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LOCAL THETA CORRESPONDENCE: THE BASIC THEORY BINYONG SUN AND CHEN-BO ZHU Abstract. We give an elementary introduction to Classical Invariant Theory and its modern extension “Transcending Classical Invariant Theory”, commonly known as the theory of local theta correspondence. We explain the two funda- mental assertions of the theory: the Howe duality conjecture and the Kudla- Rallis conservation relation conjecture. We give a status report on the problem of explicitly describing local theta correspondence in terms of Langlands-Vogan parameters. We conclude with a discussion on a certain problem of automatic continuity, which manifests unity of the theory in algebraic and smooth settings. 1. Classical invariant theory Classical invariant theory is the study of the polynomial invariants for an ar- bitrary number of (contravariant or covariant) variables for a standard classical group action. We will focus on the case of the compact orthogonal group. The basic references are [Ho2] and [KV]. So let G = O(n), the compact orthogonal group in n variables. It has the standard action O(n) yRn, and it induces a representation O(n) yC[Rn] given by (g · f)(x) := f(g−1x) for all g ∈ O(n), f ∈ C[Rn], x ∈ Rn, where C[Rn] denotes the space of complex valued polynomial functions on Rn. n Denote by {xi}1≤i≤n the standard coordinates of R . It is elementary to show that the space of O(n)-invariant polynomials is C[Rn]O(n) = C[q], arXiv:2006.04023v1 [math.RT] 7 Jun 2020 where n 2 q := xi Xi=1 is the quadratic polynomial on Rn preserved by O(n). 2010 Mathematics Subject Classification. 22E46, 22E50 (Primary). Key words and phrases. classical invariant theory, local theta correspondence, Howe duality conjecture, Kudla-Rallis conservation relation conjecture, automatic continuity. 1 2 BINYONGSUNANDCHEN-BOZHU It is a part of classical invariant theory 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 THETA CORRESPONDENCE: 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 symplectic group, 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 orthogonal group (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.
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