence is given by Moeglin and Paul [Mo, Pa2] (for p + q =2n, 2n + 2), and Adams-Barbasch [AB2] (for p + q =2n + 1). • The dual pair (U(p, q), U(r, s)) in the “equal rank” case: complete corre- spondence is given by Paul [Pa1], where p + q = r + s. • The dual pair (Sp(p, q), O∗(2n)) in the “equal rank” case and beyond: com- plete correspondence is given by Li-Paul-Tan-Zhu [LPTZ], where p+q ≤ n. • Other cases: complete correspondence is given by Przebinda [Pr] for the dual pair (O(2, 2), Sp(4, R)), Bao [Ba] for the dual pair (Sp(p, q), O∗(4)) and Fan [Fa] for the dual pair (O(p, q), Sp(2n, R)) (where p + q = 4).

Remark. Common techniques in determining the explicit correspondence include (a) correspondence of discrete series as in the work of Li [Li]; (b) the induction principle of Kudla [Ku1]; and (c) the correspondence of K and K′-types in the e f LOCAL THETA CORRESPONDENCE: THE BASIC THEORY 9 space of joint harmonics [Ho3], and arguments to identify representations using minimal K-types.

4. Automatic continuity In the literature, the theta correspondence is mostly studied in the algebraic setting of Harish-Chandra modules. In fact Howe proved his famous duality the- orem in the algebraic setting which he shows to imply the duality theorem in the smooth setting [Ho2]. Ever since this, an “automatic continuity” result has been expected, namely that the algebraic version of theta correspondence agrees with the smooth version of theta correspondence. We shall highlight this “automatic continuity” problem. Our exposition follows that of [BS]. We start with the following Proposition 4.1. Let (G,G′) be a reductive dual pair in the real symplectic group Sp(W ). Up to conjugation by G × G′, there exists a unique Cartan involution θ of Sp(W ) such that (13) θ(G)= G and θ(G′)= G′. Now let θ be as in Proposition 4.1. Then θ restricts to Cartan involutions on G and G′. Denote the fixed point groups by C := (Sp(W ))θ, K := Gθ and K′ := (G′)θ. They are maximal compact subgroups of Sp(W ), G and G′, respectively. Write K → K, K′ → K′ and C → C for the covering maps inducede bye the covering maps e G → G, G′ → G′ and Sp(W ) → Sp(W ), respectively. e e f As in Theorem 2.4, let ω be a smooth oscillator representation of (G × G′) ⋉ H(W ). Denote e e

Ω := {π ∈ Irr(G) | HomGe(ω, π) =06 }, ′ ′ ′ ′ Ω := {π ∈ Irr(eG ) | HomGe′ (ω, π ) =06 }, ′ ′ ′ R := {(π, π ) ∈ eIrr(G) × Irr(G ) | HomGe×Ge′ (ω, π⊗π ) =06 }.

It is known [Wei] that the representatione ωe|H(W ) uniquely extendsb to a representa- tion of Sp(W ) ⋉ H(W ). Write ωalg for the space of C-finite vectors in ω. For any π ∈ Irr(G), define f e alg alg alg (14)e Hom (ω, π) := Hom e (ω , π ), Ge g,K 10 BINYONGSUNANDCHEN-BOZHU where g denotes the complexified Lie algebra of G, and πalg denotes the (g, K)- module of K-finite vectors in π. We define Homalg(ω, π′) and Homalg (ω, π⊗π′) Gf′ Ge×Gf′ e ′ ′ in analogouse ways, where π ∈ Irr(G ). In this algebraic setting, we let b Ωalg := {π ∈ Irr(G) | Homf alg(ω, π) =06 }, Ge Ω′alg := {π′ ∈ Irr(eG′) | Homalg(ω, π′) =06 }, Gf′ Ralg := {(π, π′) ∈fIrr(G) × Irr(G′) | Homalg (ω, π⊗π′) =06 }. Ge×Gf′ It is clear that e f b Ω ⊂ Ωalg, (15)  Ω′ ⊂ Ω′alg,  R ⊂ Ralg. As a prelude of Theorem 2.4, Howe proved the following algebraic analog. As in [Ho3, Section 2], this implies Theorem 2.4. Theorem 4.1. The relation Ralg defines a one-to-one correspondence Irr(G) ⊃ Ωalg ↔ Ω′alg ⊂ Irr(G′). Moreover, for any (π, π′) ∈ Irr(e G) × Irr(G′), we havee dim Homalg (ω, π⊗π′) ≤ 1. e Ge×Ge′ e We state the following folklore conjectureb on the coincidence of the smooth version and the algebraic version of theta correspondence, which has been expected since the publication of [Ho3]. Conjecture 4.2. The inclusions in (15) are all equalities. Recall from Section 2 the construction of a reductive dual pair (G,G′) ⊂ Sp(W ) from a mutually centralizing pair (A, A′) of τ-stable semisimple R-subalgebras of EndR(W ), as well as the notion of type I and type II dual pairs. Let H denote a quaternionic division algebra over R, which is unique up to isomorphism. The following theorem is the main result of [BS]. Theorem 4.3. Conjecture 4.2 holds when the dual pair (G,G′) contains no quater- nionic type I irreducible factor, that is, when A contains no τ-stable ideal which is isomorphic to a matrix algebra Mn×n(H) (n ≥ 1). Remark. For the quaternionic dual pair (G,G′) = (Sp(p, q), O∗(2n)), it is also shown in [BS] that Conjecture 4.2 holds if p + q ≤ n. We note that the main tool of [BS] showing the conjecture on coincidence is the conservation relations established in [SZ]. In the following we sketch a proof for orthogonal-symplectic dual pairs. LOCAL THETA CORRESPONDENCE: THE BASIC THEORY 11

Let G = O(p, q) and π ∈ Irr(G). Define the first occurrence index n(π) as in (12), and its algebraic analogue

′ alg R(p+q)×n n (π) := min n ≥ 0 | HomG (S( ), π) =06 , n o alg R(p+q)×n alg where HomG is defined as in (14), using an obvious subspace S( ) of S(R(p+q)×n). It is clear that n′(π) ≤ n(π). Similar to the conservation relation n(π)+n(π ⊗ det) = p + q, we may prove the algebraic analog: n′(π) + n′(π ⊗ det) = p + q. Thus n(π) = n′(π). This easily implies that Conjecture 4.2 holds for orthogonal- symplectic dual pairs.

References [AB1] J. Adams and D. Barbasch, Reductive dual pair correspondence for complex groups,J. Funct. Anal. 132, (1995), no. 1, 1-42. [AB2] J. Adams and D. Barbasch, Genuine representations of the , Compositio Math. 113, (1998), no. 1, 23-66. [Ba] Y. Bao, The explicit theta correspondence for reductive dual pairs (Sp(p, q), O∗(4)), J. Funct. Anal. 271, (2016), no. 9, 2422-2459. [BS] Y. Bao and B. Sun, Coincidence of algebraic and smooth theta correspondences, Represen- tation Theory 21, (2017), 458–466. [Fa] X. Fan, Explicit induction principle and symplectic-orthogonal theta lifts, J. Funct. Anal. 273, (2017), no. 11, 3504-3548. [GS] W. T. Gan and B. Sun, The Howe duality conjecture: quaternionic case, in “Repre- sentation Theory, Number Theory, and Invariant Theory”, 175–192, Progr. Math., 323, Birkhauser/Springer, 2017. [GT] W. T. Gan and S. Takeda, A proof of the Howe duality conjecture, J. Amer. Math. Soc. 29, (2016), no. 2, 473–493. [Ho1] R. Howe, θ-series and invariant theory, in “Automorphic Forms, Representations and L-functions”, Proc. Symp. Pure Math. 33, (1979), 275–285. [Ho2] R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313, (1989), 539–570. [Ho3] R. Howe, Transcending classical invariant theory, J. Amer. Math. Soc. 2, (1989), 535–552. [EHW] T. Enright, R. Howe, and N. Wallach, A classification of unitary highest weight modules, in “Representation theory of reductive groups” (Park City, Utah, 1982), Progr. Math., vol. 40, Birkhauser Boston, Boston, MA, 1983, pp. 97-143. [Ku1] S. S. Kudla, On the local theta-correspondence, Invent. Math. 83, (1986), no. 2, 229–255. [Ku2] S. S. Kudla, Splitting metaplectic covers of dual reductive pairs, Israel J. Math. 87, (1994), 361–401. [KR] S. S. Kudla and S. Rallis, On first occurence in the local theta correspondence, in “Au- tomorphic Representations, L-functions and Applications: Progress and Prospects”, Ohio State Univ. Math. Res. Inst. Publ., vol. 11, 273–308. de Gruyter, Berlin, 2005. 12 BINYONGSUNANDCHEN-BOZHU

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Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and School of Mathematical Sciences, University of Chinese Academy of Sci- ences, Beijing 100190, China E-mail address: [email protected] Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076 E-mail address: [email protected]